In this full lecture, Professor David Kung, Ph.D. delves into the voting paradoxes that arise in elections at national, state, and even club levels. You’ll also study Kenneth Arrow’s Nobel prize-winning impossibility theorem, and assess the U.S. Electoral College system, which is especially prone to counter intuitive results.

Taught by Professor David Kung, Ph.D.

Imagine an experiment in randomness. Take a coin and flip it 200 times, and each time record whether it’s a heads or a tails, putting down Hs for the heads and Ts for the tails. Now, suppose you ask a person to just write down a random list of 200 Hs and Ts, and you put up both lists on a blackboard, one made by actually flipping a coin, and the other made by a human. Even though they may both look like an ocean of Hs and Ts, there is a way to tell which one is truly random, and which is human generated.

Learn more: Our Random World—Probability Defined

The thing to do is look for strings of long sequences where there are all Hs in a row or all Ts in a row. In the 200 Hs and Ts generated by randomly flipping a coin, you might see at least four or five long sequences of Hs or Ts: six Hs in a row here, five Ts there—a lot of streaks of many things in a row.

How often will a human being write more than four strings of the same letter in a row when they’re trying to be random?

Now consider the list generated by the human being. How often will a human being write more than four strings of the same letter in a row when they’re trying to be random? Well, we sort of resist this, because we don’t think that’s very random. They think you’ve got to sort of alternate—H-T-H-T—and so here in a human generated one you would see very few strings of Hs and Ts in a row.

This is a transcript from the video seriesWhat Are the Chances? Probability Made Clear. Watch it now, on The Great Courses Plus.

As a matter of fact, when you flip a coin 200 times, the probability of having at least one string of six or longer of Hs or Ts is roughly 96 percent—very likely. The probability of having at least one string of five is 99.9 percent—it’s essentially certain. You’d be very unlikely to flip a coin that many times without getting these long strings, and if you actually simulate this on the computer, you’ll see that this plays out, that you just almost always get long strings.

One of the common misconceptions that a lot of people have about randomness is illustrated by the coin flipping experiment. Let’s say that you flip a coin many times, and just randomly it happened that 10 times in a row you got heads. Well, doesn’t it seem like the next time it’s more apt to be a tails? It does to most people. And the answer is, of course, that the coin doesn’t know what it’s just done. To the coin, every flip is a new flip, and it’s just as likely to be a heads as a tails after it’s done 10 heads in a row, as it was to get a heads than a tails if it had done none of them.

Take a coin, and more than a million times, you flip the coin 11 times. Obviously you do this with a computer.

To demonstrate this, you can simulate the following experiment. Take a coin, and more than a million times, you flip the coin 11 times. Obviously you do this with a computer. Computers are great, by the way; they don’t care—a million times, they’ll just go ahead and do it. So you just do it a million times, and what do you get? To make it easy, you actually flip the coin 11 times for 1,024,000 times, because every 1,024 times is the probability of getting 10 heads in a row. In other words, if you do the experiment of flipping the coin 1,024,000 times, and each time you flip it 11 times, you expect that the first 10 will all be heads about 1,000 times.

Learn more: Probability Is in Our Genes

So you run the computer simulation a first time, and the number of times you get 10 heads in the first simulation is 1,008: extremely close to 1,000. What happened to the 11^{th} coin? Well, 521 times it turned out to be a head also, and 487 times it turned out to be a tail. There’s no memory. Approximately half the time heads, half the time tails.

If you do it again, the first 10 might be heads 983 times, and then the 11^{th} flip heads 473 times and tails 510 times. During a third experiment, 1,031 times it came out heads 10 times in a row, and of those, 502 had the next coin be a heads, and 529 a tails. The coin has no memory. After it’s gotten 10 heads in a row, it’s just as likely to be heads the next time as it was the first time you flipped that coin.

There is another counterintuitive aspect of probability, and it’s really interesting to think about what is rare, and how we view rarity in probability. Suppose you got dealt the following hand: the two of spades, the nine of spades, the jack of clubs, the eight of spades, and the five of hearts. Well, it probably doesn’t strike you as an impressive hand, one you write home about, but it is. One out of 2,598,960—that’s the probability of getting that hand.

Now if you were dealt the ace, king, queen, jack, ten of spades—a royal flush in spades—what’s the probability of getting this royal flush in spades? Exactly the same—1 out of 2,598,960—and yet you would write home to your mother about this hand for sure. Your previous hand was just an average hand, and yet in your whole life of playing cards, you know what? You will probably never get that hand again, because its probability is almost zero—1 out of 2,598,960. So this is one of the counterintuitive concepts of probability: that rare events happen all the time, but you may not recognize them as significant.

Learn more: Probability Everywhere

Rare events absolutely happen by chance alone. The most-common rare event that you see mentioned in the newspapers every day is the lottery. The probability of winning the Powerball lottery is approximately 1 out of 146,000,000. This is the big multistate lottery in some states. One out of 146,000,000. That chance is so remote you’d think it would never happen; but it happens regularly. Why? Because a lot of people try. A lot of people buy random numbers and some of them then occasionally win. If you try something that’s rare often enough, then it will actually come to pass.

This concept—that rare things will actually happen if you repeat them enough and you look for them enough—was encapsulated in an observation that was first made by the astronomer Sir Arthur Eddington in 1929, and he was describing some features of the second law of thermodynamics. He wrote the following:

If I let my fingers wander idly over the keys of a typewriter it might happen that my screed made an intelligible sentence. If an army of monkeys were strumming on typewriters they might write all the books in the British Museum. The chance of their doing so is decidedly more favourable than the chance of the molecules returning to one half of the vessel.

However, you can find patterns in random writing, and in fact an enterprising author made a lot of money a few years ago when he wrote *The Bible Code*. What the author of *The Bible Code* did was take the Bible, written in Hebrew, and find patterns of words by skipping a certain number of letters, and in that pattern of skips they would find words written out. One example was “Atomic holocaust Japan 1945.” He said that this was an example of how the Bible showed the future.

