This lecture is part of Berkeley Math 115, an online course on number theory for undergraduates. This is a continuation of the previous lecture on primes, and will be mostly about arithmetical functions. So an arithmetical function is just a function from the integers, or the positive integers 1, 2, 3, 2, well, anything, it could be the complex numbers, or the reals, or even just the integers again. So it's just a function defined on positive integers. And there are some obvious examples. We could just have fn is some power of n, like n to the k. This is the property that f of mn is equal to f of m times f of n. And functions with this property are particularly interesting. They're called strictly multiplicative. We'll see what non-strictly multiplicative functions look like a little bit later. So powers of n are the most obvious strictly multiplicative ones, but there are also some other examples. For instance, we can have a function which looks like this. So fn just looks like this. So here's n. And fn is just going to go 1, 0, minus 1, 0, 1, 0, minus 1, 0, 1, and so on. So it just repeats with period 4. So it's equal to 1 if n is equal to 4m plus 1, minus 1 if n equals 4n plus 3, and 0 if n is even. And you can again notice that this is multiplicative. It has this property here. Later on, we will have some very important examples of strictly multiplicative functions called the Legendre symbol given by fn is equal to np, where this is not a binomial coefficient. It will be something we define later that tells you whether or not n is a square or not modulo p. This example and this example are both examples of things called Dirichlet characters, which will again be turning up later on in the course. So the Dirichlet character is strictly multiplicative and it also takes values that are complex numbers of absolute value 1. Another example of a strictly multiplicative function is the Liouville function, which is sometimes known by lambda of n, and lambda to the n is easier to define. It's just equal to minus 1 to the power of omega of n. So what's omega of n? Omega of n is equal to the number of prime factors counted with multiplicities. For example, if you take the number 60, this is equal to 2 squared times 3 times 5, so omega of 60 is equal to 2 plus 1 plus 1, so there should be an exponent 1 there and so on. So it's got 4 prime factors except you have to count 2 twice because it's got an exponent of 2. If you don't want to count primes with multiplicity, there's another function called little omega, where little omega of 60 would just be 3, because here you don't count primes with repetition. Anyway, you notice that big omega of n is equal to omega of m plus omega of n, which easily implies that lambda of n is equal to lambda of m times lambda of n. Little omega doesn't have this property, so it's rather harder to deal with than the big omega function. Well, these are all strictly multiplicative functions, and it turns out that this condition is really too strong. What we really want are multiplicative functions which have the following property. They have the property that f of m n is equal to f of m times f of n if m and n are co-prime. That means they've got no common factors. This seems to be a slightly funny-looking condition, but it turns out there are lots of natural functions with this property, so I'll now give some examples of them. Here's the first example. We could take the divisor function denoted by d of n or sigma zero of n, which is just the number of divisors of n. Let's just look at a few examples. If we take n equals 1, 2, 3, 4, 5, 6, we look at the divisors. Here we've got 1, 1, 2, 1, 3, 1, 2, 4, 1, 5, 1, 2, 3, 6. So d of n is just given by 1, 2, 2, 3, 2, 4. More generally, if n is prime, the only factors are 1 and p, so d of n is just 2. So it can be quite small. On the other hand, if n has a lot of factors, for instance, if it's the factorial of something or if it's something like 60, well, how many divisors does 60 have? Well, this is a little bit complicated because you can see there's 1, 2, 3, 4, 5, 6, 7, 0, 8, 0, 9, 0, 10. This is getting a little bit tedious to count them. Is there a better way? Well, yes there is, because what we do is we use the fundamental theorem of arithmetic and write 60 as a product of prime powers. And now you can see a divisor of n must be able to perform 2 to the a times 3 to the b times 5 to the c, where a is less than or equal to 2, b is less than or equal to 1, and c is less than or equal to 1. There are three choices for a, because you remember we allow a to be 0 as well, and there are two choices of b and two choices for c. So d of 60 is going to be 3 times 2 times 2, which is 12. And obviously we can do the same for any integer. If we know the prime factorization of an integer, so it could be p1 to the n1, p2 to the n2 and so on, then d of this number is equal to n1 plus 1 times n2 plus 1 and so on. So if we know the prime factorization, then it's easier to work out the number of divisors. You also see that this immediately shows that the number of divisors of n is multiplicative. So this is equal to d of m times d of n, if m and n are co-prime. Because if they're co-prime, then the formula for d of m times the formula for d of n is just the