Okay, good afternoon. Can everyone see my slide? Hear me? How's everyone doing today? Beautiful day here in New York. We had a, oh, we missed the hurricane. We got a lot of rain, but we pretty much didn't get the hurricane everyone's expecting. And now the sun is shining. So that's weather. Welcome everyone. How's everybody doing today? Interesting times we're in. For those of you that have heard me before, my name is Melissa Armo. As you know, I own the Stock Swoosh. And if you're interested in a trial to the live training room after today's lecture, you can email me at melissa at thestockswoosh.com and I can send you a trial for the week. But today we're gonna talk just in the next hour, just gonna give an overview of kind of what I do, my strategy and really what I do is I focus on institutional money. Now, I've talked to a lot of people over the years as I've been teaching who want to earn a living trading. For some people, 100 grand a year is a living. For some 250 grand a year is a living. For some 50K a year is a living. It depends where you live in the country or in the world actually. But as far as trading goes, the amount of money that you risk per train is what really determines your, which is your sizing, okay? Number of share quantity, which really will determine how much you're gonna make as far as a beginner, intermediate or even advanced. We'll go through some of those differentiations today, but you have to be aware of the size of your account. So let's say your goal is to earn a living trading. Let's say that's what you wanna do. And let's say you have a small account, say $5,000. Well, you can't risk $2,500 a train. You can't risk half your account in one train. Okay, that's silly. And you should never do that. So you have to look at it in steps.   You have to be willing to take it easy to go up the steps when you're doing stuff with progress, with trading. Because if you don't, you can hurt yourself. And again, it's a process where you will maybe take 10 steps forward, two steps back, three steps forward, one step back. But if all along the way, you're making progress and your account is growing, you're gonna get to the point where you can take a $5,000 account and build it into a $25,000 account and take a $25,000 account and hopefully build it into a $100,000 account.  Okay, your amount of cash. Any questions as we go along here? Just plop them in the room, okay? So I'm gonna talk today about earning $250,000 a year trading on the side of institutional money. Really, my strategy looks for institutional money in the market.  But July was a very strong month. December, I mean December. Oh my God, I'm thinking December. August. I just said December. August has been a very strong month, okay? But for the month of July, we have an 88% win ratio in the live trading room, okay? Now, what is the live room? The live room is where you come in the morning and it will call the trains, the entry, the exit, the stop. If you wanna do the trains, you do them. That's the purpose of being in the room, okay? Anyways, this was an advanced trader risk from $2,500 per train.  So this is an average. When I'm sizing myself, I'll risk a different share quantity in each trade because the stop is different. If I say 50 by 70, that's 20 cents, okay? If I say 20 by 80, that's 60 cents. So it depends on the number of shares, the difference between the entry and the stop. If you decide to come in the room for the trial this week, you'll see that, okay? And again, if anyone wants to trial, just email me. So the first day of the month of July, we did Mew. Was a nice profit again. Average risk per trade was $2,500. Mew was a profit of $4,200. Mew was a win. I prefer to do one trade a day. If I do one trade a day and I make money, I'm done. July 2nd was Apple. That was a win, $3,850. And again, all of these trains were using my Goldman Gap rating system, which I'm gonna discuss today, which pinpoints institutional money in the market. But I'm just showing you an overview how the idea of making or the goal, I should say, of making $250,000 a year is very doable. And again, this is an average risk of $2,500 a trade. Boeing was a winner on the 13th of July, 2018-16. BAC was a nice one that was earned in the 14th, 4,050. 15th didn't do any trades. 16th, we had one loser. 22-15, Netflix was a winner. Same day. Sometimes I'll do a retake, okay? Sometimes if I lose on the first trade, I'll do a second trade. But again, we didn't have many losers in July. Diamonds was a winner on the 19th, 5375. No trades in the 20th. If I get something and I get up in the morning, I don't have any good quality picks, then I won't trade. That's another thing that's extremely important. You don't have to trade every single solitary day. I didn't do anything this past Friday. I didn't like anything. Sometimes you have to be careful.  You get paid if you do the right actions. Okay. Very important. Netflix was a winner on the 21st, 2700. LVS was a really nice one on the 22nd, 2460. There was a big move for that stop. INTC was a winner on July 23rd, 2214. LVS again on the 26th was a winner. UPS, I have this chart in here, was a big winner. On the 27th, 2980. Microsoft lost on the 28th. Then this is cake, sorry. This trade is in here. We'll talk about it. It's not Kale, it's cake. 3725 was a winner. And PayPal on the 29th and 30th. So overall, again, the benefit of being in the live trading room with me is I'm calling the trades live when you can do them with me. But again, the strategy is what I do every single day when I'm making the picks in the morning in the pre-market. Then I get on the mic in the room, then I'm calling the trades. So how can you get these kinds of results? You have to have a strategy that works consistently in the market. We were discussing this today in the room. Many, many people just buy support and short resistance. That's not a strategy that will result in consistent results, but people do it. Right now the market's rallied. It rallied the last few days. If you bought support in the market, you were up. Does that work every time? Absolutely not. Does it work more than it loses? No, it doesn't. So consistency counts, which means you have to have a strategy that you know what you're looking for. And then you repeat it over and over and over over and over and over and over again. And that's how you get somewhere with trading. And that's how you can build your account like I was talking about earlier. You have to use a good strategy. And again, I think one of the big fails for a lot of traders is they absolutely do, they're making trading choices and making trading picks thinking that something is strategic or that it is a strategy and it is not, is actually not a strategy. It is called a setup. For example, if you wanna go long into support in the market on the 20-period moving average, that's a setup. That's not a strategy. And it's not gonna be a consistent winning setup either. So that's a setup. That's not a strategy. A lot of people just fail in their thought process of what a strategy is. So this is just very, very basic, but it's very, very important. So it's important to spend time learning and not just trading. Everybody wants to make money, I get it. I was there when I started out too. I wanted to make money really, really quick and really fast, but I didn't know what I was doing at the beginning and consequently I lost before I realized I had to figure out my own strategy because I didn't have one and that this was the important. Okay, so I taught myself, the market was my teacher, but if you come to me, I will be your teacher. Learning is important. It's not just trading. It's great to be in the room and duplicate what I'm doing, that's true, but ultimately you have to understand it yourself because what are you gonna do if I ever leave your life? So you've gotta learn it, it's important, not just trading. Christine, I don't know if you have a question, I see your typing is stuck. I don't know, Kathy, if you can help. And anybody else that has any questions here?  So anyways, you do not have to feel frustrated in the market daily, even though I know a lot of people have been for the last few weeks in the summer because the market's been sideways, and then it seemed volatile, then it faked like it was gonna drop now, it's rallying, made new highs. Honestly, knowledge helps you be less stressed. Then when you lose in a trade, you won't be so worked up because you know you have the knowledge to replicate a winner, the next trade, the next trade, the next trade. Again, trading is about odds. You're trying to keep the odds in your favor, high odds, I'm looking for high odds. I love the idea of being able to make money in every single trade. You saw the results in July, but I did have two losers. So I use stops or limit order stops when I take a trade. Why? Because I know there's a possibility that I may take a trade and it may lose. This is about high odds, which means more winners than losers. If you're doing that, you can make money. And then you also don't need to have just gigamongous winners every time you win. While that's great and sometimes I get big winners, it's not something that can be planned. I can't always plan necessarily that something's gonna continue when a bigger move than I anticipated. That's just the luck of the draw, okay? Thank you, Kathy.  And it has to be a strategy that works consistently in the market, which means what? It has more wins than losses. That's one of the reasons why I wanted to show the July results. So you can see more wins than losses. That's good. That means you're up. And there were a few days in there where we didn't even trade and then I took off for July 4th for a week. So I mean, but there's times in the market where again, there won't be a good pick. And then the best thing you could do as a disciplined person is to stand back and say I'm not gonna trade today, okay? Anyways, for me, what I do is a system called golden gaps. It's gaps, but I coined the term golden gaps and named my system that because it's like finding high odds or gold in the market because I'm looking for something that has 20 points of a 26 point rating system and it's a system that I created. So it's a system that I sit down and calculate and figure out every single morning on a checklist before the open, before the run, before I do anything at all, before I even put any money on in the trade, I will rate the gap. I get up early. You do not have to get up as early as me. I don't like to rush. Plus I'm an early morning person and I live in the Northeast. So it works for my schedule with the market which opens at 930 AM Eastern time. But for me, I'm always looking for the best, best, best, best, best thing that I could possibly do in the dead. I only need one. That's what I always say to everybody. You only need one. One good trade day, that's it. And if you're doing this turn of living, you only need one. It's not about going after it, after it, after it, after it, after it. Because again, the more you trade, if you trade for six and a half hours from 930 to four, which I do not do by the way, you can give money back. You could take a great trade right out of the game and then lose, lose, lose, lose. And then you're down for the dead. A lot of people do that. A lot of trading rooms are open all day. Mine is not. It closes as soon as we're done. Could be 10 o'clock, could be 1015, could be 1030. It's rare that I'll keep the room open until 11 anymore, to be honest with you. Sometimes I close the room at 945. If I do a trade, boom, quickly, right in the first five minutes and I don't