The truth is that this is just a matter of probability. If you take all possible sequences of different lengths, you can by randomness alone find surprising things, and just to demonstrate it, people debunking this analysis found patterns in *War and Peace* and so on. This is another challenging part of probability, namely that if you look for rare things but you have a lot of places to look, you’ll tend to find them.

These are some of the challenges of looking at and asking what is random in the world.

In probability, randomness refers to events that occur in no apparent order and are not causally related.

True randomness means that something unfolds purely by chance rather than intentionality, free from human interference.

Cryptography, gambling, statistical sampling, and computer simulation are all purposes for using a random number generator.

Many people claim that they can “outsmart” the lottery or predict winning combinations. People even sell tools to this aim, but these tools are most likely a waste of money. To the best of anyone’s knowledge, the process of choosing the winning lottery numbers operates on the principle of randomness.

Our Random World—Probability Defined

Can You Trust Polling Results?

Mind Expanding Ideas of Metaphysics

With only 10 symbols, we have the machinery to describe new numbers that grow beyond our imagination. Here, we’ll explore the origins of zero and the development of our modern decimal system. With a powerful positional number system in place, humankind was finally equipped with the tools necessary to begin the development of modern mathematics.

Let’s begin with the downside of the ancient additive systems. Most of the systems required the repetition of symbols. For example, the Roman numerals XXIII equal 23, and they’d add up the two Xs (10 each) and then the three Is, and get 23. The Babylonians used dovetails and nails, which they would add up. Although computation with the additive systems was fast using tools such as the abacus, those systems required a very long list of symbols to denote larger and larger numbers, and this was a problem in practice.

Additive systems made it difficult to look at more arithmetically complicated questions and thus slowed the progress of the study of numbers. In order to move to what we call a positional system, they needed a new number. This inspired a philosophical question: How many items do you see in an empty box? Is your answer a number? This is the question about zero. In the Rhind Papyrus from 1650 B.C.E., the scribe Ahmes referred to numbers as “heaps.” This tradition actually continued through the Pythagoreans, who in the 6th century B.C.E. viewed numbers as “a combination or heaping of units.”

This notion of having zero be a quantity didn’t make any sense at all because they were thinking in terms of heaps. This lack of zero caused many challenges.

Even Aristotle defined number as an accumulation or heap. Also, the word “three” derived from the Anglo-Saxon word

The Mayans also had an eye-shaped symbol for zero that they also used only as a placeholder.

Zero first appeared as an empty placeholder rather than a number. The Babylonians had a symbol for zero by 300 B.C.E. It was a placeholder rather than a number because, again, they were thinking heapings, but they needed to distinguish between numbers. The Mayans also had an eye-shaped symbol for zero that they also used only as a placeholder. The evolution of the symbol for zero is actually very difficult to chart. The modern symbol “0” may have arisen from the use of sand tables that were used to calculate things, whereby pebbles would be placed in and moved back and forth for addition or subtraction. When a pebble would be removed, there would be an indentation or a dimple in the sand, which reflects the “0” that we see today. In fact, calculations performed on the sand tables may have actually led to the development of the place-based number systems.

Later, in the 2nd century C.E., Ptolemy used the Greek letter omicron, which looks like an “O,” in fact, to denote “nothing.” So this is the symbol for zero, the “0” that we see—the circle. But I want to make it very clear that Ptolemy did not view this as a number, but merely as the idea of nothing. But you can see, again, that these things were slowly coming together. Zero as a number really occurred in India, most likely.

By the 7th century, the Indian astronomer Bhramagupta offered a treatment of negative numbers and actually understood zero as a number, not just as a placeholder. In fact, he actually studied 0 divided by 0, and 1 divided by 0, and he decided erroneously that 0 divided by 0 equals 0 but just didn’t know what to conclude about 1 divided by 0.

Here again we see a couple of things. First of all, we know today that we can’t divide by 0. If we divide by 0, it does not yield a number, so we leave the realm of number. So we can’t do that—no dividing by 0—and we learn that in school. But we also see a wonderful thing. Bhramagupta, this very important, great mind, was making a mistake, again—something that is to be celebrated rather than to feel embarrassed about. He didn’t get it quite right. That’s okay; his contributions were enormous. So finally, humankind expanded its view of number to actually include and embrace zero.

A few words about this “nothing” number in terms of language: from the 6th to the 8th centuries, in Sanskrit there was “sun-yah,” which meant “empty,” to represent zero as we think of it. By the 9th century, in Arabic there was “sigh-fr.” By 13th-century Latin, there was “zef-ear-e-um.” From 14th-century Italian there was “zef-ear-row.” By 15th-century English, we have “zero.” So we can see the evolution of just that word.

Because of zero’s power in computation, some viewed it as mysterious and nearly magical. As a result, the word zero has the same origins as another word that means “a hidden or mysterious code,” and that word, of course, is “cipher.” We can see that “cipher” actually came from the mysterious qualities that zero possessed in the eyes of our ancestors.

** **

The earliest known example of cryptography was found in Egyptian hieroglyphics around 2500 B.C.E. This may have been more for amusement than actually for secret communication. The earliest simple substitution ciphers, known as monoalphabetic substitution ciphers, may have been used by Hebrew scholars around 550 B.C.E.

In a monoalphabetic substitution cipher, one letter is substituted for another.

In a monoalphabetic substitution cipher, one letter is substituted for another. For example, every occurrence of the letter A might be replaced by the letter W, all Bs might be replaced by the letter M, all Cs might get replaced by the letter R, and so forth, down the alphabet. With this particular scheme that I just made up, the word CAB would be encoded to read “RWM.”

A Caesar cipher is a special case of the monoalphabetic substitution cipher in which each letter is replaced by the letter a fixed number of positions down the alphabet. For example, if we replace A by C—notice that C is two letters away from A—then B would be replaced by two letters away from it, which would be D; C would be replaced by two letters away, which would be E; and so forth, all the way down to Z. When we get to Z, we come back to the beginning of the alphabet; so for Z, we go two letters later, which would be B. The Caesar cipher is named after Julius Caesar, who, in the 1^{st} century B.C.E., used such a cipher with a shift of three to communicate with his generals. Such monoalphabetic encryption schemes are very easy to break.