same as dmn. If m and n are not co-prime, then this fails. For example, if we look at d of 4, this is equal to 3. But d of 2 is equal to 2 and d of 4 is not equal to d of 2 times d of 2. The trouble is that the factor 2 kind of overlaps when we write the factor 2 there kind of interferes with the 2 there. So the divisor function is multiplicative, but it's not strictly multiplicative. So let's have some more examples of multiplicative functions. Another one is the sigma of n, which is the sum of the divisors. So let's try and work out what is sigma of 60. Sorry, my sigmas look a bit like 6s. I'm not quite sure what to do about that. So 60 we saw is equal to 2 squared times 3 times 5. So the divisors are of the form 2 to the a times 3 to the b times 5 to the c. And we want to sum over all a's from 0 to 2 and all b's from 0 to 1 and all c's from 0 to 1. So we want to sort of sum these numbers. So we're summing all numbers. So the numbers of the form can be 1 plus 1 or 2 or 2 squared times 1 or 3 times 1 or 5.  Because if we expand this out we see we get one of these possible factors each time. So this is the sum of the divisors of 60. We just take 7 times 4 times 6, which is much easier than just finding all the divisors and adding them up. And again, exactly the same thing works for any integer if n is equal to p1 to the n1 times p2 to the n2 and so on. Then by the same argument, sigma of n, we get by taking 1 plus p1 plus all the way up to p1 to the n1, then we do the same for p2 and the same for p3 and so on. And we just multiply these up. And you notice this is a geometric series and we know how to add up geometric series. This is just p1 to the n1 plus 1 minus 1 divided by p1 minus 1 and this is the same thing. It's p2 to the n2 plus 1 minus 1 over p2 minus 1 and so on. So this gives us a nice formula for sigma of n provided you know the prime factorization. In fact, working out sigma of n is quite similar in difficulty to working out the prime factorization. For example, if n is a product of 2 primes, then sigma of n is equal to p plus 1 times q plus 1, which is equal to n plus p plus q plus 1. So if you happen to know n is a product of 2 primes and you know the sum of the divisors of n, then you can work out p plus q and you also know p times q. So p and q both satisfy a quadratic equation whose coefficients you know and you can easily use that to solve for p and q. So finding sums of divisors of a number is about as difficult as finding the prime factorization. So another example of a multiplicative function that we're going to see later is Euler's phi function. So phi of n is the number of numbers less than n that are co-prime to n. So let's just work out a few values. If n is 1, 2, 3, 4, 5 and 6, then the numbers co-prime to n between 1 and n.  So phi of n becomes 1, 1, 2, 2, 4, 2 and if n is a prime then numbers 1 up to p minus 1 so phi of n is p minus 1 and you can see that this is the biggest it can possibly be. Phi of n is obviously at most n. In fact it's equal to n only if n is equal to 1. And we will see later that phi of m, n is equal to phi of m times phi of n if m and n equals 1. I'm not going to prove this now but when we do the Chinese remainder theorem a bit later we will see this. For instance you can see that phi of 6 is equal to phi of 2 times phi of 3. So it at least works in that case. You also notice that this is false if m and n are not co-prime in general. For instance the value for 4 is not the square of the value for 2. So all of these multiplicative functions it's fairly easy to see their multiplicative. So here's an example where it's rather harder to see that the function is multiplicative. Here what I'm going to do is I'm going to take the following funny looking function. I'm going to take q times 1 minus q to the 24 times 1 minus q squared to the 24 times 1 minus q cubed to the 24 and carry on like this and I'm going to multiply this all out. Well I'm not going to multiply it all out because I'm lazy. I'm just going to look up what the answer is and the answer is it's equal to q minus 24 q squared plus 252 q cubed plus 1472 q to the 4 plus 4830 q to the 5 minus 6048 q to the 6. Actually I think I got a minus sign wrong there but never mind. And you notice that if we write this as sum of tau n q to the n so this is tau 2 and this is tau 6. You might notice that tau of 6 is equal to tau 2 times tau of 3. And Riemannogen noticed that tau of mn seems to be equal to tau of m times tau of n whenever m and n are co-prime. So this was actually proved by Mordell shortly afterwards so it's an example of a multiplicative function where it's not at all obvious that it's multiplicative. And you might think what happens if you change 24 to some other number and if you change 24 to some other number it's nearly always not multiplicative. I mean I think there's one or two cases when you can sort of get it to be multiplicative but this is basically a funny property that doesn't work for anything bigger than 24. Another example of a useful multiplicative function is the Moebius function. So this function is denoted by mu of n. That's actually a Greek letter mu. You can see it actually does look a bit like an m. That's where our m comes from. So mu of n is defined as follows. It's 0 if n is divisible by square greater than 1. And it's equal to minus 1 to the omega n if n is divisible by omega of n primes.  