Appearance
1.1 — What a Number Actually Is
A shepherd twenty thousand years ago had a real problem. In the morning he let the sheep out; in the evening he had to know whether they had all come back. He could not count, because counting had not been invented, and he certainly could not write. So he did something clever: he kept a pile of pebbles, and as each sheep walked out he moved one pebble from one heap to another. In the evening he moved them back, one pebble per returning sheep. If the heap emptied exactly, every sheep was home. If a pebble was left over, one sheep was missing.
Notice what he has done. He never needed to know how many sheep he had. He only needed to know that the sheep and the pebbles could be matched up, one to one, with nothing left over on either side. That matching is the whole idea, and it is still the definition mathematicians use. Two collections have the same number of things exactly when you can pair them off perfectly.
The word for that pairing is a one-to-one correspondence. It sounds like a technicality, but it is the foundation of everything in this part, and in Chapter 1.7 the very same idea will tell us that some infinities are larger than others.
1. The leap from "three sheep" to "three"
The shepherd's pebbles are a model of the flock, not a count of it. The leap — and it took our species a very long time — is noticing that three sheep, three pebbles, three fingers and three days all share something, and that this shared something can be given a name and thought about on its own.
That shared something is the number three. It is not a thing you can point at. Nobody has ever tripped over a three. It is a property that collections can have, extracted from every particular collection that has it.
This is the first act of abstraction in the history of thought, and every other abstraction in mathematics is built the same way: notice that many different situations behave identically, strip away everything specific to each situation, and give the leftover pattern a name.
Tally marks are the oldest written numbers we have, and they are still the ones you scratch on a notepad when you are counting cars going past. They have one enormous virtue: to add one, you make one more mark. They have one fatal flaw, which we will get to.
2. Numbers and numerals are not the same thing
This distinction trips people up for their whole lives, so let us nail it down immediately.
- A number is the idea — the "how many".
- A numeral is the written mark that stands for it.
The number seven is the same number whether you write it 7, VII, seven, ৭ (Bengali), ٧ (Arabic-Indic), or |||||||. Those are six numerals for one number. Changing the numeral changes nothing about the quantity, exactly as calling a dog chien does not change the dog.
Why belabour this? Because almost every important advance in arithmetic for three thousand years was an advance in numerals, not in numbers. The Romans and we have exactly the same numbers. We simply have a notation that lets us calculate with them, and they did not.
3. Why Roman numerals were a disaster
Take a moment and multiply, on paper, in Roman numerals:
\text{XLVII} \times \text{XXIII}
You cannot. Not because you are bad at it, but because there is no procedure. There is no "carry the one", no column to line up, no digit-by-digit method, because the symbols do not encode position. X means ten wherever it stands. XL means forty only because of an extra subtraction rule bolted on top. Roman arithmetic was done on a counting board — a physical abacus — and the numerals were used only to write the answer down afterwards. The notation was for recording, not for computing.
This is not a small inconvenience. It is why Roman engineers, who could build an aqueduct that still stands, could not do the arithmetic a modern ten-year-old does in their head.
The flaw in tally marks is the same flaw, taken to the extreme: to write a thousand you make a thousand marks. The size of the written number grows in proportion to the number itself. That is unusable, and fixing it is the single greatest invention in the history of arithmetic.
4. Place value: the invention that made arithmetic possible
Here is the fix. Instead of giving each quantity its own symbol, give a small fixed set of symbols and make the position of a symbol carry a multiplier.
In our system the symbols are 0 1 2 3 4 5 6 7 8 9 — ten of them — and each step to the left multiplies by ten:
3407 = 3 \times 1000 \;+\; 4 \times 100 \;+\; 0 \times 10 \;+\; 7 \times 1
Written with powers, which we will develop properly in Chapter 1.4:
3407 = 3 \times 10^3 + 4 \times 10^2 + 0 \times 10^1 + 7 \times 10^0
Read that aloud as "three times ten cubed, plus four times ten squared, plus zero times ten, plus seven times one". The little raised number is how many tens are multiplied together, and 10^0 is one — Chapter 1.4 explains exactly why.
Two consequences follow, and they are the entire payoff.
The written number grows like the logarithm of the number. A million needs seven digits, not a million marks. Every time the quantity gets ten times bigger, the numeral gets one character longer. That is the difference between writable and unwritable.
Arithmetic becomes a mechanical procedure on digits. Column addition, long multiplication, long division — the algorithms you were drilled on in school — exist only because position carries meaning. You add the units column, and if it overflows past nine you carry one into the tens column, because ten units is one ten. The procedure is not arbitrary; it is place value being obeyed.
Chapter 1.3 in Volume I showed the same trick with only two symbols instead of ten, which is how every computer on Earth stores numbers. Base ten is not special. What is special is position.