In the basic ciphers, to decode an encrypted message, one reverses the encryption process. Thus, if people know how to encode a message sent to us, then they also have the power to decode other messages.

Wouldn’t it be great to have a coding scheme such that when people use it to send us messages, the encoding process is easy for them to use while at the same time, we’re absolutely certain that we’re the only ones able to decode the messages?

In this fantasy cipher, we wouldn’t have to trust our friends at all. If they lose the codebook and it gets into the wrong hands, it would not jeopardize the coding scheme. In other words, in our fantasy cipher, knowing how to encode messages would not provide any information as to how to decode it.

If this fantasy were real, then there would be no need to keep the encoding process a guarded secret. Instructions describing how to encode messages could be made public, and only the decoding process would need to be kept secret. In fact, in this fantasy, the encoded messages themselves could be made public as well. Our friends could take out ads in *The New York Times* with an encrypted message directed to us. Everyone would see it, but we’d be the only people who would know how to decode it.

The problem is, if a nemesis of ours sees a secret message sent, why couldn’t he take the encryption process—which we ourselves made public—and just run that process backward to decode the message made just for us? This is a problem. To make this fantasy a reality, we would need to have a secret hidden within the public encryption process. So, even though we make this process public, there’s something secret.

Such amazing ciphers are known as public key codes, because the key for encryption is made public.

We’re now ready to apply number theoretic concepts to show that just such a crypto-fantasy can be a reality. The main question remains: How can the encryption scheme be at once public—everyone knows how to encode messages—and private—only the rightful receiver can decode the messages? Such amazing ciphers are known as public key codes, because the key for encryption is made public.

We’ll make our fantasy a reality by combining the concepts of prime numbers together with modular arithmetic in an extremely clever and elegant way. We’ll begin with a metaphor that captures the idea of this modern encryption scheme. Take a brand-new deck of 52 playing cards. If you were to take it and perform eight perfect shuffles, also known as faro shuffles—you cut the deck exactly in half: 26 and 26—and then shuffle without making a mistake. If you make eight perfect shuffles, then look at the cards, magically, they’ll return to their original order. It’s absolutely amazing, and I urge you to try this for yourself, but if you try it, you have to be able to perform eight perfect shuffles in a row.

Suppose now that we performed just five perfect shuffles: the order of the cards would look thoroughly mixed up, without any semblance of pattern or structure. However, we know a systematic method that would return this mess back to a familiar, less chaotic pattern. We’d perform three more shuffles, bringing the number of shuffles up to eight, and voilà—the cards are transformed from a random mess back to their original order.

If anyone looks at them, it looks jumbled, but we know exactly what to do.

Notice that we could employ this shuffling idea to produce an encryption scheme. Our friend could write her message to us, one letter on each card; so she could say: M, A, T, H, and so forth. Then she would just shuffle a certain pre-agreed amount of times. Let’s say five. So she performs five perfect shuffles, and then she delivers the deck of cards to us. If anyone looks at them, it looks jumbled, but we know exactly what to do. We would shuffle three times and then we would be able to read the message. Of course, if we were to use this encryption scheme, anyone sending us a coded message could decode any other message sent to us as easily as we could. Easy, assuming that we can do perfect shuffles. To have such a scheme truly fulfill our encryption fantasy, we would need to first figure out how to mathematically shuffle our message and then how to make that shuffling process public without allowing others to unshuffle our message.

Here’s the moment where we introduce our number theory. The public feature arises from the fact that factoring extremely large natural numbers is impossible, for all practical purposes, despite the reality that we know that such a factorization is possible in theory. So now we’re going to start to make a distinction between practice and theory.

To see the basic idea behind this public-versus-secret dichotomy, suppose that someone announced the number 6 and also revealed a secret. The secret is that this number is the product of exactly two primes. Can we uncover the secret? Of course:

6 = 2 × 3. There. In some sense, we just broke the code. What if, instead of 6, the announced number that’s the product of two primes was 91? Can we break this code? With some thought, maybe a little bit of arithmetic, we could figure out that 91 is 7 × 13, and thus we’ve broken this code as well, although it took us a little bit longer.

What if the announced number was 2,911? Can we break this code? No, not so easily. But if we use a calculator or a computer, we’d be able to discover that 2,911 equals 41 × 71, and we’ve broken that code, too.

What if the announced number was a 100-digit number? For all practical purposes, even knowing that this number is, in fact, a product of exactly two primes, we would have no way of determining what the two factors are. In fact, even computers have limits to the size of numbers that they can factor. In this way, notice that we can both announce a piece of information publicly—namely, this enormous number—and yet, from a practical point of view, within that public information is a secret that only we, as the receiver, know.

This reality is how individuals will be able to announce an encryption process without revealing the decryption process. To encrypt messages, people need only use the huge natural number. However, to decrypt or decode an encoded message, the receiver will need the prime factors of that huge number, which, for all practical purposes, is a true secret.

Taught by Edward B. Burger, Southwestern University

The focus of this article is on sums. You will learn how to quickly add all the numbers up to 1000 and back down, learn about sums of odd numbers and of even numbers, and even establish Galileo’s results on ratios of sums of numbers—all through the use of a single picture.

We can circle groups of dots in pictures to make sense of division. For example, the division problem 18 ÷ 3 is asking the following question: How many groups of 3 can you find in a picture of 18 dots? There are 6 of them, so 18 ÷ 3 = 6.

We can push this visual picture further and make sense of some complicated division problems. For example, what is 808 ÷ 98? We can see that the answer has to be 8 with a remainder of 24.

You can imagine looking for groups of 100, rather than 98. (The number is 98 is too difficult.) If we visualize this, we see that there will be 8 of these groups, with 8 dots left over.