I mean why on earth would you define a function in this rather bizarre looking way? Well before we explain why first of all you notice that mu of mn is equal to mu of m times mu of n whenever m and n are co-primes. So it is at least a multiplicative function. It's a bit like the Louisville function which was strictly multiplicative except you've made it 0 whenever n is divisible by a square. Well now I have to justify why you should introduce such a funny looking function. I'll just give one justification for it. So suppose we take the Riemann Zeta function. Zeta of n equals 1 over 1 to the n plus 1 over 2 to the n plus 1 over 3 to the n and so on. So it's the most notorious function in number theory. Now if you work out 1 over Zeta of s. I'm going to change n to s here because I'm thinking of s as being a complex variable. This turns out to be 1 over 1 to the s plus mu of 2 over 2 to the s plus mu of 3 over 3 to the s and so on. So Moebius's function turns up in the coefficients of the inverse of the Riemann Zeta function. Since the Riemann Zeta function is perhaps the most important and mysterious function in number theory this justifies why you should take the Moebius function seriously. Now I want to give an application of multiplicative functions to perfect numbers. So you remember a perfect number is a number that's the sum of its proper divisors. So 6 for example is the simplest perfect number. As I mentioned in the introduction perfect numbers are not really a terribly serious part of number theory but this will be a sort of good exercise in working with some of these multiplicative functions. And the problem is to find perfect numbers. And 6 is perfect and 28 is perfect. So this is 1 plus 2 plus 4 plus 7 plus 14. So the sum of all proper divisors is half of the sum of the divisors. So what we're doing is trying to solve 2n is equal to sigma of n. So we want the sum of all divisors to be twice the number. And we know what sigma n is in terms of the prime factorization of n. So if n is equal to p1 to the n1 times p2 to the n2 then sigma of n you remember is 1 plus p1 plus p1 to the n1 times 1 plus p2 and so on and so on. And now Euclid in his book on the elements everybody says it's all about geometry but that's actually about a lot of things other than geometry. He gave a way of finding perfect numbers. He said if you take a number 2 to the p minus 1 times 2 to the p minus 1 such that this is prime and p is also remember if 2 to the p minus 1 is prime then p also has to be prime so p is going to be some prime. Anyway this number here is now perfect and this is actually a rather easy calculation. All we do is we expand out this. So since this is prime we know the prime factorization of this number and we just have 1 plus 2 plus 4 plus all the way up to plus 2 to the p minus 1. So that's the that's this bit for the 2 to the p and then we have 2 to the p minus 1 plus 1 here. So that's that bit there and this is equal to 2 to the p minus 1 and this bit is equal to 2 to the p and if you compare that with this see if n is equal to that number here this is just equal to 2n. So this number is perfect and this gives you the first few perfect numbers for example for p equals 2 we get 2 times 2 squared minus 1 and if p equals 3 here we get 2 squared times 2q minus 1 which is 4 times 7 which is 28 and you can easily find the next one for instance if we take p equals 5 we get 2 to the 4 times 31 which is the next perfect number. Well there's actually a nice converse to this found by Euler. Any even perfect number is of this form and let's just show you how to prove this using using the formula for the sums of the divisors. So suppose our number is 2 to the a times p to the b times q to the c and so on where p and q and so on are various odd primes. So we're going to take our number n to be this and we're assuming it's perfect and we can look at all the factors of it and the factor is going to be we take 1 or 2 or 4 up to 2 to the a times 1 or p or p squared up to p to the b and some factor of this and so on so these are the divisors of n and what I'm going to do is I'm just going to take some of the divisors so I'm going to take the divisors where the part here is either p to the b or p to the b minus 1 and for this one I'm only going to take q to the c and let's work out what we get here. The sum of these special divisors is 1 plus 2 plus 2 to the a times p to the b plus p to the b minus 1 times q to the c and so on and I'm now going to add these all up so this is just 2 to the a plus 1 minus 1 and this is just p plus 1 times p to the b minus 1 and you can see that this number is equal to 2 to the n times 1 minus 1 over 2 to the a minus 1 times 1 plus 1 over p so now we notice that this bit here must be at most 1 because the sum of all divisors has to be 2n and the sum of some of the divisors is this and the sum of all of them is going to be at least that. On the other hand we notice that this number here is odd and since this product here is equal to 2n this must be divisible by some prime factor of n so if p is the smallest prime factor of n then p is less than equal to 1 plus 2 plus all