have anything else to talk about or teach about, then we're done fast. And that's okay. That's actually good, okay? The faster you can make money and the sooner you can be done, then you don't have to worry about the agirations of the market. You also don't have to worry about White House press conferences, economic news coming out or anything else that may screw up your trade of the market, okay? So for me, the system that I use is called the Golden Gap. And anyone that's trading, be honest with yourself and ask yourself if you really have a system. If you really have a strategy or are you just doing what I explained earlier that is a setup. It's really not a system. It's a setup, okay? So think about that, because that may be why your results, you're not getting the results you want and you desire, okay? So again, I coined the term golden because it's like finding gold in the market because it has big moves and that's what you want and that's what I'm always looking for. But I really think it's important for people to have a strategy and I think the focus on one strategy for me has helped me to become successful because I replicated over time and I've been doing it for 13 years so I'm very good at it. So it has become second nature to me. Like when you play golf, if you play golf, if you do a sport or something and you do it for a long time it just becomes second nature, you're good at it. You don't think about swinging the club, you just do it naturally, okay? That's how I am with my trading. That's how I am with the rating system. That's how I am with my dating trades even when I'm looking at an option or when I'm managing the trade, when I'm looking to get out, the target, all of those things. There's no substitute for experience.  Even if you trade before you're new to what I do, so you have to learn it from me and then you gain the experience. But again, that's the benefit of being in the live room. Excuse me. Any questions here where I get a sip of water? No questions, okay. Anyways, you do not need a general overall broad base view to make money. Like I don't read reports, earnings reports when they come out, I don't go through all that. Sometimes I'll scan the Fed minutes, but I don't spend a lot of time on it, okay? I don't have a lot of time to spend on that stuff. Tons of people have read and read and read and they lose money. So what's the point? When you're reading price action, which is what I do, it's based on technical analysis, I'm getting an immediate thing of what's happening. For example, today, the market gapped up today and rallied and made new highs on what? On information that there was an FDA approval for one of the vaccines. The market had an immediate, in real lifetime, positive bullish move today and rally because of that. Not, the market didn't care about anything else. Didn't care about COVID, the Delta variant, what's happening in Afghanistan, the market couldn't care less. The market cared about right now, today, that's it. Today, not tomorrow, not Friday, not Wednesday, not next week, okay? So the beautiful thing about trading using technical analysis, which is what I do, is all I care about is making money today, right now, in the moment, reading the price. And that's what I'm doing when I'm reading institutional money, okay? I'm reading the footprints of institutional money that are coming in and the price patterns in gaps. And again, I'm gonna explain in a minute what a gap is. But that's it, you don't need to do anything else. It's really that simple, but it's knowing the points to look for and analyzing the points that takes the learning. That's the process of the learning process and the skill set and the knowledge, okay? Because ultimately, if your reason for doing this is to make money, you really don't care about all these other factors. You care about what's really happening with the price action. If something's rallying, you wanna be long. If something's falling, you're probably losing if you're long. And that's the problem, okay? And getting back to what I was saying about high odds, everything is about risk when you trade. So that's why I use a stop. And when you take an options trade, which we'll talk about a few options here, an options trade is basically the stop is what you risk. So if you take one contract, you can't lose any more than that. You spend a thousand dollars, you can't lose any more than a thousand dollars on that one particular trade. So when I'm doing trades, I'm looking at the odds. I'm weighing the cost benefit of it. I might lose this much money in it, or I might make this much money in it. Does it have a good target? Things like that. I'm always weighing the pros and cons. So you can't make money if you don't take risk. One thing I find really, really interesting about traders, and this is just a general, general, general statement here, is that a lot of traders really say that they wanna take risk when they trade, but they really, they say that they want to. They say they're okay with it. They say they understand it. They say they're fine, fine, fine, fine, fine. But in reality, they're not fine. They're not fine taking risk. They really don't wanna take risk. They can't withstand even one loss, whether they can't really in actuality in their account or emotionally they can't. The fact is when you trade, you have to understand that some trades will lose. That's the whole purpose of sizing yourself correctly with trades and also using stops. You have to know that. How do you ever get somewhere with this, make 250 grand a year more is because you have to have more winners. That's what you gotta focus on, the odds.  I have losers, okay? And you have to be okay with risk. I don't wanna say you have to be comfortable with risk, but I guess you do. I guess you do. And when people do things like spreads and options, which I don't do, they're trying to alleviate their risk. They can't really make a lot of money in trades like that. When trades go in the right direction of the one they thought, then they never really benefit from it and they never make a lot of cash. The fact is if you wanna get ahead, it involves risk. That's it. You have to embrace it and move forward with it and understand there's a risk associated with taking trades. There's a risk associated with taking my class. My class is $7,000.  That's a risk, okay? Everything that you do in life, if you want to get ahead requires risk. And that is something that you have to be excited about and embrace instead of confronting it in a way that you feel negative and you're trying to say, well, I'm okay with it, but in reality, you're not okay with it, so you're lying to yourself. Don't lie to yourself. If you're really not okay with risk, then don't trade. But by all means, don't say that you are okay with it. Trade, lose money, lose money, lose money, lose money. And I mean, that's a disaster. What is a disaster for you financially and to it's a disaster for you emotionally and mentally? Why would you do that to yourself? Not everyone is cut out to do this. That is the truth. And after teaching people for a number of years, I realized that. And that's why I'm bringing up this discussion because I know now enough about people. I'm very good at reading people that people, when I talk to them, they were really not okay with taking risk, okay? But I can also read people who are okay with taking risk and those are the people that are doing well. And even if they make a mistake or losing a trade, they get right back on the horse and they get right back on the horse and they can do it. And sometimes when I do that, sometimes if I have a losing trade on a losing day, I get right back on the horse and then I have a huge week or a huge day or a huge trade after that. It makes me push forward even more. That's an unusual quality that I have. But pushing forward, taking risks will get you ahead in life and you have to come to terms with that if you wanna trade. And the sooner you come to terms with that, the better you're going to be. The better you're gonna do instead of kidding yourself, okay? Success means not just taking risks for the sake of taking risks, which is what a lot of people are doing in those Reddit chat rooms right now, taking ideas from strangers and not understanding what's going on and taking risks for the sake of taking risks like buying a stock like GME, which people are thinking are gonna double in price again, which it may or may not. Nobody knows. There's absolutely no calculator risk involved in that trade whatsoever at all. You have to look at it using calculator risk when you're taking a trade. And how do you do that? You implement a system and you utilize a system that allows you to put the odds in your favor so that you can say, well, if I'm risking $1,000, I have a 22 point gap rated gap, which tells me that this has high odds, okay? About 80% odds that it's gonna work. Therefore, this thousand bucks that I'm gonna risk is a good risk for me because I got better than 50, 50 odds, high rated gap, okay? And that's how you think it through and you make the process. It's not just throwing money at the market and throwing money at trades for the sake of taking risks. It's calculated risk. I want to take a class because I really want to learn how to trade because I really wanna do this and I wanna quit my job and I wanna do something else. I'm deciding to do this because I wanna learn and I wanna learn what I'm doing and I need to learn a system and I don't have a system now and I know that this class costs money but I'm gonna get this right and I'm gonna do it and I'm gonna give it all I have. That kind of thought process, do you see? You know, this is how your mind works. Your mind works in ways that many, many people don't even understand and it's interesting. I understand so much about the mind. I've read a lot about it and I'm really in touch with myself and I'm highly intuitive. But I will tell you right now that one of the reasons why people lose money in the market is that they simply are not willing to take risk. They risk money, but they're not really willing to. I don't know if that makes any sense. Any questions? Anybody there am I talking to myself? I know Kathy's there. Unless she's feeding the cat. Hello, it's 455 in New York City. Am I talking to myself? No one's writing any comments so I must be talking to myself. Oh, there's Kathy. Kathy's alive and she's not feeding the cat. All right, getting back to what I was talking about. Charles, there you are. Charles, I saw you in the open house last week. Are you gonna get back to trading?  