But each group of 100 is itself off by 2 dots—we wanted groups of 98—so we have an extra 16 dots floating around. That makes for 8 groups of 98 and a remainder of 16 and 8, which is equal to 24 dots. Therefore, 808 ÷ 98 = 8 with a remainder of 24.

When asked to do 34 − 18, we can certainly do the traditional algorithm and get the answer, 16.

But can’t we just see in our minds that the answer has to be 2 + 10 + 4, which is 16? Line up a row of 34 blocks and a row of 18 blocks side by side.

Now we can see that the 2 rows differ by 2 and 10 and 4 blocks, so the difference is 16.

In the same way, 1012 − 797 has to be 3 and 200 and 12—which is 215. From 797 to 800 is 3, from 800 to 1000 is 200, and there is an extra 12, for a total of 215.

This flexibility of thought helps with subtraction in general. For example, consider 1005 − 387.

We have a lot of borrowing to do if we follow the traditional approach: 5 − 7, 0 − 8, and 0 − 3 all need borrows.

But we can make this work simpler.

We are looking for the difference between 1005 dots and 387 dots. Let’s make 1005 friendlier and turn it into 1000. Remove 5 from each and just compute the difference between 1000 and 382 instead. Now we can see the answer: 8 + 10 + 600, or 618.

But if we still want to do the traditional algorithm, then we can remove 1 more dot from each pile and make the problem 999 − 381.

Now we can do the algorithm without any borrows: 9 − 1, 9 − 8, and 9 − 3. This way, we’ve made the problem much easier to do, even if someone insists that we use the algorithm.

Isn’t multiplication really a geometry problem? Isn’t 24 × 13, for example, just asking for the area of a rectangle that is 24 units wide and 13 units high?

Then why not just chop up the rectangle into pieces that are manageable? For example, think of 24 as 20 and 4, and 13 as 10 and 3.

Then we see that 24 × 13 must be the areas of the individual pieces added together: 200 + 40 + 60 + 12 = 312.

In a 5-by-5 grid of squares, there are 25 small 1×1 squares within the grid. But we can count 2×2 squares as well. There are 16 of these in total. If we count the 3×3 squares, there turns out to be 9 of those. And there are 4 of the 4×4 squares. Finally, there is 1 large 5×5 square.

So, there are 25 1×1 squares, 16 2×2 squares, 9 3×3 squares, 4 4×4 squares, and 1 5×5 square. Each count of squares is itself a square number!

Why does counting squares on a square grid give square-number answers? Let’s focus on the lower-left corners of the squares we’re counting. For example, of the 2×2 squares, the following are some possible lower-left corners can be seen in figure 1.14.

Let’s draw all of the possible lower-left corners. Now we see that there is a square array of them, 4 × 4 of them, which is 16. Thus, there are 16 2×2 squares.

Let’s view the 5‑by‑5 grid as an array of dots as in figure 1.16. This is certainly a picture of 25 dots, but can you see in this picture the sum 1 + 2 + 3 + 4 + 5 + 4 + 3 + 2 +1?

Look at the diagonals: 1, 2, 3, 4, 5, 4, 3, 2, 1.

The sum we seek matches the diagonals of the square. There are 25 dots in all, so without doing any arithmetic, we can say that the value of the sum must be 25.

What is the sum of all the numbers 1 + 2 + 3 + … up to 10 and back down again?

This sum must come from the diagonals of a 10‑by‑10 array of dots. Again, without any arithmetic, the value of the sum must be 10 squared (10^{2}): 100.

What is the sum of all the numbers from 1 to 1000 and back down again? It must be 1000 squared, from a 1000-by-1000 array of dots. That’s 1 million.

If you were to compute this on a calculator—1 + 2 + 3 + …—it would take forever. But the answer is available to us quickly via this picture.

1 + 2 + 3 + … + 998 + 999 + 1000 + 999 + 998 + … + 3 + 2 + 1 = 1000 × 1000 = 1,000,000

There is a general formula for the sum of numbers.

1 + 2 + 3 + … + n = n2 + n ÷ 2

The sum of the first *n* numbers, 1 + 2 + 3 all the way up to some number *n*, is (*n*^{2} + *n*) ÷ 2. For example, the sum of the first 5 numbers, 1 + 2 + 3 + 4 + 5, is 5^{2} + 5 = 25 + 5 = 30, and 30 ÷ 2 = 15. And we can check that 1 + 2 + 3 + 4 + 5 is indeed 15.

Where does this formula come from, and why is it true?

Our 5×5 array of dots gave us something akin to this result. We have that 1 + 2 + 3 + 4 + 5 + 4 + 3 + 2 + 1 = 25. Can we get from this answer to just 1 + 2 + 3 + 4 + 5?

If we look at what we have, we see that the sum we want, 1 + 2 + 3 + 4 + 5, is the left half of the equation.

1 + 2 + 3 + … + n = n2 + n ÷ 2

1 + 2 + 3 + 4 + 5 + 4 + 3 + 2 + 1 = 25

Actually, half is not quite right. The right portion of the equation is missing a 5. It’s just the sum 1 + 2 + 3 + 4. We want to see an additional 5, so let’s add a 5 on the left—and to keep things balanced, we need to add a 5 to the right as well.

1 + 2 + 3 + 4 + 5 **+ 5** + 4 + 3 + 2 + 1 = 25 **+ 5**

Now we see 2 copies of what we want. Twice the sum we seek is 25 + 5. So, this means that the sum itself is half of this. 1 + 2 + 3 + 4 + 5 is indeed (5^{2} + 5) ÷ 2. And this matches the general formula. There is nothing special about the number 5. The same ideas show that the sum of the first *n* counting numbers must be half of *n*^{2} + *n*.

2 × (1 + 2 + 3 + 4 + 5) = 25 + 5

1 + 2 + 3 + 4 + 5 = 25 + 5 ÷ 2

Look at the 5-by-5 grid of dots again. Do you see the sum 1 + 3 + 5 + 7 + 9, the sum of the first 5 odd numbers?

We can certainly circle these groups randomly and make them fit.