the way up to 2 to the a now from the fact that p is less than or equal to 2 to the a plus 1 minus 1 we see that this factor here must be greater or equal to 1 if p is equal to 2 to the a plus 1 minus 1 you can easily check this expression is 1 and if p is less than that that makes this even bigger so on the one hand this expression must be at most 1 and on the other hand it must be at least 1 we see that must be exactly equal to 1 well if it's exactly equal to 1 then we can't have any other divisors so in particular this implies b must be equal to 1 it implies we can't have any other primes q dividing it so q, r and so on do not exist and it also implies that p is equal to 2 to the a plus 1 minus 1 so in other words our number is equal to 2 to the a times 2 to the a plus 1 minus 1 in other words it's just one of these mo-sen primes that we discussed earlier so this gives us all the even perfect numbers and it leaves two problems are there an infinite number of even perfect numbers? well since these correspond exactly to primes of the form 2 to the p minus 1 we can ask are there infinitely many primes of the form 2 to the p minus 1 and we don't know it's just probably yes and people have found in about 40 or 50 of them by computer the largest that have been found absolutely enormous the largest known at least according to Wikipedia this morning was 8, 2, 5, 8, 9, 9, 3, 3 minus 1 which has about 25 million digits if you try and write it out in other words you need an entire bookshelf of books just to write down this prime number as you see you can't possibly check these things of primes by checking all prime factors up to their square root I mean that would be just absurd fortunately there are much faster ways of checking whether mo-sen numbers are actually primes in fact there used to be something called the sort of mo-sen internet prime search where people would use spare computer time on their computers to help search for new mo-sen primes I think people have stopped doing this to a large extent because if you've got spare cycle time on your computer you don't use it for looking for mo-sen primes anymore you use it for mining bitcoin instead which is more profitable so whatever this is another example of one of these very simple questions that anyone can ask are there infinitely many primes of the form 2 to the P minus 1 and it seems to be hopelessly out of reach of any techniques we have at the moment there's just none of our techniques come even close to this so we can't tell whether they're an infinite number of perfect numbers there's also another question that you can ask are there any odd perfect numbers and nobody knows people have tried to answer this and you can put some very restrictive conditions on your perfect number for example suppose your perfect number is P1 to the N1 times P2 to the N2 and so on then sigma of n must be equal to 2n which must be divisible by exactly 1 power of 2 because n is odd and if you expand this out you see this is 1 plus P1 plus all the way up to plus P1 to the N1 the same for P2 and you notice that if N1 is odd this sum is even and if N1 is even this is odd so let's say this is odd if and only if N1 is even now if this product is 2 times an odd number exactly one of these must be even and that means that almost almost all these exponents here must be odd so this says exactly one of the Ni is odd so in other words the number N can actually be written as a square times a prime because one of these numbers is odd and it's going to be that prime times something with all exponents even so it's a prime times a square and you can push this sort of argument much further and find much more restrictive conditions on odd perfect numbers but it's very frustrating if you do this because you never quite get a contradiction and nobody's been able to prove that odd perfect numbers don't exist and people have done computer searches and haven't been able to find any so nobody knows and this is another of these really annoying problems that are easy to state and in some sense are kind of almost pointless but we just have no idea how to solve them in fact this is a good candidate for one of the oldest problems we don't know how to solve people were discussing this problem about 2,000 years ago and we still don't know the answer I'm not quite sure if it's the oldest unsolved problem I'm not an expert on Greek mathematics but I don't offhand know of any older problems we don't know the answer to so I've just finished by listing Landau's collection of really hard problems so these are Landau's problems about primes that he stated about the beginning of last century and just show us how little we know about primes and some more questions that are quite easy to state but we have no idea how to answer so first of all we have the Goldbach conjecture is any even number greater than 4 the sum of 2 primes so next we can ask the Twin prime conjecture so can we find infinitely many primes p1 minus p2 with difference 2 thirdly we can ask suppose we take a number n squared can we find a prime between n squared and n plus 1 squared so that's saying is the gap between prime between numbers at most the square root of the number in fact we think the gaps are a lot less than that but even this case seems to be out of reach of what we can do and finally can we find infinitely many primes of the form n squared plus 1 and again the answer to this seems to be the answer to all these four questions is almost certainly yes we can give probabilistic arguments suggesting they're rather likely to be true but they're all just out of reach of any techniques we've got these days ok next lecture will be on binomial coefficients


