Professor Stewart's Incredible Numbers

Small Numbers

The most familiar numbers of all are the whole numbers from 1 to 10.

Each is an individual, with unusual features that single it out as something special.

Appreciating these special features makes numbers feel familiar, friendly, and interesting in their own right.

Soon, you’ll be a mathematician.

1

The Indivisible Unit

The smallest positive whole number is 1. It’s the indivisible unit of arithmetic: the only positive number that cannot be obtained by adding two smaller positive numbers together.

Basis of the Number Concept

The number 1 is where we start counting. Given any number, we form the next number by adding 1:

2 = 1 + 1

3 = (1 + 1) + 1

4 = ((1 + 1) + 1) + 1

and so on. The brackets tell us which operations to perform first. They are generally omitted, because it turns out that in this case the order doesn’t matter, but it’s best to be careful early on.

From these definitions and the basic laws of algebra, which in a formal logical development must be stated explicitly, we can even prove the famous theorem ‘2 + 2 = 4’. The proof fits on one line:

       2 + 2 = (1 + 1) + (1 + 1) = ((1 + 1) + 1) + 1 = 4

In the twentieth century, when some mathematicians were trying to put the foundations of mathematics on a firm logical basis, they used the same idea, but for technical reasons they started from 0 [see 0].

The number 1 expresses an important mathematical idea: that of uniqueness. A mathematical object with a particular property is unique if only one object has that property. For example, 2 is the unique even prime number. Uniqueness is important because it lets us prove that some slightly mysterious mathematical object is actually one we know about already. For example, if we can prove that some unknown positive number n is both even and prime, then n must be equal to 2. For a more complicated example, the dodecahedron is the unique regular solid with pentagonal faces [see 5]. So if in some piece of mathematics we encounter a regular solid with pentagonal faces, we know at once, without doing any further work, that it must be a dodecahedron. All the other properties of a dodecahedron then come free of charge.

One Times Table

No one ever complained about having to learn the one times table. ‘One times one is one, one times two is two, one times three is three . . . ’ If any number is multiplied by 1, or divided by 1, it remains unchanged:

       n × 1 = n    n ÷ 1 = n

It is the only number that behaves in this manner.

In consequence, 1 is equal to its square, cube, and all higher powers:

1² = 1 × 1 = 1

1³ = 1 × 1 × 1 = 1

1⁴ = 1 × 1 × 1 × 1 = 1

and so on. The only other number with this property is 0.

For this reason, the number 1 is generally omitted in algebra when it appears as a coefficient in a formula. For example, instead of 1x² + 3x + 4 we write just x² + 3x + 4. The only other number treated in this manner is 0, where something even more drastic happens: instead of 0x² + 3x + 4 we write just 3x + 4, and leave the 0x² term out altogether.

Is 1 Prime?

It used to be, but it isn’t any more. The number hasn’t changed, but the definition of ‘prime’ has.

Some numbers can be obtained by multiplying two other numbers together: for example, 6 = 2 × 3 and 25 = 5 × 5. This type of number is said to be composite. Other numbers cannot be obtained in this way: these are called prime.

According to this definition, 1 is prime, and until 150 years ago that was the standard convention. But then it turned out to be more convenient to consider 1 as an exceptional case. Nowadays it is considered to be neither prime nor composite, but a unit. I’ll explain why shortly, but we need a few other ideas first.

The sequence of primes begins

       2  3  5  7  11  13  17  19  23  29  31  37  41  47

and it appears to be highly irregular, barring a few simple patterns. All primes except 2 are odd, because any even number is divisible by 2. Only 5 can end in 5, and none can end in 0, because all such numbers are divisible by 5.

Every whole number greater than 1 can be expressed as a product of prime numbers. This process is called factorisation, and the primes concerned are called the prime factors of the number. Moreover, this can be done in only one way, aside from changing the order in which the primes occur. For example,

       60 = 2 × 2 × 3 × 5 = 2 × 3 × 2 × 5 = 5 × 3 × 2 × 2

and so on, but the only way to get 60 is to rearrange the first list of primes. For example, there is no factorisation into primes that looks like 60 = 7 × something.

