

The Ways of Counting

--- Essay by Philip Emeagwali


"Can you do addition?" the White queen asked. "What's one and one and one and one and one and one and one and one and one and one?" --- Lewis Carroll, Alice Through the Looking-Glass.

The word "computer" is often thought to have been coined with the advent of the first electronic computer in 1946. Actually, according to Webster's dictionary, the earliest recorded use of "computer" came exactly 300 years earlier, in 1646.

From the 17th century until the turn of the 20th century, the term was used to describe people whose jobs involved calculating. In his 1834 book Memoirs of John Napier of Merchiston, Mark Napier wrote:"Many a man passes for a great mathematician because he is a huge computer. Hutton and Maseres were great calculators rather than great mathematicians. When their pages were full of figures and symbols, they were happy. [John] Napier alone, of all philosophers in all ages, made it the grand object of his life to obtain the power of calculation without its prolixity."

The logarithms of numbers from one to 10.

The exponential and logarithmic representations of numbers.

Human computers have both strengths and weaknesses. We humans have great powers of reasoning. Most of the things we do come from knowledge that we have developed culturally or that we have learned as so-called common-sense rules. We also have great tolerance for ambiguity. Confronted with a sentence that has several possible interpretations, humans can choose the correct interpretation for the given context. Humans also have the capacity of categorization, such as identifying and classifying birds, understanding regional variations in accented speech and so on.

At the same time, we are unable to perform more than one calculation at a time (sequential computing). We compute at a very slow rate, and with a very small memory. Even worse, our limited memory sometimes causes us to forget intermediate results and thus fail to reach solutions. We need to write down many intermediate results and can use only simple procedures to solve mathematical problems.

Since the human computer preceded the electronic computer, and since humans developed the electronic computer, we created it in our own image. This means that early electronic reflect our limitations. Just as the human computer has only one brain, so the early electronic computers had only one processor. Since our brain can perform only one calculation at a time, most computers were built to perform only one calculation at a time.

The world's first large-scale general-purpose electronic computer, the ENIAC (Electronic Numerical Integrator And Computer), was designed with only one processor.

Two men (in uniform) being trained to maintain the ENIAC computer. The two women in the photo were programmers. The ENIAC occupied the entire thirty by fifty feet room.

One of the primary problems that motivated the design of ENIAC was to calculate the trajectory of ballistic missiles at the U. S. Army's Ballistic Research Laboratory in Aberdeen, Maryland. Before the invention of ENIAC, Army engineers used 176 human computers to perform the calculations needed to produce firing and bombing tables for gunnery officers. Such calculations must be sequential, and therefore they mapped well onto the early electronic computers and their sequential processing.

Another example of a technology matched to the human image is robotics. Early robots tended to have arms and/or legs and one centralized brain. This design has shifted recently to one in which robots now have a network of small, simple brains (or processors), each devoted to a single job such as lifting an arm.

As mathematical reasoning progressed, however, it became apparent that sophisticated mathematical techniques suited for electronic computing diverged from those suited for the human brain. For example, the 10-based decimal system has proven to be the best for human computers, while the binary system based on the number 2, although too awkward for humans, is a natural match for electronic computers, where strings of bits with values of either 0 or 1 are used to store data.


"`Firstly, I would like to move this pile from here to there,' he explained, pointing to an enormous mound of fine sand,`but I'm afraid that all I have is this tiny tweezers.' And he gave them to Milo, who immediately began transporting one grain at a time." Norton Juster, The Phantom Tollbooth.

As mathematics progressed, the tasks undertaken by human computers grew increasingly complex. By the late 19th century, certain calculations were recognized as outstripping the calculating capacity of hundreds of human clerks.

The 1880 United States census posed one of the biggest computational challenges of that time. To analyze the 1880 census results, the U.S. government employed an army of clerks (called "computers") for seven years. The population changed significantly during those seven years, however, rendering the results out of date before they were published. More timely results could have been obtained by employing even more clerks, but that would have been too costly.


The human computers worked independently and simultaneously on the 1880 census data to produce numerous and voluminous sets of tables. In the field of computation their method is termed "working in parallel." It was the best technique known at that time, but since the results were still unacceptably slow in coming, the Census Bureau held a competition to find a more effective way to count the population. This competition led to the invention of Hollerith's Tabulating Machine, which was subsequently used to compute the U.S. population more accurately in 1890.

The tabulating machine and its card sorter.


The century-old idea of holding a competition to solve computation-intensive problems was revived in 1986 by the Institute of Electrical and Electronics Engineers Computer Society with its annual Gordon Bell Prize Competition. As in 1890, competition has spurred advances in computing.