Sometimes it's good to take a break every once in a while and come back. Anyways, let's talk about institutional money. Footprints, big ones, big ones, big ones. Think about when you go out into the woods and you see a bear. Well, hopefully you wouldn't see a bear, but if you saw the footprints of a bear, it's something that isn't gonna escape your notice. Institutional money is big, big footprints of money that come into the market. It could be buying, it could be selling. It could be shorting, okay? Either way, it's the idea of money that moves stocks in one direction or another. Either up or down, okay? Here's an example. This was a trade we did today. We did NVIDIA. This was a nice move. I called calls and in the options letter, NVIDIA, it ran up right out of the gate. A beautiful, beautiful bullish move. So what happened in NVIDIA and NVIDIA got bought? This was an institutional loan that money came in and bought the stock. So let's go over to GAP as I've been talking about it. A gap is due to the close in the open. NVIDIA closed here on Friday. Boom, open Monday morning here and rally. So it gapped up. Again, I will do bullish gaps and I will do bearish gaps. This in the case of NVIDIA was a bullish gap and it ran straight up, okay? It got bought. Everyone see this here. So anyways, there's the rally. Dream Target actually was 220. This is a beautiful move. It was like a $10 move in one day. So what else do you get with institutional money? You get big moves. That's how you can make money in the market as one individual trader. Big moves because you are not going to move a stock like this, 10 points. You could throw your whole account and do it. You're not going to move it. You've got to have big money. Big, big money that comes in and scoops it up and buys it or pushes it down and sells it. But in the case of NVIDIA here, this was a long. This was a nice call. It was a calling options newsletter, okay? So here was another one. Institutional money that sold the stock gap down. This was UPS, this was back in July. Stock closed here gap down. Again, closed the night before up here, around 2, 10 and change. Opened in the morning here around 198 and change. Fell, sold off, fell 10 points here on the day. We shorted this and we did puts. It was a nice straight.  Had institutional selling. So what you wouldn't have wanted to do here shorted, okay? What you wanted to do here is go long, okay? So institutional money can move in both directions. Again, I do bullish gaps and bearish gaps. But the reality is that you can make money going in both directions, okay? You can. But I'm looking for the gap, like I got up in the morning, the morning of UPS and I rated the gap. I rated the gap using my golden gap rating system. And I determined that UPS in fact would continue falling. And it did. Just like I rated NVIDIA this morning and determined that this was going to get bought. And it did. And by the way, even though the market rallyed today, this would have gone today without the market. Didn't even need the market today. Any questions here so far? Yeah, Kathy, I don't know what you're doing. All the chat's gone. I don't know what you're doing or if that was the cat on your keyboard. Just put my information back in the room for people. Anyways, getting back to what I was saying earlier, benefits versus risk. The benefit if you learn how to train and if you learn how to make money and you can do it and you're successful, there's many, many benefits. You can quit your job, you can work for yourself, you can work from home, you have unlimited income. It's fun too. The risk is that you're going to spend money taking trades, risking money and trades, not knowing the outcome until you're out of the trade. The risk is that you will take a class like mine and you will spend money not knowing what the information is of the system, what it's about. Okay. The risk is also you may learn it, do the class, start trading and it may take you longer than you thought to actually learn it. Or maybe shorter, I don't know, but the risk is you don't know. Could take a day if you're gonna learn it, could take a week, could take a year, you don't know. That's the risk. When are you going to make the money that you want to make? You don't know. There's risk involved. Nobody knows, there's no guarantees. The timing of things isn't always necessarily what we want it to be. Sometimes it's better than expected, sometimes it's not. That's life in anything, okay. So it's always benefits versus the risks. You weigh them. When I quit my mortgage job, there was, you know, risk involved. But for me, the benefits of quitting my mortgage job and trading far outweighing then the risk. And so I ended up doing that because I really didn't like my mortgage job. I wasn't happy doing it. In fact, it was miserable. So you have to weigh the benefits versus risk of anything that you do and choices that you make in life and particularly trading. And you know what else I find? For me, I'm very black and white with my decisions. I either like something or if I don't, like Friday, I didn't like anything, I didn't trade. There were some things that people liked in the room, some things people did in the room that worked. I didn't like them. I'm very black or white. I either like something or I don't like anything at all or I don't like it, okay. But I will tell you that when you get to that point where you can make a defined decision, I want to do this and I'm going to do this, that's it. Many, many people are looking lose and they're on the sidelines and they're waiting and they're hanging and teeter tottering. Like they wanna do it but they haven't done it and they think about training but they haven't opened an account yet and they're maybe gonna do it but they don't really know. And they're just kind of in limbo. Well, as soon as you make the decision what you're gonna do, you'll be far, far better off. It's like if you ever wanted to lose weight and you're just like, well, I wanna go on a diet but I don't know when I should start and maybe I'll start here and then maybe I'll start after Labor Day and maybe I'll wait till then it's Christmas, then it's Thanksgiving, maybe I'll do, but do. The sooner you make the decision to start the diet the sooner you're gonna lose weight and the better you're gonna feel about yourself and everything else or an exercise program or anything like that. Making a decision and just starting it and taking action is just such a huge step and any big goal that you have any future endeavor, whatever it is, you know. Any questions here so far? So again, it's risk to reward for me. So I have a 26 point rating system. If the gap rates under 20, the rule is not to do it. If it's 20 or more, then I can do it, okay? So that's where I balance the risk to reward. And as I was discussing with you earlier just like with you saw the NVIDIA today I'm looking for a good risk to reward payout. I wanna get paying. I think one to one is a good payout but many trades I do are more than that. Like you saw the July results, we're gonna look at some here in a minute. So I want a good payout. Good payout for me is 100% and that I consider a big profit. Considering the fact that I'm just one individual I'm not running a hedge fund in trading, you know. To be able to make a couple thousand dollars a day that's a good trade, that's big profits to me. Especially when many people are losing and some people can barely even make $50 a day. The other thing I like about trading is in reference to gaps the moves usually happen quick. Now I wouldn't say all the time, the most of the time, most of the time, the high, high majority percentage of the time. The moves happen pretty fast. What do I mean in an option 24 to 48 hours and in a day trade in the first 30 minutes to half an hour are hour of the day. Just to give you some kind of timeframe. Most of the trades that's how they work for both the day trades and the options. And again, I'm focusing on one thing. I'm focusing on the gap, that's it, boom. Now, you can use my Golden Gap system for options and day trades. The options is a newsletter. If you wanna subscribe, the newsletter is emailed to you. Like everybody got from the video trade today in the morning, pre-market. If you want to do the day trades, you join the room. In order to join the room you have to be a student of the Golden Gap course. There is no prerequisite though for the options newsletter. You can just sign up and join if you're gonna manage the trades yourself and not know the system. I do have targets in the letter I would follow along that. And I would also follow along with a 50% return in investment or a 100% return in investment profit margin. So say you take a trade, say it costs a buck. You could just put an option in and sell it at about 50 and walk away as soon as you buy the option. You don't have to sit there and watch it, watch it, watch it. Now I like to watch stuff. Like I like to see what's going on. I'm here, I'm watching stuff. But if you're busy or you're doing one of the things you don't have to babysit the options. The day trades you do have to be in the room. You do have to be there life that goes so fast you can't really walk away. Well, no, a stock is gapping and then I can do it as an option. It's the same system. That's my point. It's the same concept like the UPS. I'm gonna go back. This gap. We shorted this as a day trade and we did a put that both worked. This gapped down. Close to the day before it gapped down. I rated it in the morning. It rated per my golden gap system that it would fall. And I said, okay, this is a good one. We're gonna do a put in it and we're gonna do a day trade in it. And we did and they worked and it was good. It's the same gap. It's the same system. It's two different ways to do it. Which you could do both or one or the other. Some people do both. Some people do one. Some people do just the day trade. Some people do just the options. People of jobs that can't be in the room in the morning at 9.30 Eastern time do the options. People that can be in the room I find are in the room. Even if they're just doing options and not day trades and options, they like being in the room to hear me call it. But anyways, there are some people in the room that are just day trading. They don't do options. They never did options. They may in the future. It has to do with the rating system. Everything has to do with that. Everything, everything, everything. That's the system. Again, the idea of following a system and focusing on one thing. Now we did this one. This was Facebook. I called a late trade in this to be honest with you. I could have called this earlier. I called this on July 30th. I called the Facebook 350 putts. So I'm gonna go back to the daily chart. This is the newsletter. You would have gotten this at 9.14 in the morning on Friday. So I called the 350 putts. Here. So I could have called this like a couple of days before. Anyways, I called this late. It worked. It dropped. Boom. Take it to the right. You see where we are here. That was the area. Fell into it. Again, boom, drop, boom, dropped. The cost of this was fairly, fairly inexpensive for a stock of this price point. One option would have cost you $260. An advanced trader risk of 30 contracts or risk of 7,800. That's a lot of money to risk. But if you've got it to spend, you can really get to town on these things. Shoulder 450, not quite double, but close enough. Profit 5,700. Again, in and out. In and out. You take it. You get the move. 24 to 48 hours out. Okay. Return of investments, 73%. Remember, for every trade that you're in, I don't care if it's an option. I don't care if it's a day trade. You're using your money. It's your money's being sucked up by the trade. Especially if you're in an option and you're in it overnight, it's being sucked up by using it. Whatever you have at risk. Can't use it to do anything else tomorrow until you're out of that one. Hopefully you get the move and get out with the profit. And when you're in a day trade, you're using your margin or buying power or the money that you have in there to take the trade, even if it's for 15, 20 minutes. Okay. So the idea of trading is chunking it out. Chunk it, chunk it, chunk it. Take it, get the move, get out. Take it, get the move, get out. Boom. That's what we're doing here. This is not long-term investing. I'm not shorting Facebook to the end of time. I didn't look at what this did today, but I'm sure it flipped and rallied like everything else. This is just getting a move that is a move to the downside or a move to the upside in a momentum fashion where you