This property is called ‘uniqueness of prime factorisation’. It probably seems obvious, but unless you’ve taken a mathematics degree I’d be surprised if anyone has ever shown you how to prove it. Euclid put a proof in his Elements, and he must have realised that it’s neither obvious nor easy because he takes his time working up to it. For some more general number-like systems, it’s not even true. But it is true for ordinary arithmetic, and it’s a very effective weapon in the mathematical armoury.

The factorisations of the numbers from 2 to 31 are:

The main reason for treating 1 as an exceptional case is that if we count 1 as prime, then prime factorisation is not unique. For example 6 = 2 × 3 = 1 × 2 × 3 = 1 × 1 × 2 × 3 and so on. One apparently awkward consequence of this convention is that 1 has no prime factors. However, it is still a product of primes, in a rather strange way: 1 is the product of an ‘empty set’ of primes. That is, if you multiply together no prime numbers at all, you get 1. It probably sounds crazy, but there are sensible reasons for this convention. Similarly, if you multiply one prime number ‘together’ you just get that prime.

2

Odd and Even

Even numbers are divisible by 2; odd numbers aren’t. So 2 is the only even prime number. It’s a sum of two squares: 2 = 1² + 1². The other primes with this property are precisely those that leave remainder 1 when divided by 4. The numbers that are a sum of two squares can be characterised in terms of their prime factors.

Binary arithmetic, used in computers, is based on powers of 2 instead of 10. Quadratic equations, involving the second power of the unknown number, can be solved using square roots.

The distinction between odd and even extends to permutations—ways to rearrange a set of objects. Half the permutations are even and the other half are odd. I’ll show you a neat application: a simple proof that a famous puzzle can’t be solved.

Parity (Even/Odd)

One of the most important distinctions in the whole of mathematics is that between even and odd numbers.

Let’s start with whole numbers 0, 1, 2, 3, . . .  . Among these, the even numbers are

       0  2  4  6  8  10  12  14  16  18  20  . . .

and the odd numbers are

       1  3  5  7  9  11  13  15  17  19  21  . . .

In general, any whole number that is a multiple of 2 is even, and any whole number that is not a multiple of 2 is odd. Contrary to what some teachers seem to believe, 0 is even, because it is a multiple of 2, namely 0 × 2.

Fig 8 Even and odd numbers.

Odd numbers leave remainder 1 when they are divided by 2. (The remainder is nonzero and less than 2, which only leaves 1 as a possibility.) So algebraically, even numbers are of the form 2n where n is a whole number, and odd numbers are of the form 2n + 1. (Again, taking n = 0 shows that 0 is even.) To extend the concepts ‘even’ and ‘odd’ to negative numbers, we allow n to be negative. Now −2, −4, −6, and so on, are even, and −1, −3, −5, and so on, are odd.

Even and odd numbers alternate along the number line.

Fig 9 Even and odd numbers alternate along the number line.

A pleasant feature of even and odd numbers is that they obey simple arithmetical rules:

no matter what the actual numbers are. So if someone claims that 13 × 17 = 222, you know they’re wrong without doing any sums. Odd × odd = odd, but 222 is even.

Smallest and Only Even Prime Number

The list of prime numbers starts with 2, so 2 is the smallest prime number. It is also the only even prime number, because by definition all even numbers are divisible by 2. If the number concerned is 4 or larger, this expresses it as the product of two smaller numbers, so it’s composite. These properties, simple and obvious as they may be, give 2 a unique status among all numbers.

Two Squares Theorem

On Christmas day 1640 the brilliant amateur mathematician Pierre de Fermat wrote to the monk Marin Mersenne, and asked an intriguing question. Which numbers can be written as a sum of two perfect squares?