The solution of important computation-intensive problems, like those required in weather-prediction, demand the calculation of trillions, quadrillions and even more calculations, whether by human or mechanical or electronic devices. Solving these problems one at a time or in a step-by-step fashion is analogous to moving a pile of sand one grain of sand at a time. It takes forever.

Fortunately, many computation-intensive problems do not have to be performed one at a time. In fact, performing thousands and even millions of calculations at the same time is the only way to solve many computation-intensive problems quickly enough for the results to do some good. The latter approach is called massively parallel computing --- but with machines instead of the human computers that used this system in the 1880's.


Researchers are now learning that many problems in nature, human society, science and engineering are naturally parallel, that is, that they can be effectively solved by using mathematical methods that work in parallel. These problems share the common thread of having a large number of similar "elements" such as animals, people and molecules. The interactions between the elements are guided by simple rules but their overall behavior is complex.

An individual ant is weak and slow, but ants have developed a method of foraging for food together with other ants. Their massively parallel approach is well described by the scientist and writer Lewis Thomas in The Lives of a Cell:


Massive parallelism can also be found in human society. We see it in wars, elections, economics and other endeavors characterized by the simple, independent and simultaneous actions of millions of individuals. This parallelism is so natural that people aren't even aware of it. Adam Smith described it in The Wealth of Nations:


In science and engineering the common thread that makes many problems naturally parallel is that they are governed by a small core of physical laws that are local and uniform. Local means that to know the temperature in say, Detroit, in the next few minutes, we only have to know what is happening right now in the nearby suburbs of Detroit. Uniform means that the laws governing weather-formation in Detroit are the same as in Timbuktu, Vladivostok or anywhere else.

In weather forecasting, five uniform and local laws are used: conservation of mass; conservation of momentum; the conservation equation for moisture; the first law of thermodynamics; and the equation of state.


Constructing and running a computer model with the level of spatial resolution required for accurate computer-based weather forecasting takes quadrillions of arithmetical operations. Yet even this spatial resolution is too coarse to image tornadoes and other damaging local weather phenomena.

To improve the spatial resolution so that it images local weather increases the amount of calculation required by 10s of times and consequently makes the original problem computationally intractable even for a modern supercomputer.

The U.S. government has identified weather forecasting as one of 20 computation-intensive "grand challenges." Others include improving extraction of oil deposits, modeling the ocean, increasing computers' skill at understanding human speech and written language, mapping the human genome, and developing nuclear fusion. Many other problems in mathematics would take trillions of years to solve with the faster computers currently available.


Seminal ideas in science usually occur in a process that has been described as a series of paradigm shifts, that is, a shift in the dominant patterns or examples which humans use in thinking or portraying "how things are." A well-known example of paradigm shift is the transition from the belief that the Earth is flat to the belief that it is round. With this paradigm shift, Magellan became aware that he could travel around the globe.

If only one human computer had been used to analyze the 1880 census, the work would have taken forever to complete. Similarly, if only one computer were used to solve a computation-intensive problem of today, it would take forever to complete it. This paradigm of using one computer to solve a problem is called sequential computing.

We are now living through a great paradigm shift in the field of computing, a shift from computing in the image of the human brain (sequential computing) to massively parallel computing, which employs thousands or more computers to solve one computation-intensive problem. Just as a paradigm shift in the belief about the shape of the Earth led to routine circumnavigations on the high seas, we would soon routinely solve important societal problems that are so computation-intensive that we had previously only dreamed of attempting them.

Examining the roots of this paradigm shift will show why continuing efforts to solve computation-intensive problems with sequential computers will succeed only as well as the early aircraft did by attempting to fly by flapping bird-like wings.

Since many computation-intensive problems are inherently parallel, it only makes sense to build and use a computer that exploits their inherent parallelism. Such a computer will give the best performance when it "looks" like the inherently parallel problems that it is trying to solve. The problems --- such as modeling atmospheric conditions, as discussed earlier --- arise from myriad simple interactions between thousands or millions of elements. As a result, the total calculations required carry us into the realm of "teraflops" computing.

The cart pulled by the ox represents a conventional supercomputer. Intuition might lead one to think that the bull can outperform the cart pulled by a multitude of well-trained, harnessed chickens. But the coordination of many smaller units results in better performance. Similarly, a massively parallel computer is faster and more powerful than a conventional supercomputer.


"Teraflops" may sound like the name of a dinosaur, but it does not describe extinct creatures, rather a level of computing yet to be achieved --- computing on the trillionfold level. The Holy Grail of large-scale computation is to attain a sustained teraflops rate in important problems.

As Elizabeth Corcoran put it in January's Scientific American, "From the perspective of supercomputer designers, their decathlon is best described by three Ts--a trillion operations a second, a trillion bytes of memory and data communications rate of a trillion bytes per second."