are taking the trade, booking the profit and getting out, and then you're going to do another trade and another trade and another trade. And that's how you book money and that's how you build your account. These are not long-term swing trades here. Okay. Makes sense. Is that makes sense, people? Now let's talk about Boeing. Beautiful Boeing. Actually, let me go show you this. I called same day, it was July 30th at the 225 puts in Boeing. This was that same day we did the Facebook. Find it over here. Gosh, this is a while ago now. Closed here, gap down, fell, boom, dropped, fell, boom. Again, called the 225s. I didn't even hold some of these things that long. They were profitable pretty fast, but they kept going. Anyways, you can see how this fell. The red bar here depicts what selling. Then we had continued selling and actually this was continued selling here in the tail. This open fell. Okay. Anyways, this was pretty cheap too. Again, for the cost of the stock over $200, 240 for one. One contract, you could have spent $240. And no one, anyone in the trade should be able to afford $240 in a risk. Okay. You would have made 96%. Should have 470, profit 6,900 on a risk of 7,200. It's a nice trade. This was an advanced trader risk of 30 contracts. Again, if you have the money, you can do it. You could take half. You could take 3,500 risk, you could take 1,000. Like I said, for $240. Either way, it's the idea of taking the trade, getting the move, getting the profit. Now, here's a day trade. Again, I wanted to show you some day trades and options in here that we've done. Just so that you can see it's the same system applied in the manner of taking the trade in whatever way that you want to apply it. This was back on 728, we shorted cake. This was a 15 minute of cake. And again, how did I know this would drop? How did I know this would be a short because I raided the gap? And again, this is a 15 minute chart. Night before was up here around 55. Boom, gap down in the morning around under 51. It was above 50, but it was under 51. Open, dropped, boom, we shorted it. Drop, drop, drop, drop, drop. The next day it fell too. Actually fell this 29th and the 30th, but we did it on the 28th. So this was the one that was a 728. Here's the daily. Here you can see the gap better. Closed here, okay? Gapped down, fell, okay? Didn't even hold this all the way, okay? We exited this 4886. Once it broke 49, here's 49. This kept going, this kept going. Look where it did. And I did not do an option in this, but I could have, look what that did. Actually entry was 5035, 2,500 shares, risk was 3,000, exit 4886, profit 3,725. This was a nice move in and out quick. But again, you could have made more and you also could have done it as an option, which I did not do. I did not do. So I don't do everything that's a day trading option together. Sometimes I'll give me them up or I don't think I wanna do an option in it, but you could have if this has options. I didn't even look it up. I've never done an option in this. But like I won't day trade Amazon, I'll do options in that. So it depends, that's very expensive. But this kept falling and you could have actually held this overnight.   I know people that have done options that have risked their whole account of one trade, then they just told me after the fact. Luckily the trade worked, but that's so silly. Don't ever do that. That's just silly, silly, silly, okay? If I call an Amazon and it happens across $50 for one and you have $5,000 in your account, you'd be risking your whole account. That's silly, okay? You have to consider the fact that you want your account to last. You wanna be able to grow. Think of going up the ladder. 10 steps up, two steps back. Three steps up, one step back. You've gotta make progress, okay? But it is about the share quantity of the size. So whether you take 1,000 shares, 2,500, 5,000, whatever you wanna take, I'm usually looking to have common days. And again, if your risk is really small, that's okay. And if your risk is really small in the options, maybe you don't even try to get to 100%. If you're risking 500 bucks in an option, put the sell order at 50 cents and make 250. Because if you have a small account, you gotta really chunk it, okay? You gotta grow it. And what you wanna be focusing on is growing it as fast as you possibly can. So you wanna be green, green, green, green. You don't wanna be a piggy in holding too long and letting trade go against you. Especially if you have a small account, okay? You have a little more wiggle room to do that if you have a larger account or you're taking more size. Any questions here? So you can trade the system with an advanced risk or beginner risk. If you have questions about that or the size of your account, ask me. You could take the class and trade in a demo and not trade live money at all. I don't suggest doing that for months and months on end, but you certainly could do that for the first week if you've never traded before or you're nervous about how to do this or you could do it for the first month. I'd say after a month though, you gotta get to real live money. That's my two cents, okay? The idea is to learn it and then start to implement the program. But if you have a fat finger, you don't know what you're doing, you're new to platform charts, whatever, then trade in a demo, okay? Until you get used to doing it. But one month is pretty much long as long as anybody should be in a demo, in my opinion. You can open up an options account with little $2,000, okay? And there's profit accounts too for those of you that wanna do the active day trades on margin, you need $25,000 at a retail place, but you can open up a prop account with $2,500, okay? So there's lots of different ways and types to trade out there were smaller amounts of money. There are even places you could trade with smaller amounts of money, although I don't necessarily suggest any of those. I think $2,000, $2,500 is a good starter account. And you can trade in a demo till you have that saved, okay? Any questions here? BA was a big winner for us in the month of August. Day trades options, we did it all. In fact, we did this Thursday and it worked. I mean, this again has just given us, given us, given us tons of profits. We did BA a lot in 2020, when the COVID period began in February. In fact, even before that, I think we had shorted BA before COVID hit. But this stock, it's an airline, as many be known Boeing, with many, many things that got hit in 2020 with COVID, this got hit a lot. In fact, this chart has never recovered, never recovered. And I don't know if and when it ever will recover, it may someday, but it hasn't yet. So we've been taking advantage of this. We did decide some beautiful, beautiful, beautiful short moves. They're all done, we're all out of the trains now, but we got this a lot in the month of August. So this was back, oh no, this was in July. July 15th, I call the 220 puts in Boeing. Oh, this was before the 30th. I'll go back and show you this gap. Class was 250, 30 contracts, risk was 7,500, sold at 14. You could have made a 460% return on investment in this trade. I will show you the chart in a minute. But that's $34,500 for the risk of 7,500. What if you took a beginner risk? What if you risk $1,000? You're gonna make 4,600 in this. This is even before last week's move. I forgot this one, July 15th. So you could have risked $1,000 in Boeing for contracts and made $4,600. That's a specific way to take even an example, a $5,000 account and end up basically doubling it. And there are people on my letter that have done that, really, because I've been on such a roll with the options really since Memorial Day. Now let's go back to the 15th and look at the chart. Right here, 13, 14, 15 was here. So again, close to your gap down. 220s, okay? Fell, boom. And that's the nice thing about what? About gaps doing options is that you can get bigger moves overnight, okay? Gosh, that seems like a long time ago now, but that was really, well, it was a month ago. It was more than a month ago now. But we did this all last week too. We did all these. Close to your gap down, fell, close to your gap down, fell, close to your gap down, fell, close to your gap down, fell. We day traded this, we did puts and it was fabulous. But anyways, the one I have in here was huge.  Now, again, day trades, I like because I'm in that fast and I know where I am by the end of the day.  With options, if you hold options overnight, you never know where you're gonna be or the market the next day. And the market absolutely does affect trades, okay? This was a huge move. So if you had been on the options newsletter, you would have gotten this trade before the open on July 13th, 220 puts. Any questions on options, any questions on day trades, any questions on the differences between them? Here was the BA trade that we did on Thursday. BA, here is the daily chart going back a year, Boeing dropped off, fell, fell, fell, fell, fell, rallying and again, we did this on August 19th called the train in the live room. Entry was 217, 10, boom, 1800 shares which was an advanced trader risk of $2,790, exited at $2,290. I had a good exit on this but it wasn't even the low of the day. I'll have to pull it up and see where it went but I did have a good exit on this. I held this a little bit longer. I think it was on TV that day, yeah. I was on TV Thursday. Some people got out of this earlier but I was on TV and I stayed with it and a couple other people did too. 7,560 bucks, how, why? It dropped basically $4 plus. This stock Boeing, as you can see, I'll go back here to the blown up chart here where you can see this is the candle from Thursday. Again, this is even a medium move for the stock but we had a perfect entry. We shorted this sucker right at the tail, almost near the high and got the drop. Look at that, that was beautiful. But this stock can move even bigger. Like this bar over here is a green bar but you can see this bar is almost double the size of this bar. So, again, the benefit of trading stocks and specifically stocks that I like to trade, things that we know, things that we do, Facebook, Apple, whatever, they move. They have big moves. They're fun to trade. If you take 1,000 shares or 2,000 shares it's something to make a lot of money because they have big moves. That was a great call, great calls. And then what if you did a smaller amount on Thursday if you had been in the room and took half the size, 900 shares, it was $1,395. Got the drop, boom. Again, this was Thursday the 19th. You would have made $3,780. Okay, here, big drop. Any questions from anyone about day trades or options or anything we're talking about here so far? Talking, talking, talking. I'm gonna leave time for questions. Anyways, I have a daily focus. I don't know what I'm doing tomorrow. I usually don't know until I get up in the morning. There are some earnings this week. I'll be looking at and taking a view of. But I focused on making money. That's what I do. I'm focused on the gap that leads me to the direction of what's happening with, again, the institutional money.  