Literally, you would know that I am a professional YouTuber because of my camera issues. I think I'm literally perched on a jar right now. Anyway, welcome to a new video. You're sort of precariously perched, but I think that'll work. Okay, so for this first part of this video, I just wanted to do a little explaining. We're going to start a new series that has to do with doing art for therapy for mental health.  I'm not a doctor. I'm not a counselor. I do encourage you to seek out professional help if you're struggling with your mental health. I'll try to find some links for some online places where you can research getting some help down in the description below. I'm doing what's working for me on the advice of my professionals that I'm talking to. I do have a counselor and I am on medication. I have generalized anxiety disorder and depression for those that don't know. So anyway, I forgot when I first started my YouTube journey that mental health and doing art for mental health was one of the things that I did do and that it worked and I forgot to keep doing that and now I'm back to doing it and I do love it. So anyway, I'm going to show you some new things that I'm doing and share my experience with you. Hopefully you decide to follow along and if you do, please share. I do have a Facebook group, a life of art and self expression. The link is in my link tree list of links down below. I encourage you to share it over there and tag me in the post if you want to make sure I see it. I'm also on Instagram and all that stuff too. And if you do videos on this, I would love to see it. So make sure that you tag me or tell me or something. Anyway, let's go down to the table and let's get started. I'll explain what this is in a minute. So years ago, and I mean like 2016, 2017, 2018, I took this Moleskine kayak and I used, I was trying to use up ink sprays or yeah, like delusion sprays.  And I did have some and I wanted to use them up. So I took one of these journals and I just sprayed random colors on every page, squished the pages together and then stuck something in between them while they dried. And then I went through and drew over whatever was on there. Now, when I first started it, I really tried to like force an image into what was on there in the background. This, of course, is being what was in the background. Most of the time, not all the time. I really tried to not think about it too much and just draw what I was inspired to draw or when I was feeling most of the time, I felt like I was forcing it and I really wasn't opening up. Sometimes I just was inspired with words. Sometimes I was really, yeah, I was just really trying to figure out my voice and what I wanted to say. I did do a photo shoot for Yahoo Small Business a long time ago and I did have this journal at the photo shoot and they had me do a couple of drawings in it, this being one, this being another one. They weren't really what I was feeling. At some point, I picked this up recently. I rediscovered it and said, you know, I really should finish this. And I started again with doing drawings that I felt like I was forcing myself to do. And then I slowly moved towards seeing what the ink splots spoke to me, what they were inspiring me to draw or in this case, write or journal, what was I feeling about what was in the page. And slowly it became more about what I saw on the page and less like thinking and analyzing and just drawing what I was feeling. And it became a very therapeutic process for me, sort of like in the Rorschach ink blot test where the therapist will have you look at ink blots and what do you see in the ink blot. And instead of just saying what I would see, I started drawing what I would see. And I found that very so therapeutic that I'm going to, I think keep doing it. So I'm working on a couple more journals with ink blots or, you know, spots already distressed on the page. And we are going to, I think that was my phone, we are going to prep another couple of journals. Now, again, in this one, I, and I'm going to show you how I do this is going to be part of the process is going to be a multi part video series. This first one, we're going to just prep our journal and I'm going to just do this like quick flip through. You can see what I'm talking about like towards the end. I was just, you know, oh, that looks like bubbles to me and I drew bubbles on a fish. Oh, I see a face. Draw face. Oh, I see abstract like triangle slices of pie, like whatever. Yeah. So I'm going to show you how I prep the journal in this video. And I'm going to find someplace safe to put that one. Okay. So you can use any journal. I want you to use what you have.  