The abacus is the oldest computing machine. It was invented in China about 2,000 years ago. Two millennia ago, a massively parallel supercomputer will be 64,000 people operating 64,000 abacuses.

The effort to increase computing speed is an old one. It is the primary motivation that led to the invention of the abacus, logarithmic table, slide rule and electronic computer. However, these devices share one fundamental limitation: they are based on sequential computing, and thus are subject to the speed limits imposed by computing singly. As a result, some investigators now believe that massive parallelism is the only feasible approach that can be used to attain a teraflops rate of computation. Only massive parallelism can allow us to perform millions of calculations at once.

The general formula for logarithms of numbers. Three hundred and fifty years ago, a massively parallel supercomputer will be 64,000 mathematicians aided by 64,000 tables of logarithms.

The most massively parallel computer built so far, known as the Connection Machine, uses 64,000 processors to perform 64,000 calculations at once. This approach makes it 64,000 times faster than using one processor.

The Connection Machine has attained computational speeds in the range of five to 10 billion calculations per second. The next generation of the Connection Machine is expected to use one million processors to perform one million calculations at once. That could make it one million times faster than using a single processor. The target computational speed of this machine is one trillion calculations per second.

The Pickett linear slide rule. Note that the Pickett manual dated 1960 used the word computer to denote a person. The manual reads: "A computer who must make difficult calculations usually has a slide rule close at hand." In the past, a massively parallel supercomputer will be 64,000 people operating 64,000 slide rules.

Massively parallel computers also have properties that make them far less costly than conventional computers. When the number of processors are doubled, the performance doubles, but the cost increases only by a few percentage points, since many components are shared by all the processors.

A massively parallel computer also runs cooler than a conventional supercomputer. You can rest your hand on it while it runs; it doesn't require the $10,000-a-month energy bills needed to keep a mainframe supercomputer from overheating.

We have come a long way from the individual with an abacus to today's massively parallel computers. The motivation along the path from then to now has been the existence of computation-intensive problems requiring computing resources of larger magnitude than those available.

Despite humanity's great progress in computing, the national grand challenges compiled by the U.S. government identify computation-intensive problems that will inspire us to find even more new ways of computing. In five years, we should be computing at the teraflops level. It will be fascinating to see what achievements and new challenges will come after that.

NOTE: The preceding article was published in the February 1991 issue of MICHIGAN TODAY, a quarterly publication that describes the most innovative research conducted at the University of Michigan in Ann Arbor. The later issue focused on Emeagwali's research and was mailed to the university's 400,000 alumni. Nine of the original illustrations and photographs were not available and, for this reason, we have added four images for your benefit.


  • The march of technology is towards oodles of calculations in almost no time at all.
  • Simulation of seismic waves propagationg through a model containing a salt dome --- an arching formation in sedimentary rock with a mass of rock salt as its core. Such models require massive amounts of computations at many sites over regular intervals of time. The demands of a whole class of such computation-intensive problems is ushering in a new era in computing.
  • Quadrillions of calculations are needed to model Earth's climate. A conventional supercomputer (diagram #1) tries to use one processor to perform the quadrillions of calculations. It's very powerful, but not powerful or fast enough to complete the calculations in time to yield a useful forecast of approaching weather.
  • The natural laws governing Earth's weather--and the mathematical equations derived from those laws--operate at all locations in our atmosphere, and in the same way (diagram #2). Therefore, the quadrillions of calculations required to solve weather equations are distributed evenly throughout the globe, whether the climatic conditon is a blizzard in Boise, a thunderstorm in Oslo, torrential rains in Bangladesh or bright son over Harare. When a natural law applies to all locations in the same way, the problem is inherently massively parallel.
  • The architecture of a massively parallel computer is designed to match the computer with the problem. To perform calculations in weather prediction, a massively parallel computer assigns different locations around the globe to different processors inside the computer (diagram #3). Today, the most powerful massively parallel computer has 64,000 processors. A massively parallel computer with a million processor is under construction.
  • The 1922 dream of Lewis F. Richardson: 64,000 human computers to calculate the variables required to predict the weather numerically. He foresaw that mathematical forecasting was computation-intensive, but didn't realize it would have taken his calculating army 1,000 years to predict the next day's weather! Illustration by A. Lannerback, Dagens Nyheter, Stockholm.
  • Emeagwali with the Connection Machine in Cambridge, Massachusetts. Massively parallel computers are a young technology. Only a few universities have acquired their own models. The U-M is now considering venturing into this field, Emeagwali says.
  • There's a supercomputing race between --- The Chickens and the Ox

Philip Emeagwali in his office
The author, Philip Emeagwali, programmed the Connection Machine to perform the world's fastest computation of 3.1 billion (3,100,000,000) calculations per second in 1989.