That's what I'm trying to pinpoint. It's the idea of predicting it. See, I'm like, NVIDIA went to the dream target today. My trade, I called early in the morning. So my rating system predicts something before it happens. I'm not predicting the gap. I'm predicting the move on the day after I see the gap. That gap helps me do the prediction. The gap is what I'm rating. And that is telling me, oh, this has a high odds that it is going to continue higher today. Therefore, I'm just talking about in the video right now. Therefore, it's a good idea to go along this. That's what we should do. You see how it goes. Or it would be low odds, low odds not to go long. And then we wouldn't do a trade in it. There are more trades not to do in any given day than there are trades to do. Do you know what I'm saying? I mean, there's more things that are cracked than there are things that are good to do. And that's also why many days I'll do something different every single day. Now, sometimes I might do a repeat if I'm looking for something like a continuation. Like we did the PayPal two days in a row and that's only when I'm struggling to find something and if I like it. But many days we do something different, you know? Anyways, there are many benefits to trading. If you want a day trade, you're working less than an hour a day. You can work from home. That's good too. And again, the information, the knowledge that you have in your experience will help you do well. And then you can build a small account into a larger account or whatever you need to do. You can take a larger account and build it into an even larger account. And I think that's the benefit of doing options of the day trades as well. So what do you need to do to get there? You need to learn the system, listen to me lecture in the room, ask me questions if you don't understand stuff after the class as well. And basically understand that if you want to get ahead it's going to take risk and you have to be willing to take the risk. I look at it as something that's exciting. I was excited when I started trading. I was excited to learn something new. I was excited to do it. I was excited every time I took a trade and put risk on. I'm still even excited now. I didn't know exactly that the video would go as big as it did today, but I had hoped that it would. I rated the gap. I followed the system. I had a good feeling. And then the excitement is of course that it does. Trading is exciting when you get it right a lot. No one likes losses. I get pissed when I lose even if it's twice a week or twice a month or whatever it ends up being. I don't like losing, but I fight back then to win. And that's again my personality, but that's one of the reasons why I think it makes it beneficial to trade with me. I had that type of personality, which is what? I'd like to win. I want to succeed. I have an enormous drive to succeed. That's what makes it very advantageous to trade with me. So when I'm looking for things, I'm looking for again weighing the cost of the benefits versus the risk, looking for high odds in my trades. Wanting them to see if I'm getting the bigger move, the medium move, looking at the market. What else is going on in the world? There's many, many things right now as we know that are going on in the world. What's going on today? It's always a benefit versus the risk. I think that people, just like I said earlier, don't accept that. Like they're just blindsided by it. They're just kind of in La La Land about it. Once you understand and accept that trading involves risk and you're willing to embrace it, you're going to do a lot better. You're going to do a lot better. You're not going to be so upset with yourself and cringe when you lose. And you're going to get right back on the horse then when you do lose to win. Understanding that you're putting the odds in your favor and that this is part of the process and that the benefits and your future, once you get this and you learn it and you do it and you use the system, it's going to be a lot brighter, a lot better off by using it. So that's part of learning as part of like when you go out again, when you learn to sport and you start out. You know, you're not winning at the beginning every single race. You're running a marathon. You never did one before. What, how do you expect to win? There's a process, kind of like you can call it, you're paying your dues or whatever it is. But it's really just the learning. It's just the process of going through the action of trading, implementing the system, learning it, doing it, implementing the system. Again, asking questions, learning and doing it. Okay? Just like anything you would learn. Really. So again, you have to want it if you want to do it. And I think people go through, and I see Charles in here, I think people go through ups and downs of other things that have to do with their life. People go through personal things that happen in their life. Relationships, whatever.  Acknowledge that, that's okay. You can go back to trading when you're ready. But the system that I'm doing now is the same system I've been doing since day one. So it's really nice that people have taken the class, trained, sometimes they go awry, come back, doing the same thing, just doing the same thing as I've ever, ever, ever done. Any questions here so far? And yes, if anybody wants a trial, you can email me at melissathestockswish.com for a trial. Anyways, if you want to learn my system and just call it the golden gap, it's looking for institutional money in the market. I'm measuring gaps or rating them in the daily chart to find stocks to trade that have, number one, a high probability of directional bias for the entire day. Ideally, you know, like the NVIDIA. A big move on the day, again, like NVIDIA. Early confirmation of the bias and move between nine and 30 and 10, again, like NVIDIA. And precise entries with follow-through and a good wish-to-reward target potential. I'm looking for that. I want a good wish-to-reward. Again, one-to-one on a day trade is what I think is good. And an option, I think 50% to 100% is good. And if you get more than 100%, it's fabulous when it happens in an option. And a lot of times what happens is you can't predict that either. It just happens. Either you get the market with you like today with NVIDIA or you're in it overnight and you get the follow-through continuation move like we did in the Boeing. It just happens. It's not like you plan it when you get these 400% moves. It just happens. Okay? Any questions? Okay. But anyways, be consistent. Be consistent with your sizing. If you're gonna risk $1,000 in every trade, that's what you're doing. You can't risk $1,500, $2,000, $5,000. No, it has to be the monetary risk has to be the same or close to the same on every trade you take. That's how you're gonna be consistent. Otherwise, you can take five trades and have four winners and one loser and if the one loser, you risk too much money, you could be down even with that many wins. So you have to risk close to the same in every trade. Okay, that's my doorbell. Can you guys hang on? Just give me one second. I think it's the package time. Sorry about that. We're almost out of time and I just have a couple more slides. Anyways, the class is called the Golden Gap course. It is a system. It's a 26 point professional bearish gap rating system. The purpose of the system is to help you evaluate which gap to trade each morning using a checklist. The checklist tells you what to trade when and in what direction. The 26 point checklist predicts directional bias in a stock. Again, I'm not predicting the gap. I'm predicting the move it's going to have after I see the gap. So you can learn the system and use it for options or day trades. If you wanna be in the day trade room, you have to do the class. If you wanna just join the letter, you can join the letter and subscribe for a year. There's no prerequisites for that. I'm using my gap rating system to call the trades on the newsletter. I'm looking at institutional money. We do longs, we do shorts in the letter. We do puts, we do calls. We do both. Okay? Thanks, Kathy. Anyways, one strategy is all you need to be successful in the market. You do not need a general overall broad base view to make money. Lots of people have that and fail all the time. I was talking about that earlier. So if you learn how to read institutional money and price patterns and gaps, you don't need to do anything else. Because if your reason for doing this is to make money, you will make money if you use the system. Don't skip it. Don't fudge it. Don't make it up. Don't change the rules. I created the system. It took me three years to do it. Don't make shifted it into something else. You follow it, follow it, follow it. You get up, there's no good gaps. You're not doing anything. You don't trade that day, okay? It's 20 points or more per the 26-point system. Don't fudge it, okay? Again, everyone knows you can trade from home. And if you want to trial for this week, you can email me at melissa at thestockswitch.com. The last class for the summer is this weekend. Weekend before Labor Day, August 28th and 29th. If you want to learn before the fall, this is the class to do it. 9 a.m. to 5 p.m. Eastern time costs the class of $6,999. It's U.S. dollars. You must email me for sign-up forms. The next class is probably gonna be the end of September. It's gonna be a big gap between this class and the next class, because I have my birthday in September. It's Labor Day, I'm going out of town, so if you want to trade for the fall, this would be the class to learn and do it. I'm doing a special, a pre-fall special, actually, getting in a fall mood here in New York. It's running through Friday. You get the training room in the outskirts who's been afraid through the end of the year with this weekend's class. This is a really nice deal, because then if you want to do both, you'll see if you want to do the options and the day trains. If you just want to do one or the other, you'll see.  It's a good offer. And I'm running this through Friday. And again, if you want to travel the room this week, email me, and I'm running this special through this Friday for this weekend's class. Class is online. Any questions from anyone at all? Only a couple minutes over here. Even after the package, guys. How's everybody doing? I think we're gonna have some good fall training. And again, market could be bullish, market could be bearish. We take it day by day. I look for the picks. Look for the picks, look for the gaps in the market, rain it. Any questions, new people, old people? Anybody there? Quiet group tonight. Have a wonderful, wonderful, wonderful night. Have a good evening. If anyone has questions, email me. Nice to see some of you. Have a good evening, and thank you so much, Kathy.


I'm Ashley Dorf, and I'm here at the Rightfully Hers exhibit, which commemorates the hundredth anniversary of the ratification of the 19th Amendment by exploring the diversity of women and strategies that were critical to winning women the right to vote. You may have heard of famous suffragists like Elizabeth Cady Stanton, Susan B. Anthony, and Alice Paul, but winning women's suffrage required the activism of millions of women whose stories often go untold. Women like Marie Louise Botnou Baldwin. She was born in 1863 and was a part of the Turtle Mountain Band of Chippewa Indians in North Dakota. In 1904, she was appointed as a clerk in the Office of Indian Affairs and was a respected and outspoken member of the Society of American Indians. Baldwin wore her native dress in her official employee file photo at a time when the federal government urged Native American assimilation, but that's not all. In 1912, she started law school at the age of 49 and finished in two years instead of three. Baldwin also marched in the 1913 Women's Suffrage Parade in Washington, D.C., along with other female lawyers. For more information concerning minority women and the suffrage movement, come visit Rightfully Hers here at the National Archives Museum in Washington, D.C., or visit us online at museum.archives.gov.