But such a random picture isn’t enlightening. We want to see a picture that isn’t locked into this particular example of 25 dots. We want a picture that speaks to a higher truth and clearly holds for all possible square arrays. Mathematicians are always on the lookout for this sort of thing, and symmetry is often a pointer to higher truths.

Do you see 1 + 3 + 5 + 7 + 9 in the 5-by-5 array of dots in a way that speaks to a higher truth? Think L shapes.

The sum of the first 5 odd numbers is hidden in the 5-by-5 array as Ls. The sum 1 + 3 + 5 + 7 + 9 must be 5^{2}, or 25.

In the same way, the sum of the first 10 odd numbers, 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19, sit in a 10-by-10 array of dots and therefore must have an answer of 100, the count of dots in that array.

In general, the sum of the first *n* odd numbers must be *n*^{2}.

Galileo lived at the turn of the 16^{th} century and is revered today for his work in science and mathematics, thought to make fractions out of the odd numbers. For example, take the first 5 odd numbers and use their sum for the numerator of a fraction and the sum of the next 5 odd numbers for its denominator. This gives a fraction that simplifies to 1/3.

Do the same for the first 2 odd numbers, followed by the next 2. You get 1/3 again.

Do it again for the first 10 odd numbers, and the next 10. It’s 1/3 again!

Galileo observed that all the fractions made out of the odd numbers this way are equal. They all equal 1/3. These fractions are today called the Galilean ratios. There is a connection between the ratios and the L shapes in squares. Figure 1.23 is purely visual proof of the Galilean ratios.

The first 5 L shapes, the sum of the first 5 odd numbers, makes 1 block of 25 dots. The next 5 L shapes for the next 5 odd numbers makes 3 blocks of 25 dots. So, the first 5 odd numbers make for 1/3 of the next 5 odd numbers.

Are there results about sums of even numbers, too? For example, we have a picture for the sum of the first 5 odd numbers. Can we get from this a picture of the first 5 even numbers, 2 + 4 + 6 + 8 + 10?

Just add a dot to each L shape!

This has turned the 5-by-5 square into a rectangle. The sum of the first 5 even numbers must be the 5×5 we had before plus 5 more, 5^{2} + 5, which is 30.

In general, the sum of the first *n* even numbers must come from the picture of *n*^{2} dots plus an extra *n* dots: *n*^{2} + *n*.

We’re coming full circle, because we have seen the expression *n*^{2} + *n* before.

Take the sum of the first 5 even numbers. It equals 5^{2} + 5.

Now divide everything by 2: 2 ÷ 2, 4 ÷ 2, 6 ÷ 2, 8 ÷ 2, 10 ÷ 2, and (5^{2} + 5) ÷ 2.

And we’re back to the formula 1 + 2 + 3 + 4 + 5 = (5^{2} + 5) ÷ 2.

We have come full circle. We’re back to the general formula for the sum of numbers.

- a. What is the sum of the first 1000 counting numbers?

b. What is the sum of the first 1000 odd numbers? (What is the thousandth odd number?)

c. What is the sum of the first 1000 even numbers?

2 Draw a picture to show that the sum of the first 3 odd numbers must be 1/8 the sum of the next 6 odd numbers.

- a.

b. The one-thousandth odd number is 1999 and the sum of the first 1000 odd numbers: 1 + 3 + 5 + … + 1999, is 1000^{2 }= 1,000,000.

c. The sum of the first 1000 even numbers: 2 + 4 + 6 + … + 2000, is 1000^{2} + 1000 = 1,001,000. (Divide this by 2 and get back to the sum of the first 1000 counting numbers!)

- (See FIGURE 1.27.) In general, we have:

Taught by Professor James Tanton, Ph.D.

Sir Isaac Newton was a mathematician and scientist, and he was the first person who is credited with actually developing calculus. As I say, it really is an incremental development, and many other mathematicians had part of the idea. In fact, Newton’s teacher, by the name of Barrow, actually said “the fundamental theorem of calculus” in his writings but somehow didn’t realize the significance of it and didn’t actually highlight it. But he was Newton’s teacher and presumably Newton learned things from him. Fermat invented some of the early concepts associated with calculus, finding derivatives and finding maxima and minima of equations. And other mathematicians, many mathematicians contributed to both the development of the derivative and the development of the integral.

Newton was, apparently, pathologically averse to controversy. He really didn’t like to be involved in controversy. And because of his aversion to controversy, he was involved in probably the biggest controversy in the history of mathematics about a discovery in mathematics.

Well, Newton was, apparently, pathologically averse to controversy. He really didn’t like to be involved in controversy. And because of his aversion to controversy, he was involved in probably the biggest controversy in the history of mathematics about a discovery in mathematics. So it’s ironic that the person who was so averse to it actually ended up being embroiled in the biggest one in history. It was cause and effect. It wasn’t just an accident. It was his aversion that caused the controversy.

The reason that it caused it is that Newton actually developed the concept of calculus during the middle of the 1660s. And in 1664, ’65, ’66, in that period of time, he asserts that he invented the basic ideas of calculus. And in fact, in 1669, he wrote a paper on it but wouldn’t publish it. In 1671, he wrote another paper on calculus and didn’t publish it; another in 1676 and didn’t publish it. In fact, these papers were actually published. The one he wrote in 1669 was published in 1711. That’s, what, 42 years later? The one he wrote in 1671 was published in 1736; nine years after he was dead. And then the paper he wrote in 1676 was published in 1704. So none of his works on calculus were published until the 18th century. But he did circulate them to friends and acquaintances, so it was known that he actually had this. This wasn’t just hearsay, and he used the techniques of calculus in his scientific work.

But Leibniz, Gottfried Wilhelm Leibniz, independently invented calculus. He invented calculus somewhere in the middle of the 1670s. So he said that he thought of the ideas in about 1674, and then actually published the ideas in 1684, 10 years later. His paper on calculus was called “A New Method for Maxima and Minima, as Well Tangents, Which is not Obstructed by Fractional or Irrational Quantities.” So this was the title for his work. It was six pages and was extremely obscure and was apparently very difficult to understand.