Please visit http://emeagwali.com
for more technology essays by Philip Emeagwali


I WAS very much intrigued by the parallel computation article in the February '91 issue. It seems to me that this is one of those technological breakthroughs that can have a profound effect in so many areas of our society that we can make quantum leap forward. Thanks for telling us all about it.

As a former structural engineer and machine designer, I shuddered when I saw your illustration on the cover of the same publication. While I realize that the picture is illustrative only, may I suggest that you consult with a good structural engineer before you build that chicken harness. As the picture stands right now, I would bet on the ox.

Keep up the good work.

John T. Hall
Williamsville, New York
May 1991

THE DIVERSIFICATION of the articles in the February issue was quite impressive, ... particularly "One of the World's Fastest Humans" featuring Philip Emeagwali.

Being an electrical engineer myself, I found Mr. Emeagwali's work and research captivating and motivating. I am sure the detailed description given of the parallel supercomputer will be substantial in the development of my engineering knowledge as well as that of my fellow engineers of the National Society of Black Engineers (NSBE). NSBE is the largest non-profit student-operated organization in the country dedicated to the academic retention, excellence, graduation and cultural consciousness of technical degree students.

Please continue to enrich our nation's scientific contributors.

Sylvia L. Wilson, NSBE President
Alexandria, Virginia
May 1991

Editors Note --- The NSBE presented its Scientist of the Year award to U-M graduate student Philip Emeagwali at its convention in Los Angeles March 29. The award goes to an individual who has made a "significant contribution in a scientific or engineering related field to benefit all mankind."

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WHILE READING the feature article by Philip Emeagwali and the background article on his work, I was impressed both with the excellence of the topic and exposition and the superlative value of the article for use in secondary and some college classrooms, in a range of disciplines from mathematics to natural sciences, geography and social science.

I teach part time in the math department at Saginaw Valley State University, substitute in secondary schools in Saginaw, and have taught adult ed computer literacy classes, as well as being certified in English and social sciences.

Thus, particularly as an Anglo teacher in predominantly non-Anglo classrooms, I have become personally aware of the needs (including role models) expressed in current studies on the achievement of students in science and mathematics. Philip Emeagwali's words and example --- as well as the vitality of his work in computer modeling --- could serve multiple pedagogical purposes in terms of motivation and consciousness-raising --- notoriously difficult accomplishments in the area of mathematics.

Reading guides and discussion topics with a rich potential for critical reasoning would spring naturally from these articles. They also constitute excellent materials for learning about mathematics in expository and narrative, social and historical contexts --- perspectives that are often excluded from the skill-mastery, problem-oriented environment of school mathematics.

I can only state for myself that, if I had a high classroom of my own, I would strive to utilize such materials of which these present an outstanding example, in order to connect themes, encourage students and legitimate interdisciplinary learning experiences.

Andrew Tierman
Saginaw, Michigan
May 1991



MICHIGAN TODAY highlighted parallel processing for computationally intensive computer problems. The reader was informed of the importance of speed and problem size. The reader was not cautioned that the results of immense computer problems, done very fast, may still be wrong.

The most important problems in physics and engineering are highly nonlinear differential equations. These state equations are based on a continuous flow model, but computers require these equations to be modeled as discrete formulations. The latest research shows discrete math models are subject to "spurious solutions." This means despite extensive parallelism to do huge problem sizes, the answers are not correct. Chaos in numerical simulations is probably the most important research issue confronting scientists and mathematicians. Until this problem is recognized and solved complicated computer simulations, including teraflops, are just very expensive video games.

Allen P. Kovacs
Ypsilanti, Michigan
May 1991

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Philip Emeagwali replies: I agree that chaos can occur in special situations and can cause spurious (useless) solutions. Interestingly, massively parallel computers are also used to simulate, study and understand the spurious solutions arising from the phenomenon of chaos.

Most important problems are non-chaotic, however, and scientists and engineers use supercomputers to solve them. For example, the differential equations used by the petroleum industry are non-chaotic. Although supercomputers are very expensive, an oil company can discover (or recover) enough oil in just one simulation to pay for the $6 million to $30 million cost of a supercomputer. This is just one example out of many. Hence, it should not come as a surprise that the governments of the United States, Japan and several European countries have all put supercomputing at the top of their list of critical technologies that will help improve their economic competitiveness. Clearly, governments and industries believe that supercomputer simulations are not merely expensive video games.

I might add that shortly after the first electronic computer was introduced in 1946, one expert proclaimed: "Surely, the whole world will never need more than 10 of these machines."



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