Hey there, Radiant. I have some very exciting news. This September, Radiant Church is turning 25 years old. And in celebration of our 25th anniversary, we're going to do something really special. We are going to have one United Service on September the 12th, right here at Growlers Stadium. And we want you to join us, Portage, Richland, online, and friends and family coming together for one United Service right here at Growlers Stadium. There's going to be outstanding food and fun games for the kids. It's going to be a time of celebration and remembering all that God has done and dreaming and worshiping together of all that God's going to do in the next 25 years. We want you to be here. We want you to mark it down on your calendar and go to the website, radiant.church and register to be a part of this 25th anniversary celebration.


So, the Fermat's little theorem says that if you have a P to be a prime and you take any norm 0 element in G P, then a power P minus 1 is congruent to 1 modulo P.  What happens is that if you have a number A in Zn which is co-prime to n, only then we can expect a power P minus 1 to be 1 modulo n. Because if the GCD of A and n is bigger than 1, then of course all the powers of A will have GCD with n bigger than 1 and therefore all these powers cannot be equal to 1. So, indeed Fermat's little theorem can be used to find the whether a given number is prime or not. However, there are some numbers n which have the property that there is some A less than n which has the possibility that A power n minus 1 is 1 modulo n. So, if you were to start with the given n, you were to start with choosing a random number A in Zn and then checking A power n minus 1, it may happen that you get it to be 1 modulo n, but n need not be prime. I will let you find more information about this by working out some examples. For the moment we will go ahead with something else. What we now want to do is to observe that there are elements in Zn which behave in a nicer way. So, Zp has very nice arithmetic because whenever you take any two elements in Zp which are not 0, their product gave you a non-zero number. This is not true in general for Zn. Again, for instance if we were taking n equal to 8, there are these two numbers 2 and 4, these are not 0 modulo 8, but their product is 0 modulo 8. So, these are the elements the 0 divisors in Zn which create problem for the arithmetic in Zn. So, we would like to take these elements out or in other words we want to find elements which have the property that they are not 0 divisors. What does it mean to say that something is not a 0 divisor? It means that when I take such an element A in Zn and I multiply by various B's then I do not get 0. I would always get non-zero elements when can such a thing happens. So, for instance if your element A happens to be equal to 1 then 1 into any number will give you that number 1 into A always gives you A. It will never give you 0 unless A itself is equal to 0. So, when you have such a number one more just an example, we would like to find elements such that when you multiply to them by any non-zero number you should get a non-zero number modulo n you should not get something which is 0. But after all you have only finitely many elements and when you are going to look at it modulo n you have the property that A into I minus A into J is A into I minus J and if your A has the property that A into I minus J is 0 implies I minus J is 0 that would tell you that whenever you have A i equal to A j you will have that i equal to j. So, the elements which have the property that they are not 0 divisors these are also the elements which can be cancelled and we want to find the set of these elements. So, in other words these are elements which are invertible in Z n so that is there is some another element B in Z n with the property that A into B will give you 1. So, what we are looking for are the elements A modulo n such that A x congruent to one mod n has a solution. We should have a solution for the linear congruence A x congruent to one mod n. Now, linear congruence is something that we are trained so well in that even if I wake you up in your sleep you should be able to give me the condition for the existence of solution here. You should be able to tell me right away that the GCD of A and n divides 1. Now, GCD of A and n divides 1 in natural numbers we tell you that the GCD be better equal to 1. So, these are all the elements A with the property that the GCD of A and n is 1. So, starting from something which had something to do with the multiplication in Z n asking for elements which are invertible in Z n we have now come up with a number theoretic concept that the GCD of these numbers and n should be equal to 1. If you observe that whenever n is a prime p this condition a, n equal to 1 holds for all numbers less than p except of course 0. So, these are the numbers that we are looking for these are the numbers which are all the elements in Z p all the nonzero elements in Z p these are the nice elements in Z n which we are looking for. As mathematicians are trained to do perhaps what we then do is that we define a function you may think that it would be easier to compute this for particular n's but mathematicians will tell you by experience and rightly so that it is good to have this function defined for all n and then study the property of this as a function from natural numbers to natural numbers. So, we define this function phi n this is a function and what does it do it computes the number of elements in Z n remember this is Z by n Z which are co prime to n that means these elements are the ones which are not the 0 divisors when you are looking at them in Z n. So, phi n which is the function that we have just now defined is the cardinality of all the numbers from 0 to n with the property that the GCD of these numbers with n is equal to 1. It may look like a very daunting thing we are going to study the property of all properties of this function which is now defined for all n but that is the beauty in mathematics that when you generalize something then studying it becomes much simpler. So, for the moment let us try to compute some values of phi n of course phi 1 is 1 because there is the element 1 which is less than or equal to n and you have. So, here the number of elements that you are looking for is simply the single turn 1 this is the element which goes from 0 to n and has the GCD 1 with n n itself being equal to 1. Second is phi 2. So, we are going to look at this set 0 or you may say we are looking at the set 1, 2 this is the set Z 2 and you remove the elements which are not co-prime 2, 2. So, again here we are left with only one element which is single turn 1. If you go one step further phi 3 this happens to be 2 let us write all the elements from 1 to n and now we remove the element which is not co-prime 2, 3. 1 and 2 happen to be co-prime 2, 3. So, here the value of the function phi at the number 3 is equal to 2. Phi 4 once again here we write all the elements of Z by 4 Z this will have to be removed and 2 will also have to be removed because 2 is also not co-prime to 4. The GCD of 2 and 4 is 1 is 2. The GCD of 2 and 4 is 2 which is bigger than 1 as well as the GCD of 4 and 4 is 4 which is also bigger than 1. So, these are the 2 elements which we need to remove and so phi 4 is indeed equal to 2. You may now think that we had 1, 1 and then 2, 2 and if you ask anybody to complete the sequence ahead or compute the sequence ahead the most natural answer would be 3. But if you want to compute phi of 5 the Euler phi function of the number 5 you will get it to be 4 and the reason being that all 1, 2, 3, 4 are co-prime to 5. 5 is the thing that you will have to remove so you are left with 4 elements. So, phi of 5 is 4 and now brace yourself phi of 6 is equal to 2 it is a smaller number than phi of 5. The sequence is not quite an increasing sequence. So, let us write all the numbers up to 6 and let us see where we are getting the GCD to be not equal to 1 GCD of 6 with itself is 6. So, that has to be removed 4 has to be removed because the GCD of 4 and 6 is equal to the GCD of 2 and 6 which is 2. 2 also needs to be removed 3 also needs to be removed because 3 and 6 the GCD is 3 all the GCDs are bigger than 1. So, the only set that is left is the set 1, 5. Therefore, phi of 6 is equal to 2. Now, from the example of phi 5 you should be ready for the answer of phi 7 and that answer is indeed equal to 6. In fact, you should be able to generalize this by saying that whenever we are looking at a prime P phi of P is equal to P minus 1. In fact, we have also seen this that in ZP all the nonzero elements are invertible you can invert all the nonzero elements and therefore, phi of P is P minus 1. Let us check one more example phi of 8 and this happens to be 4. So, this is because when you write all the elements modulo 8 we can write only the odd elements once you have even element that will have a non-trivial GCD with 8. So, the GCD of 2, 8 the GCD of 6, 8 are equal to 2 which are bigger than 1. The GCD of 4, 8 is 4 which is bigger than 1 and of course, the GCD of 8 with itself is 8 which is bigger than 1. So, here we are left with only 4 numbers from 1 to 8 which are co-prime to 8. This is how the values of all these elements can be computed. And as we have observed when you have phi of P to be P minus 1 we can actually write down the values for most of these numbers. For instance, the value of 2, phi of 2 is 2 minus 1, phi of 3 is 3 minus 1, phi of 5 is 5 minus 1 and phi of 7 is 7 minus 1. These could always be, these could of course, be written without even have to worry about it. These other quantities phi of 4 and phi of 8 these are prime powers and then this is something which is still something complicated which is that it is a product of 2 different primes. So, to be able to compute the values of this Euler phi function what we are going to do is to define 2 is to have a result which allows you to compute the phi values for a prime power which will explain all the terms here in this table except the term 5, 6 the Euler phi function at 6 which is equal to 2. So, that is a very nice result. The result simply says that whenever P is a prime and you want to compute the Euler phi function on P power E for any E here there is no restriction on E any power E then phi of P power E is simply P power E minus P power E minus 1. So, this explains all the values on the previous slide except the value phi of 6. This is because if I wanted to find phi of 4 which is phi of 2 square this would be by our previous formula phi of 2 square minus 2 power 1 which is 4 minus 2 and therefore it is equal to 2. Similarly, phi of 8 which is phi 2 cube is 2 cube minus 2 square which gives me that it should be 4. This should tell you that if you wanted to compute the next 5 value which we did not but if you wanted to compute this this would be phi of 3 square which would be 3 square minus 3 and therefore this will be equal