This is a transcript from the video series

Change and Motion: Calculus Made Clear. It’s available for audio and video download here.

And one thing that you have to understand, by the way, is that at that time, what we today think of as absolutely fundamental to even starting to think about calculus, some of those ideas simply didn’t exist at all, for example, the idea of function. We’ve talked about function—you know, at every time we know where we are, at every time there’s a speed—those are examples of function. That concept itself wasn’t actually formulated until the 1690s, after calculus was invented. So people were a little vague on these concepts.

Well Newton and Leibniz didn’t understand it in any more formal way at that time. This was a big problem for all of the people in that century because they were unclear about such concepts as infinite processes. That was a huge stumbling block for them. They were worried about infinitesimal lengths of time. Both Newton and Leibniz would think about infinitesimal lengths of time. How far does something go in an infinitesimal length of time? Well that kind of thinking leads to all sorts of paradoxes, including Zeno’s paradoxes.

I wanted to read you a very famous couplet from a poem by Alexander Pope. These are the kinds of things one would like to have written about oneself. “Nature and Nature’s laws lay hid at night; God said, Let Newton be! and all was light.” So this was Alexander Pope on Newton.

“Taking mathematics from the beginning of the world to the time when Newton lived, what he has done is much the better part.” This was Leibniz talking about Newton.

The controversy between Newton and Leibniz started in the later part of the 1600s. 1699 was a date associated with a start of a tirade, which just went downhill. It was a tremendous controversy. But Leibniz had this to say about Newton. And I’ve served on many committees reading letters of recommendation for mathematicians, you know, for positions, and I can assure you that there are huge numbers of mathematicians who are the best three mathematicians in the world in any given field, but if a sentence like Leibniz’s sentence on Newton appeared in a letter, one would take notice. Leibniz said about Newton, “Taking mathematics from the beginning of the world to the time when Newton lived, what he has done is much the better part.” This was Leibniz talking about Newton.

Even a mathematician wouldn’t know from the actual translation of the sentence exactly what it was that he had done.

But when Newton began to realize that Leibniz had the ideas of calculus, which he began to realize in the 1770s, Newton’s response to make sure that he got credit for calculus was to write a letter to Leibniz in which he encoded a Latin sentence and I will—well, I won’t attempt the Latin, but I’ll attempt just a few words of the Latin. It starts out, “Data aequatione quotcunque” and so on. It’s a short Latin sentence whose translation is, “Having any given equation involving never so many flowing quantities, to find the fluxions, and vice versa.” This was a sentence that encapsulated his, Newton’s, thinking about derivatives. And what he did is he took that sentence and he just took the letters, individual letters, a, c, d, e, and he put them just in order. He said there are six a’s, two c’s, one d, 13 e’s, two f’s. He put them in order and that was what he included in this letter to Leibniz to establish his priority for calculus. And I read you the sentence, which means very little to anybody. Even a mathematician wouldn’t know from the actual translation of the sentence exactly what it was that he had done.

So he tried to establish his priority in that fashion, but then there were accusations that Leibniz had read some of these manuscripts of Newton’s work before he got the ideas. But since Leibniz had published first, people who were siding with Leibniz said that Newton had stolen the ideas from Leibniz. And it was a huge mess, which incidentally led to British mathematics being very retarded for the next century because they didn’t take advantage of the wonderful developments of calculus that were taking place on continental Europe.

Taught by Professor Michael Starbird, The University of Texas at Austin

At the beginning of this game, I give you $100 and a button. Imagine that 100 viewers in 100 rooms across the country are reading this, and that each has been given $100, just like you, and a button, just like you. In a moment, I’m going to ask you to decide whether to push your button. That’s the only decision that you’ll have to make, and in doing so, you’ll be deciding upon a strategy.

In every game, every player has a strategy. If you’re a rational player, you’re going to try to adopt the strategy that will maximize your expected payoff given what you know—or think you know—about the other players in the game.

In every game, every player has a strategy. If you’re a rational player, you’re going to try to adopt the strategy that will maximize your expected payoff given what you know—or think you know—about the other players in the game. But you don’t know enough yet to know whether to push. What does it do? Pushing this button has two effects: one that affects you and one that affects everybody else. Actually, if nobody’s actions affected anyone else, it wouldn’t be a game. Games are interactive.

When you push your button, the first thing it does is to take $2 away from every other player. You push your button, and just like that, everyone else is down to $98. You still have your $100. Sounds rather vicious on your part. But other people may press their buttons, too, you know, and every time they do, you lose $2 along with everybody else. If 60 other people pushed their buttons, you’re going to lose 2 × 60, or $120. Given that I gave you only $100 to start with, you’re going to end up $20 in debt. You’ll have to pay up. Except that there’s a way out of this for you.

I said that pressing your button has two effects, and the second one targets you. If other people press their buttons and cause you damage, pushing your button will cut that damage in half. A moment ago, I said that if 60 people push, you lose $120, but if 60 other people push and you do, too, then you only lose $60. You’re still $40 to the good. You’ve done $2 damage to everybody else, but you’ve saved yourself $60. Are you going to push?

You probably made some assumptions about this game, reasonable ones, as it turns out. You’ve assumed that everybody else’s button works the same way that yours does. It does because this game is symmetric; everyone is in the same boat. Also, you’ve probably assumed that everybody else has the same information that you do, that is, that the structure of the game is common knowledge to everyone. Actually, being common knowledge means quite a bit more than that. It’s not just that everybody knows the rules of the game; it’s that everybody knows that everybody knows the rules of the game, and everybody knows that everybody knows that everybody knows the rules of the game, and … you get the idea. And you’re right: Everyone knows the same information that you do.

I want you to think carefully now and decide what you’re going to do: push or not push.

A hundred people, I don’t know. Some of them are going to push. No matter what the other people do, I’m at least as well off pushing as not pushing.