This is a Moleskine Cayet. The paper is pretty thin, but I was okay with that. I knew I was going to work single-sided. I didn't care for blood. You can use a bigger journal. I do recommend one with not too many pages so that you just have this to work on like for a month and then see if you still like it. If it's too many pages and you're like me, you're going to get bored halfway through. You're never going to finish it or you'll get discouraged or like, I don't know what the deal is. Most of the time with big journals, I never finish them. These are traveler's notebooks. Now this, I've already started and this is coffee mixed with a little bit of walnut colored ink. And I did that and I let it dry. I did it yesterday. And now I'm looking at it and I'm feeling like it's not enough color. The ink blots, although there are blots on here, it's not enough of what I want. For whatever reason, I'm feeling the blues and I'm going to put blue. I'm going to show you what I do. This is one I did a few days ago and I started it the same way. It's the same kind of journal. Instead of doing it the way we're going to do it, I mixed up a little cup of water and some of the same coffee and walnut ink and I used a pipette and I just dropped it on. I got it really, really wet, so wet that the paper started ripping. I was okay with that. I just put masking tape on it after it was completely dry. I did know right away that the coffee wasn't enough color and these teally blue colors you see are pigment powders. And I have a bunch of those that I'm just trying to use up, so I use those on here. Now the only problem with some of these pigment powders is the glittery ones, I don't know if you can see that. They kind of rub off a little bit, but that's okay. It's totally okay. Pieces of paper are just completely missing. It's going to be part of the journey of this journal. Again, we're going to work on creating something to work out our emotions or feelings or mental health about whatever is going on in our lives based on what we see on the page, what we're feeling on the page. It's not about creating perfect art or anything. That being said, if you're just into junk journaling, this is a great way to distress a journal. Anyway, I love the way this one turned out, but it did get very wet. So anyway, this time I have decided to stick with spraying like I did the first time. So this one, I mixed up the walnut ink and coffee in this little spray bottle. I used up the whole bottle on the journal and when I was done, then I just washed the bottle out, which we'll do again. One of my issues with sprays is that they clog all the time, which drives me up the wall. Now, if you don't have, maybe you don't drink coffee, you could use tea, maybe you don't have any pigment powders or inks, anything that you can dissolve in water and get watery, a little bit of acrylic paint, a little bit of, maybe you have a re-inker, maybe you have, this is an acrylic ink, but maybe you have a little bit of watercolor paint to dissolve some in a bunch of water and use that. You can use anything, use what you have, look around your art room, what do you have that's going to dissolve in water and create a color that you want that you're feeling right now. Maybe you want to do a few of them in different colors. I do recommend that you either use spray bottles you're okay with tossing out or as soon as you're done that you clean the bottle out because if it's acrylic, it's going to dry and clog the tip, like kind of guaranteed. Get an old cookie sheet. I do line it with a piece of paper because this is going to get inky and this could be collage paper for a future project. So anyway, we're going to open it to the first double set of pages. Maybe I'll start here. Okay. So we're going to take our bottle, which already has water in it. I'm going to open my acrylic ink. There we go. So I'm in there. I want to put any other colors in there. Let's see. I want to put a little bit of gray in there because that's just what I'm feeling and I know why. This is like a purpley gray. We'll put a little bit in and see how that looks.  Yeah, that's good. I like that color. I just put a little bit in it. Just took a little bit of the brightness out of the blue. Okay, and then and then have something next to you that you can put in between the pages.  I've got a bunch of different kinds of journaling clips around the art room and you'll see as we go along, I just put them between the pages so that they don't stick together too much. You're going to stick a little bit and you'll get like little spots where the paper is going to want to rip. If it rips all the way through, just put a piece of masking tape there. It's part of creating this random series of marks on the pages that you're later going to do some art therapy over and journaling over. So anyway, we're going to just go with it, but yeah, this will help them from sticking completely together in theory. All right. So now we're going to just like that. You might find you have to do this a few times with a few different colors in order to get, you know, the kind of look that you want. Maybe you want to make a much darker color because you're not feeling these lighter tones. Do what you're feeling. Think about, you know, where your mental state of being is at the moment and what's going to work for you and then maybe turn on some music or your favorite podcast. You'll notice I'm alternating where