to 6. Indeed, if you were to look at the elements then you will have to remove 3, you will remove 6 and these are the elements modulo 9 which are all co-prime to 9. So, these are 6 in number. This is how the 5 values can be computed for all prime powers. Of course, every number is not a prime power. So, we will also later see a result which will allow us to compute general phi values. So, for the moment we go ahead with the proof of this very nice result that whenever we have p to be a prime then phi p power e is p power e minus p power e minus 1 and the proof is just the counting proof. We will write first of all we observe that a comma p power e is equal to 1 if and only if a comma p is equal to 1. So, if you have a natural number and it has GCD equal to 1 with respect to a prime power then clearly its GCD with p has to be 1 because p divides that prime power. If the GCD of a and p is not 1 then it would be bigger than 1 and then it will have to be p because p has no other divisors other than 1 and p and if p divides a then p will divide a as well as p power e and therefore the GCD of a and p power e cannot be 1. So, whenever the GCD of a and p power e is 1 the GCD of a comma p has to be 1 and if GCD of a comma p is 1 the GCD of a and p power e also has to be 1 because p is the only prime that divides p power e. So, when we are counting elements modulo p power e so we will start with 1, 2, 3 and write down all the elements up to p power e and among these we will have to find elements which are not co-prime to p power e and remove them we simply have to find elements which are not co-prime to p and remove them that is simple that is how simple it is. So, we start by writing the elements modulo p power e so we have 1, 2, 3 so on up to p. Then, we start writing the elements in the next row p plus 1, p plus 2, p plus 3 dot dot dot and then the next multiple of p will be 2 p after that we have 2 p plus 1, 2 p plus 2, 2 p plus 3, 3 p and we can just continue this way the last one before p power e would be where you will have this to be p power e minus p and the last one would be the number p power e. At each stage we are adding p we add p and get 2 p we add p and get 3 p so that here we will add p and get the number p power e and these are the only numbers which are not co-prime to p because to have some a power p not equal to 1 this is equivalent to p dividing the number and we have counted exactly the multiples of p going from 1 to p power e. How many multiples have we found? Let us count this as 1 this is 2 this is 3 and so on the total number of rows here is p power e upon p these many rows we have. So, phi p power e is p power e minus the number of elements 1 up to p power e with the property that p divides a and this number as we have seen is the number of rows in the previous slide which is nothing but p power e minus 1 and therefore, we get that phi p power e is equal to p power e minus p power e minus 1 a very simple proof. So, this is the proof that allows us to compute the phi value for any prime power. Now, to go for a general n what we have is that a general n will be product of prime powers. So, if we had a way to go from product of different prime powers product of from prime powers to their product then we would be able to compute the phi value for all such numbers. So, here is the next lemma which says that whenever a comma b is 1 then phi a b is the product phi a phi b. So, once you have this lemma this will tell you that you can compute the phi value for all numbers you will simply write n as product of distinct prime powers. For each prime power we know how to compute the phi value and to find the phi value of the number n you simply have to take the product of the phi p power e whenever p power e is the exact divisor of n for the prime p. So, even this proof is not difficult. So, one would prove this in the following way. Note first of all that some alpha is co-prime with a b if and only if alpha is co-prime to a and alpha is co-prime to b. This is the first major thing that we need to observe. How does one think about proving this? Suppose we have so this is again I have been telling you that we have whenever we have if and only if, if means that if this holds and this holds then we should get this. This is the meaning of if alpha comma a b is 1 if alpha comma a is 1 and alpha comma b is 1. So, how does one prove this? If you have that alpha comma a is equal to 1 and alpha comma b is also equal to 1 and suppose now that alpha comma a b is not equal to 1. So, we are assuming the contradictory. We are assuming the contrary we are assuming that alpha comma a b is bigger than 1 but alpha comma a equal to alpha comma b equal to 1. So, take the GCD d of alpha and a b and take a prime divisor of that since that number is bigger than 1 it should have a prime factor. If that p is the prime factor which divides alpha and also the product a b then you know what we are going to do we will take p that has to divide either a or b. p being a prime when it divides a b it will have to divide either a or b. If p divides a then alpha comma a will have a prime factor. The GCD alpha comma a will have p in it if p divides b then the GCD of alpha comma b will have p in it.    Think about this proof but it is an simple enough proof. So, when you are looking at elements which are coprime to the product a b you have to have those elements to be coprime to both a and b. This is number 1 second property that we observe is the following that if you have alpha congruent to beta modulo the number a then the GCD of alpha comma a is equal to the GCD of beta comma a and similarly when you have alpha congruent to beta modulo b then the GCD of alpha comma b is equal to the GCD of beta comma b. So, these are the two things which can be seen quite simply and therefore we will not see the proof of these things but we will just assume these both statements which says that if you have in particular it will say that if you have some number which is coprime to a by adding multiples of a to it you will still get a number which is coprime to a and the same thing will be true for the number b. So, now we want to compute the number of elements which are coprime to a b in terms of the numbers which are coprime to a and the numbers which are coprime to b. So, the proof would go again by writing all these elements as multiples of a 3 a and so on the last thing here would be a b and you are just adding a every time. So, the last but one thing would be a into b minus 1 these are this is how we would have all these a b terms these all these are all a b terms. So, you are actually looking at g a b and here we are now going to find elements which are coprime to a b. So, in this row we have to remove elements which are not coprime to a b and for that what we do is that we remove elements which are not coprime to a. So, here remove numbers which are not coprime to a. Suppose that there is an i here which is not coprime to a then all these elements in the column of i will have the property that these are going to be i plus a multiple of a. All these elements are i plus a multiple of a and therefore all these will have to be removed. So, all these going to be removed because all of these are not coprime to a and therefore they are not coprime to a b. So, when we remove a particular number i from here we are removing a particular column and thus in the table thus we have kept only phi a columns the table only phi a columns are kept. Further each column the table lists all elements of z b modulo b. Let us think about this statement for 5 seconds. We are saying that we are only going to look at the columns of the table. So, the columns remember they start with some element which is from 1 to a and then you keep adding a to those elements but a is coprime to b therefore when you look at mod b a is an invertible element. Therefore, when you start with any element module any element from 1 to a and then you keep adding a to them you are going to get all other elements in z b. First of all you have b elements because there are b rows and so you have say some j then j plus a j plus 2 a all the way up to j plus 3 a j plus a into b minus 1 and no 2 of these can be equal because j plus alpha a equal to j plus beta a modulo b we will tell you that alpha a is beta a modulo b but a is invertible modulo b being coprime to b. So, it tells you that alpha is beta so the b elements that you have obtained they are all distant modulo b. So, in this table each column if you for instance look at 2 the column corresponding to 2 you are going to get all b elements modulo b when you reduce them modulo b. And therefore each column looks like the elements of z b in a different order and hence to obtain elements which are coprime to b so we are looking to obtain elements which are coprime to a b we need to remove from each column the elements which are not coprime to b. We will have to remove the elements which are not coprime to b from each column. Earlier what we had done was that we have removed elements which are not coprime to a we observed that if there was any element from 1 to a b which was not coprime to a then modulo a it will correspond to some element from 1 to a with the property that that element is not coprime to a. This is the content of those 2 remarks that we have observed at the beginning of the proof. And similarly now from each of the remaining columns we will have to remove elements which are not coprime to b so the elements which are left in each column now is equal to phi b these are the elements which are coprime to b. So to summarize we have phi a columns in each column we have only phi b elements because we have removed all the elements which are not coprime to b. So we are left with only the elements which are coprime to b and that number is phi b and thus we get that phi a b is phi a into phi b. Just to give you an idea of this proof let me work this proof out in the very special case where we have a equal to 3 and b equal to 4. So we are looking at 12 and then we would look at 1, 2, 3 or let me put 4 next to 3 so that we have some space 5, 6, 7, 8, 9, 10, 11 and 12. And from these now for the numbers 1, 2, 3, 4 to remove numbers which are not coprime with 4 we have to remove 2 and 4 and then automatically 6, 10, 8 and 12 get removed. So we have exactly 2 columns left which is equal to the number 5, 4 remember 5, 4 is 2. And now from each of these columns we will go modulo 3 and we will see which are not coprime with 3, 9 is not coprime with 3 and here 3 is not coprime with 3. So there are 2 columns which is equal to 5 of 4 and each of these 2 columns we are exactly going to have only 2 elements which is 5 of 3. So this is how the general proof works. There is not much that I can say after this. So please think about this proof and if you have any problem in understanding this proof please write back to me and I will try to respond to all such queries. I hope to see you in the next lecture. Thank you.