You’re probably entertaining several different lines of thought right about now. One line of reasoning is this: We all know how the game works; it’s obvious. If nobody pushes the button, everybody gets $100. I might not even be concerned about being a nice person, but I don’t have to be. We can all get $100. I’d be crazy to push. That’s a good argument. A second line of argument, maybe even more compelling, is this: A hundred people, I don’t know. Some of them are going to push. No matter what the other people do, I’m at least as well off pushing as not pushing. If I don’t push, I could end up $100 in debt. If I push, at least I end up breaking even. Heck, I’m a good person, and if I’m thinking about pushing, I can imagine what the other people will do—I have to push in self-defense.

Here’s the third line: I’m not going to push. I’m not pushing because it’s the right thing to do in a moral sense. I could lose up to $100. I could go $100 in debt, but it’s worth it for the sake of my ethics.

Or you may decide: Eh, $100; it’s not that much money. It would be too much fun to just stir things up and see what happens. Push the button. Or you may have a competitive streak, and you know that if you don’t push, everybody who does will end up ahead of you. Maybe you don’t have much of a taste for being a chump. Of these five lines of reasoning, it’s interesting to know which, if any, actually are rational. That’s a question that we’ll be visiting over and over as this course goes on.

Okay, it’s time to decide. I really wish that I could tally the votes that are coming in in real time, but of course, I can’t. What I can and will do is tell you, after you make your choice, the results of similar games that I’ve played with other people. Make your choice and, please, state it out loud; keep yourself honest. Push or don’t push. Three, two, one; done.

With groups of strangers who have no training in game theory, generally somewhere between 30% and 70% of the people push the button. That’s a pretty wide range, but if you take the average, you get 50%.

If you didn’t push, that means that you’re now broke. If you pushed, you still have $50.

This might not make that much of an impact on you; after all, this was just a game. No, that’s the wrong way to say that. What I mean to say is, of course, this was just pretend, but the game is real. It’s not pretend. We’re not talking about child’s play here. We defined a set of possible moves by which players interacted with each other; they had common knowledge of the structure of the game; and they made rational decisions about strategies that led to their best expected payoff. These components—players, strategies, payoffs, and common knowledge—are what make a game a game in the game-theoretic sense. And if you change the context of this game by replacing the players with countries and by changing, pushing the button to being willing to engage in military conflict, then we have something that is much more than just a diversion.

Later in this course, we’ll find out how game theory says this game should be played. But at the moment, what we know is how it is played. The variety of responses that we’ve seen in this game—between 30% and 70% pushing—show that one of two things must be the case: Either the theory of game theory isn’t sufficiently common knowledge that people are comfortable choosing rightly, or maybe this game is an inherently dangerous one. Maybe we need to find a way to keep pushing the button from being so tempting an option, because if 30% to 70% of the people in the nuclear version of this game decide to press the button, we’re all in for a very, very bad time.

In any case, the name “game theory” may be an unfortunate one. A more descriptive name would be “strategic interaction decision making.” Game theory sounds like child’s play, and it’s not.

Taught by Professor Scott P. Stevens, James Madison University

Book VII of *The Republic *discusses the education of the guardians, the rulers and protectors of the perfectly just city. Education is paramount in Plato’s *Republic,* and the guardians are going to receive a very carefully crafted form of education. The principle subject that the guardians must study is that subject which affects their soul. Socrates is even more specific. He says, “The guardians must study a subject that draws the soul from becoming to being.” Becoming is a region, a category of reality. It expresses those kinds of things which come into being and pass out of being, finite, mortal, temporary, transient, fleeting things, the things of the world of our senses. Anything we can touch with our hands or see with our eyes is changing, and anything we can sense will eventually disappear. The other great region of reality is being. The permanent, the changeless, the purely intelligible, that which has no interaction with matter, that which must be thought, but cannot be seen. The guardians need a subject that will turn them around, from becoming to being.

What is this subject? Socrates identifies this subject by describing it as the lowly business of distinguishing the one, the two, and the three—the number. The Greek word for number is arithmos, and it’s the root of our word arithmetic. The guardians that are undergoing this rigorous form of education do not study mathematics for practical purposes. Of course, this is the way mathematics is studied in most universities today. It was the way most people even would have studied mathematics in ancient Greece; we learn a little bit of math, and then we use it. Not the guardians. The guardians study mathematics in order to turn around. They study the nature of numbers themselves. They’re interested not in commerce, they’re not interested in technical applications of mathematics, they’re interested in the pure study of numbers. In modern language, this is described as number theory. After they study arithmetic, the guardians study plain geometry, solid geometry, theoretical astronomy, and harmonics.

What is the nature of mathematics, and why it was so important to Plato? It was important because mathematics is the best preparation for dialectic, the study of the purely formal structure of the whole of reality. It’s really not so obvious what the relationship is between mathematics and the forms. That’s what I want to have us think about. Let me put the point in the following way: think of the kinds of issues in which we have very real disagreement. You and I might disagree about the painting in the museum, and I say it’s beautiful and you say it’s ugly. You and I might disagree about a specific tax policy, you might say it’s unfair to tax rich people more than we tax poor people, and I might say no, it’s perfectly just to do that; we disagree. These are the issues, of course, that human beings have always intensely engaged in conflict over. Now, contrast that realm of disagreement with the realm of mathematics.

They study the nature of numbers themselves. They’re interested not in commerce, they’re not interested in technical applications of mathematics …

None of us would ever disagree that two plus two equals four. We take that to be a simple universal objective truth. We take it to be 100 percent clear that two plus two equals four. Take us back to the museum and imagine the discussion in which we’re disagreeing about the beauty of the painting. Well, that’s a hard discussion to have because it’s not clear what you mean by beauty or what I mean by beauty. Our disagreement about the tax policy, it’s not clear what you think justice is or what I think justice is, and that’s perhaps the reason why our disagreement goes on for such a long time.

I would suggest that the very best way to think of the relationship between mathematics and the forms—and in turn to understand Plato’s deep appreciation of mathematics and the prominent place he gives it in the education of the guardians, because after all, their education seems to be almost exclusively mathematical—is to think of the platonic forms as containing many of the same qualities that mathematics has, but operating in a different sphere. Another word that might be useful here: think of the forms as a projection of mathematical qualities onto issues like beauty and justice. Socrates believes that there is a form of beauty, a form of justice, beauty itself, justice itself.