these clips are. That's just so that I can get through the whole book in one go around, but maybe you just want to do one or two pages and you don't you're not impatient like I am. And you want to just do the whole thing and get it over with and let it dry. Now once you do this, it's going to take a couple of days to dry completely. See here's a piece that has torn, but you know what, it's part of the journey of this journal and the journey of my mental health. And that's OK. The only problem with doing it this way is your fingers get sore. If you want to do more than say one color, pre-mix all your colors. You can spray all the different colors on here and then just keep going. I don't clean these clips off in between. And so they may transfer some of the coffee color or other colors onto the paper. Totally OK with me. So just keep going until you get the whole book done. Break out if something rips. Just let it go when it's done and dry. Get some tape. Maybe you have some washi tape that you're feeling needs to go in here. Maybe you just have plain masking tape and like I did and that's what you're feeling. Maybe you're feeling you want to collage some paper over the rips. Don't be super precise about it. Agonize about it. Just again, go with how you're feeling. Almost. We're almost there. I can tell you we're ready. There's a couple spots that are ripping. You can't see them, but I can. And that's OK. You might get to the point like I'm starting to get through with the sprayer. We're not wanting to spray because there's running out of water. OK, once I have it to where there's some spray on all of them, whatever's left in here before I go wash the sprayer out, I dump on there. So now we're going to just let it dry and let it go. And when it's done, we'll get started on our Rorschach test or ink blot art therapy on the journal. And we'll see where we go from there. So I want you to prep your journals, let them dry. And I'll be back in the next clip before we end this video and I'll show you what the result is of how this looks. And we'll do some taping because it's going to need some masking tape. So I'll be back when this is dry. OK, guys, I've got lots of stuff on my desk I have to deal with, but we're going to do this first. So this is the journal that we died. And yeah, that's fine. And I'm going to just flatten it a little bit, but it's only going to do so much. Anyway, we're going to go through it and we're going to find places like this where the paper tour. And we're going to just put some masking tape and tape up all of those spots. The tape is going to be part of the background, part of the spots and marks that are going to hopefully help us do some art and journaling in a way that helps us work through whatever it is we have to work through. You can use colored tape if that's what you're feeling. I am really just feeling using masking tape. So I'm going to use that. This is how I do the spines when they rip. Just FYI, push it all down good. And then I'm not worried about these little tears on the edges. But like here we go. So let's fix that. And I'm going to go through every page, page by page. And I'm going to fix each and every one that I feel like I need to. And maybe you're not going to feel like you need to put tape over any of the holes. Because maybe you want to work with the holes as part of what you end up doing on the page as part of your therapy. That might work too. But anyway, for me, I want to fix the holes and I'll be right back. So I want you to make one or two journals. And then join me in the next video where we do some ink blob art on one of the pages. And again, maybe you want to have your journals be more colorful like my first one was. Maybe you want it to just be all one color. Maybe you want to use paint instead of inks or water soluble materials. You can do that. My suggestion is use something with a fairly matte finish. There's another spot that needs tape. Use something with a fairly matte finish because it's easier to draw and write over. If you use traditional acrylic paints that are glossy, they can be a challenge to write or draw over. So I don't necessarily recommend them. So just keep that in mind. That's why I tend to use ink for maybe I would use a gouache or a matte finish paint if I was going to go that way. If you're going to do paint, too, you can only do probably one or two pages at a time without really having a huge problem with the journal and all the pages like completely sticking together. So experiment and play. Grab a journal that's just sitting in the closet that you probably know you're never going to use for anything else. Use what you have. And if you have any questions, comments or concerns, leave them down below. I'm going to prep for the next video. I'll see you over there.   Again, I am not a counselor or a therapist of any kind. I go see someone for that. There's no shame in needing help with if you have mental health issues. I will try to find some sort of a link or something down below. But if I can't and there's nothing there and you guys have a some kind of reference website that you use, please let me know and I'll add that in and go out and do something nice for yourself because you deserve it. And I'll see you later. Bye, guys.