I've got a small but kind of growing collection of old vintage textbooks, not massive ones, not incredibly rare ones yet, but I collect them mostly to see what the old style of science was like and whether there are any techniques in there that we might have lost or might be interesting to pick up. So I thought let's dive into some and see what they're like. So this one is a refresher course in mathematics by Frederick Cam, FJ Cam. So I did a little bit of digging on this guy and he is the founding editor of a number of kind of hobbyist engineering magazines, including practical mechanics, practical motorists, one of them the practical radio hobbyist, I can't remember the name, it's actually still going. And he's also the younger brother of the guy who designed the Hawker Hurricane fighter plane, so obviously a very engineering focused family. And this one comes from 1943, first published though I think this edition is 1955, so we're talking pre-computer, this is not incredibly old, it's not necessarily going to be completely different maths and it is intended to provide a refresher course for mathematics, for people who have previously mastered the subject but have forgotten the fundamental facts. And it really is just a list of the basics, decimals, how to do division, what all the signs mean, the subjects do get more advanced, we start with the addition and then we're going to plotting graphs. So we know this has not evolved a great deal since the mid 20th century, we would now use computers to do this and get the graphs plotted perfectly but they were always hand drawn back then, you could always find slides on presentations were hand drawn, they had the etched kind of at scale on the slide and doing graphs like that would have been part of that. So let's have a look at what we were talking about, averages, ratios and the one I wanted to look at here we are, the square roots and the cube roots. So this is an algorithm to work out the square root of a number, so there are multiple ways of doing this, there's the Babylonian method, Newton's method, the more generalized version and this algorithm, so if you have been taught how to multiply and divide and add up in school, you were probably taught an algorithm more than anything else, that's what the column multiplication actually is, it is an algorithm, it is a method that you repeat and you should get the right answer, it is not necessarily a mathematically robust explanation of how the system works but it should get you the right answers. So let's say if we can make sense of this, bear in mind that this is a refresher course, it is not meant to teach it, square root of the method of extracting square root is as follows, mark off the number the square root of which is to be found into periods by marking a dot over every second figure starting at the left decimal point, to draw a vertical line next to the left of the figure in a bracket you can kind of get the feeling here that this is really about the notation. Let's see, we can replicate the example for a moment, let's do, what is it, one, one, five, six, off and off every two or bracket, line down, gonna put the answer here, I think that's what it means. Find the largest square in the left hand period and place this root behind the bracket, so that is, okay so we need to find the largest, largest square, implicitly that is square number, so the largest square number, we can see it in the example is nine, so we place the root here and here for some reason, okay, next to square, next to square of this root is subtracted from the person, this is weirdly redundant, the square of the root, so we find the largest square, then you find the root, then you square it, okay, so that's three times three, that's nine, square of this root is subtracted from the first period, the next period is brought down adjacent to the remainder and used as a dividend, so 11 minus nine, that's okay, 11 minus nine, well that didn't help, right, so 11 minus nine, that's two, and then you bring the next period down, that's five, six of 256, so we've got this remaining. Now, multiply the first root find by two and place the pro to the left of the verse line, so three, multiply that by two, place it to the left of the verse line here, so now multiply the root find by two and place this product into the last of this line, then divide it into the left hand figure of this new dividend, ignoring the right hand figure, what does that mean? Now divide it into the left hand figures of the new dividend, ignoring the right hand figure, so that's 6 into 25, 6, 12, 24, okay, that's four, right, so attach the figure obtained, finally subtracting the product, multiply the latest divisor by the figure of the root last of, you just fold it in, four, so 64 times four, multiply, finally subtracting the product from the dividend. You say fold in one more time. It says fold it in. So 64 times, so what's 64, I'm gonna try it 256, subtract that, okay, so we terminated zero, don't we? So what I think this is doing is if you think about what a square number has to be, we have a square x on one side, x on the other, and the total area of that is x squared, so this total area is gonna be 1056, x is gonna be 34, what's happening is we've split this up, I'll not draw it to scale like here, so we have 30 and 4 here, and 30 and 4, so what we've got here is 30 times 30 for 900, so this nine here at the top is not really a nine, it is a nine kind of 100, so we're working out that as a perfect square, and the rest of the square are these three areas which must add up to in total 256, so we're trying to back calculate what that number must be, and if we get it exactly right, then the algorithm terminates, if not we have a little bit of a square left over and we start again on the next period, so we're multiplying it by two because we've got this area here which is 4 times 30, and this area here which is 4 times 30, and there are two of those, so what we've really got is if we ignore that square for one, just ignore that one, we've got two that are going to be four and 30, so if we times that 3 by 2 we get 6 or 60 here, and then we're trying to really back calculate what that number should be, and any remainder is there I think, so I think that's what that's getting at, that's what that's doing, let's try another example that's not in here, let's do 256, I know 256 is 16 squared, that's one positive, so well the answer to this should be 16, let's try this one, so we're going to square root 256, mark off the number of the square root of which is to be found into periods two five six, so we've got one period and one period there, draw a dot over every second figure, draw the vertical line, next find the largest square in the left hand period, well the left hand period is two, the largest square number in that is one, the square of this root is subtracted from the first period, yep that next period is brought down, so two minus one is one, we're in the next one, 156, now multiply the first root found by two, so we'll do that times two and divide it into the left hand figure, so two into 15, one, two, three, four, five, six, seven, seven, two into 15 is seven and a bit, so that's finding us the square root of 256, 17, divide it into the left hand figures of the new dividend, all in the whole ignoring the right hand figure, two into 15, two into 15 is seven, two should be six, next square of this root is subtracted from the first period, the next period is brought, this is your recipe, you fold in the cheese then, 20 sometimes, what's 27, what's seven times seven, 49, 189, 189, so 189, let's find the largest square in the left hand period, two into 15 is seven, I'm gonna have to work a pg of this, let's go, right, I know what's going on here, I have checked, there is a slight ambiguity here, let's start again 256 and why does 256 cause a bit of a problem, so 256, we want to work out that first root it is going to be a one the first figure and we have one two and we're going to do 156. So we've got to divide something into here and it's either going to be and we've got to figure out either 26 times 6 or 27 times 7 because if we put the skesh to 7 here or we put 2 into 15 to get 7 we then multiply 27 by 7 to get 189 that is 26 times 6 is 156 which rounds off perfectly so we would put a 6 here and then we do 26 times 6 256 algorithm terminates reason for marking these off as periods of groups of 2 it's because 10 times 10 is a hundred so that's we are working this out in groups of a hundred because if we have anything left over that becomes another another square that we can deal with we're doing it desk place by place and you can kind of see that that's what's going on when we talk about the screw boot because the key boot is effectively the same algorithm but the periods of placing a dot every third figure so a cube 10 times 10 10 will be a thousand there is a couple of drawbacks to it one as you can see from this example especially the numbers you have to start dividing in by get increasingly big so every time you bring another period down you're adding two figures but it must you're eliminating maybe one and if you get lucky two from the left hand side so this number every time you have to make a new iteration grows one larger and the numbers that you have to then start putting into your head if you want to do this without calculator are much much bigger it would take oh you'd have to then divide here by right you're doing what 12,563 dividing into something or multiplying by something but the advantage of this method is that it is figure by figure each figure is correct we know from the start of this this must have begun with a one that's not going to get better we know that that number is correct and we did it 16 we know that that is correct other methods are iterative so Newton's method for instance of finding roots we want to find a root of a polynomial we kind of guess and we do derivatives to figure it in a polynomial like that x squared minus 2 equals 0 is the equivalent of solving for the square root of 2 but all the numbers in those methods change every time you iterate this method the numbers you just stack on more numbers they are going to be corrected as an algorithm this is how it would have originally been done if you want the numbers to be as precise as possible as quickly as possible that's it here at all some shortcuts so practical men use a practical men there's very very possibly the most engineering words ever written use a vast number of shortcuts in calculations a few of the more useful ones are given to multiply by 5 add not the number to be altered and divide by 2 very straightforward sounds kind of weird but that is the mental shortcut you would do times in by 10 when you're in a decimal base 10 is trivial and then half is also kind of trivial so it's two operations that are kind of trivial and you get a lot of resentment towards things like that when people get confused by new math common core of you in America and the object to it they think oh why aren't you just doing the normal way that I was taught you actually do this in your head and it's more efficient and it's about learning number sense and the ability to do the shortcuts so it's nice to see that this even in the in the 1940s and 50s that was actually being taught I think towards the end we get shapes geometry some mechanics stuff so mechanics are really useful for science not because you will necessarily be using these diagrams and forces and understanding them directly but because what these diagrams are is that they have meaningful forces and directions and masses on them and you build equations from the diagrams and that act of interpreting what an equation means with the diagram and constructing the two out of each other that connection is amazing the useful if you want to do a mathematical science this is this is incredibly practical you want to relate two things together we do see it in physical chemistry we have to relate multiple things together if you want to do chemical kinetics you have to link the you have to link the macroscopic the symbolic and the sub microscopic worlds together it's not too different to how mechanics works you have a symbolic domain and a physical representation of what that means and you have to link them together so that's a really nice thing to see in a book like this and it's nice just to have all of this stuff that is designed to be the refresher of mathematics before you do any science or engineering it's it's interesting how some of its formatted because you see this is pre latex and and pre word and etc so how these were assembled I don't know probably having to typewriter them really difficultly in with a lot of precision instruments and then copying them or will one day a workout so there is nice book to have I might look at some more later