The square of a number is what you get when you multiply it by itself. So the square of 1 is 1 × 1 = 1, the square of 2 is 2 × 2 = 4, the square of 3 is 3 × 3 = 9, and so on. The symbol for the square of a number n is n². So 0² = 0, 1² = 1, 2² = 4, 3² = 9, and so on.

The squares of the numbers 0 –10 are

       0  1  4  9  16  25  36  49  64  81  100

So 4 is the first perfect square after the less interesting 0 and 1.

The word ‘square’ is used because these numbers arise when you fit dots together into squares:

Fig 10 Squares.

When we add squares in pairs we can obviously get the squares themselves: just add 0 to a square. But we also get new numbers like

which are not square. Many numbers still don’t occur, though: for example, 3, 6, 7, 11.

Here’s a table showing all of the numbers from 0 to 100 that are sums of two squares. (To obtain the number in any non-boldface cell, add the boldface number at the top of its column to the boldface number at the left of its row. For example 25 + 4 = 29. Sums greater than 100 are omitted here.)

Table 2

At first sight it’s hard to find any pattern, but there is one, and Fermat spotted it. The trick is to write down the prime factors of the numbers in the table. Leaving out 0 and 1, which are exceptions, we get:

Here I’ve underlined the numbers that are prime, because those are the key to the problem.

Some primes are missing, namely: 3, 7, 11, 19, 23, 31, 43, 47, 59, 67, 71, 79, and 83. Can you emulate Fermat and spot their common feature?

Each of these primes is 1 less than a multiple of 4. For instance 23 = 24 − 1, and 24 = 6 × 4. The prime 2 occurs in my list; again, this is exceptional in some ways. All of the odd primes in my table are 1 greater than a multiple of 4. For instance 29 = 28 + 1, and 28 = 7 × 4. Moreover, the first few primes of this form all occur in my list, and if you extend it, none seems to be missing.

Every odd number is either 1 less than a multiple of 4 or 1 greater than a multiple of 4; that is, it is of the form 4k − 1 or 4k + 1 for a whole number k. The only even prime is 2. So every prime must be one of the following:

    equal to 2

    of the form 4k + 1

    of the form 4k − 1.

The missing primes in my list of sums of two squares are precisely the primes of the form 4k − 1.

These primes can occur as factors of numbers in the list. Look at 3, for instance, which is a factor of 9, 18, 36, 45, 72, 81, and 90. However, all of these numbers are actually multiples of 9, that is, of 3².

If you look at longer lists from the same point of view, a simple pattern emerges. In his letter, Fermat claimed to have proved that the nonzero numbers that are sums of two squares are precisely those for which every prime factor of the form 4k − 1 occurs to an even power. The most difficult part was to prove that every prime of the form 4k + 1 is the sum of two squares. Albert Girard had conjectured as much in 1632, but gave no proof.

The table includes some examples, but let’s check out Fermat’s contention with something a bit more ambitious. The number 4001 is clearly of the form 4k + 1; just take k to be 1000. It’s also prime. By Fermat’s theorem, it must be a sum of two squares. Which?

In the absence of a more clever method, we can try subtracting 1², 2², 3², and so on in turn, and seeing whether we get a square. The calculation starts like this:

4001 − 1² = 4000: not a square

4001 − 2² = 3997: not a square

4001 − 3² = 3992: not a square

. . .

and eventually hits

       4001 − 40² = 2401: the square of 49

So

       4001 = 40² + 49²

and Fermat is vindicated for this example.

This is essentially the only solution, aside from 49² + 40². Getting a square by subtracting a square from 4001 is a rare event; it almost looks like pure luck. Fermat explained why it’s not. He also knew that when 4k + 1 is prime, there’s only one way to split it into two squares.

There’s no simple, practical way to find the right numbers in general. Gauss did provide a formula, but it’s not terribly practical. So the proof has to show that the required squares exist, without providing a quick way to find them. That’s a bit technical, and it needs a lot of preparation, so I won’t attempt to explain the proof here. One of the charms of mathematics is that simple, true statements don’t always have simple proofs.