Think of the forms as a projection of mathematical qualities onto issues like beauty and justice.

They would be the answer to the famous Socratic question, what is beauty, what is justice; they would be forms. They would have precisely the same sorts of qualities that mathematical truth, as we would all agree, already has. These forms would be clear and distinct and universal and objective. This is very hard to imagine. It’s very hard to imagine being in a museum and having an intense disagreement about a painting and thinking it could be resolved in the same way that an arithmetic problem can be resolved. If I ask you to multiply 75 times 152, I know we will all reach the answer if we do the steps properly or if we use a calculator, and we will end up with the same answer, and we don’t disagree. You and I will not come to blows over that mathematical problem. We may very well, however, come to blows about tax policy. We may disagree so vehemently that we can’t find a common ground.

The great platonic hope, the great platonic projection, is to project these kinds of mathematical attributes onto precisely those questions that right now seem to be so far from being resolvable. In Plato’s youth, in the 5th century, he witnessed tremendous turmoil. He witnessed his fellow citizens literally killing each other. This made, without a doubt, an enormous impression on him. Much of his thinking, I think, can be derived from this impulse. How do we resolve conflict? How do we come to harmony among ourselves? The platonic forms may be conceived, in fact, as a hopeful vision in which conflict about those most basic values, the values that people are willing to do die for, values like goodness and justice, can be resolved.

Let me shift focus a little bit and look at mathematics from another perspective. I think Plato would say that mathematics is a wonderful example of community. Here’s what I mean by that apparently strange statement. Mathematics is the great equalizer. There’s only one answer to a problem and it doesn’t matter whether you are a man or a woman or young or old or from Greece or from Persia, from Athens, or from Sparta, the answer is the same. I think this gives, for Plato, a kind of inspiration about learning in general. He can imagine a common group of students who are working together towards the attainment of mathematical truth. They’re bonded precisely by the common objective that they have, and because the objective is mathematical, it’s there to be had by all.

If you’ve ever known a mathematician, it’s likely this person will have told you that mathematics is beautiful. The greatest mathematicians have long felt this.

Here’s a last way to put this point and to make a suggestion. If you’ve ever known a mathematician, it’s likely this person will have told you that mathematics is beautiful. The greatest mathematicians have long felt this. They study mathematics not because it’s practical, although it is, not because it’s useful, but because the sheer beauty of formal structure, the sheer beauty of literally perfection, shines through in mathematical truth. To take a ridiculously simple example, the one I’ve cited, two plus two equals four is a perfectly true sentence. That has, as trivial as it is, a beauty to it. I think this notion of beauty has long inspired mathematically minded thinkers. I think it inspired Plato. As a result, in Plato’s Academy, mathematics seems to have been a prerequisite. One had to study geometry in order to enter Plato’s Academy.

The culmination of the education of the guardians is called dialectic. Dialectic is the study of forms and is inspired by the “what is it” question that Socrates is famous for asking. The first and, perhaps, the most interesting point that Socrates makes about dialectic is that it’s potentially very dangerous, and it’s especially dangerous for young people. Reading Book VII, you’ll see that the curriculum of the guardians is very rigidly regimented. Guardians, until they’re about 20 years old, do very little else but engage in physical exercise and training, called gymnastic. Between 20 and 30, these future rulers only study mathematics, but when they’re 30 and up to about the age of 35, they start to get their first introduction to dialectic. To complete the sequence, between the ages of 35 and 50, the guardians will be required to go down into the cave where they will rule the city. Then, at the age of 50, they return to the study of dialectic, and only at that very late stage of their education will they finally get a peek at the Idea of the Good, the pinnacle of their study.

The first and, perhaps, the most interesting point that Socrates makes about dialectic is that it’s potentially very dangerous, and it’s especially dangerous for young people.

Now, the dialectic is potentially quite dangerous for young people. I want to elaborate a little bit on that. Imagine that there is a young Athenian soldier, and his leaders tell him that he must go to war, and his leaders try to inspire him by telling him that this will be a just war. Perhaps, this was a soldier in the year 431 B.C.E. when the Peloponnesian War broke out. This soldier, in my hypothetical story, is on his way to serve in the army when he bumps into Socrates. What does Socrates do? He says, where you going, and the kid says, I’m going to war. Why are you going to war? Because the cause is just and I’m willing, even, to lose my life if my city requires me to do so. Socrates would then hit him with his question, what is justice?

Well, if you study *The Republic, *you know just how hard it is to answer this question. It’s very difficult to imagine that a 19-year-old boy would be able to make any real progress in answering this question. So he leaves the conversation with Socrates puzzled, confused, in a state of wonder, of bewilderment. What is justice? I thought I knew, I thought it was what my leaders told me was just, but this man Socrates has disrupted me. This man Socrates has taught me that I do not know what I thought I knew.

Well, what might happen? Maybe this boy will become a deserter, maybe he won’t serve in the army or maybe even worse. Maybe this boy will say I don’t know what justice is, maybe I’ll go over to the Spartan side. Maybe they’re just; maybe these Athenians who’ve been ordering me around aren’t telling me the truth. Socrates has taught me that I don’t know what justice is; the door is therefore open to me to do what ever it is I might want to do.

Now this story corresponds to an actual event with an actual person. His name was Alcibiades, a very famous Athenian. He was famous for two things: he was an associate of Socrates and he was a traitor to Athens in the Peloponnesian War who went over to the Spartan side. This, by the way, is no doubt one of the real reasons Socrates was executed in 399 B.C.E. He was thought to be associated with the traitor Alcibiades. The point is that dialectical inquiry, the inquiry that begins with the question “what is it” and leads to an inquiry into the forms, is potentially subversive of the city. This is why in the educational program outlined in Book VII, Socrates does not allow young people to even be exposed to dialectic until they’re at least 30 years old.