Binary System

Our traditional number system is called ‘decimal’, because it uses 10 as its number base, and in Latin 10 is decem. So there are ten digits 0–9 and the value of a digit multiplies by 10 at every step from right to left. So 10 means ten, 100 a hundred, 1000 a thousand, and so on [see 10].

Similar notational systems for numbers can be set up using any number as base. The most important of these alternative notational systems, called binary, uses base 2. Now there are only two digits, 0 and 1, and the value of a digit doubles at every step from right to left. In binary, 10 means 2 (in our usual decimal notation), 100 means 4, 1000 means 8, 10000 means 16, and so on.

To get numbers that are not powers of 2, we add distinct powers of 2 together. For instance, 23 in decimal is equal to

       16 + 4 + 2 + 1

which uses one 16, no 8, one 4 one 2, and one 1. So in binary notation this becomes

       10111

The first few binary numerals and their decimal equivalents are:

Table 3

To ‘decode’ the symbols for the number 20, for instance, we write them out in powers of 2:

The powers of 2 for which the symbol 1 occurs are 16 and 4. Add them: the result is 20.

History

Some time between 500 BC and 100 BC the Indian scholar Pingala wrote a book called on rhythms in poetry, and he listed different combinations of long and short syllables. He classified such combinations using a table, which in modern form uses 0 for a short syllable and 1 for a long one. For example,

00 = short-short

01 = short-long

10 = long-short

11 = long-long

The patterns here are those of binary notation, but Pingala did not perform arithmetic with his symbols.

The ancient Chinese book of divination, the I Ching (Yì Jīng), used 64 sets of six horizontal lines, either complete (yang) or broken into two (yin), as an oracle. These sets are known as hexagrams. Each hexagram consists of two trigrams stacked above each other. Originally the hexagram was used to foretell the future, by throwing yarrow stalks on the ground and applying rules to determine which hexagram you should look at. Later three coins were used instead.

If we use 1 to represent a complete line (yang) and 0 for a broken one (yin), each hexagram corresponds to a six-digit binary number. For example the hexagram in the figure is 010011. According to the method of divination, this is Hexagram 60 ( = jié), and indicates ‘articulating’, ‘limitation’, or ‘moderation’. A typical interpretation (don’t ask me to explain it because I have no idea) begins:

Limitation—Above: k’an the abysmal, water. Below: tui the joyous, lake.

Judgement—Limitation, success. Galling limitation must not be persevered in.

Fig 11 Left: A hexagram. Right: The eight trigrams.

Image—Water over lake: the image of limitation. Thus the superior man creates number and measure, and examines the nature of virtue and correct conduct.

Again, although the patterns of binary are present in the I Ching, the arithmetic isn’t. More of the mathematical structure of binary symbols shows up in the writings of Thomas Harriot (1560–1621), who left thousands of pages of unpublished manuscripts. One of them contains a list that begins

and continues up to

30 = 16 + 8 + 4 + 2

31 = 16 + 8 + 4 + 2 + 1

It is clear that Harriot understood the basic principle of binary notation. However, the context for this list is a long series of tables enumerating how various objects can be combined in different ways, not arithmetic.

In 1605 Francis Bacon explained how to encode letters of the alphabet as sequences of binary digits, getting very close to using them as numbers. Binary finally arrived as an arithmetical notation in 1697, when Leibniz wrote to Duke Rudolph of Brunswick proposing a ‘memorial coin or medallion’.

Fig 12 Leibniz’s binary medallion.

The design tabulates the binary representations of the numbers 0–15, with the inscription omnibus ex nihilo ducendis sufficit unum (for everything to arise from nothing, one suffices). Mathematically, Leibniz is pointing out that if you have the symbol 0 (nothing) and throw in 1 (one) then you can obtain any number (everything). But he was also making a symbolic religious statement: a single God can create everything from nothing.

The medal was never produced, but its design alone was a significant step. By 1703 Leibniz was developing the mathematics of binary, and he published a paper ‘Explication de l’arithmétique binaire’ (Explanation of binary arithmetic) in