5. Zero: the hardest idea in this chapter
Place value cannot work without a symbol for "nothing in this column". Otherwise how do you distinguish 37 from 307? The Babylonians, who had place value in base sixty as early as 1800 BCE, wrote these ambiguously for over a thousand years and relied on context to tell them apart — the same way we can usually tell whether "a coke" costs one rupee or a hundred.
The full solution came from India. In Brahmagupta's Brāhmasphuṭasiddhānta of 628 CE, zero is treated for the first time not as a placeholder but as a number in its own right, with rules for calculating with it: a number plus zero is unchanged, a number minus itself is zero, and zero times anything is zero. He also tried, honourably and wrongly, to define division by zero — a question we will settle in Chapter 1.2.
The route into Europe is worth knowing, because it explains the name. Al-Khwārizmī, working in Baghdad around 820 CE, wrote a book on calculating with the Indian numerals. It reached Latin Europe as Algoritmi de numero Indorum — "Al-Khwārizmī on the Indian numbers" — and his Latinised name became the word algorithm. Every time you say that word you are naming a ninth-century Persian mathematician who was explaining how to do long division.
Europe resisted for centuries. Florence banned the new numerals in 1299, partly because a 0 can be doctored into a 6 or a 9 far more easily than a Roman V can be altered, and partly out of simple institutional inertia — the guild of abacus-reckoners had a trade to protect. It took Fibonacci's Liber Abaci (1202) and then the printing press to finish the job.
6. Two different jobs numbers do
Notice that we have quietly been using numbers for two unrelated purposes.
Cardinal numbers answer "how many". Five sheep. This is the shepherd's use, and it is about the size of a collection.
Ordinal numbers answer "which position". The fifth sheep in the line. This is about order, not size.
For finite collections the two match up so neatly that it is easy to think they are the same idea. They are not. If there are five sheep, the last one is the fifth — fine. But when we reach infinite collections in Chapter 1.7, cardinal and ordinal come apart spectacularly, and keeping them distinct now will save confusion later.
There is a third use, which is not really a number at all: the nominal use, where a numeral is just a name. Your phone number, a bus route, the number on a footballer's shirt. Bus 42 plus bus 7 is not bus 49. The numeral has been borrowed as a label, and none of arithmetic applies to it.
7. The natural numbers, and what "closed" means
The counting numbers 1, 2, 3, 4, \ldots are called the natural numbers, written \mathbb{N} — a capital N in a doubled-up "blackboard" style, which is just the traditional way of writing the names of number collections. Whether 0 belongs to \mathbb{N} is a genuine disagreement between mathematicians, not a deep question; different books make different choices and say so. In this book, \mathbb{N} = \{0, 1, 2, 3, \ldots\}, and when we mean the ones from one upward we will say so.
The curly braces \{\;\} mean "the collection consisting of", and the three dots mean "carry on in the obvious way, forever". A collection like this is called a set, and Chapter 8.1 develops sets properly.
Now, an idea that shapes the rest of this part. Ask: if I take two natural numbers and add them, is the answer always a natural number?
Yes. 3 + 5 = 8. There is no pair of counting numbers whose sum escapes the counting numbers. We say \mathbb{N} is closed under addition. Multiplication too: 3 \times 5 = 15, always still a natural number.
Now subtraction. 5 - 3 = 2, fine. But 3 - 5 is not a natural number. There is no counting number that answers it, because you cannot have negative three sheep.
That failure of closure is the engine of this whole part. Every extension of the number system in the next six chapters happens for exactly this reason: an operation we want to do produces an answer that does not exist yet, so we invent numbers for it to land on.
- Subtraction escapes \mathbb{N} → we invent negative numbers, getting the integers \mathbb{Z} (Chapter 1.2).
- Division escapes \mathbb{Z} → we invent fractions, getting the rationals \mathbb{Q} (Chapter 1.3).
- Taking roots and limits escapes \mathbb{Q} → we invent the irrationals, getting the reals \mathbb{R} (Chapter 1.7).
- The square root of a negative escapes \mathbb{R} → we invent i, getting the complex numbers \mathbb{C} (Chapter 2.5).
Four inventions, one reason each time. Nobody sat down and decided to make mathematics complicated. Each new kind of number is the smallest repair that closes a hole someone kept falling into.
The letters, since they look arbitrary: \mathbb{Z} is from German Zahlen, "numbers". \mathbb{Q} is from quotient, since a fraction is one number divided by another. \mathbb{R} is for real and \mathbb{C} for complex.
8. The four operations, said properly
Addition, subtraction, multiplication and division are so familiar that their actual structure gets missed. Here is what each one really is.
Addition is combining. Put the two collections together and count the result. Its defining properties:
- Commutative: a + b = b + a. Order does not matter.
- Associative: (a + b) + c = a + (b + c). Grouping does not matter, so you can write a+b+c without brackets and not be ambiguous.
- Identity: a + 0 = a. Zero is the thing that changes nothing.
Multiplication is repeated addition — but only at first. 3 \times 4 is four things taken three times. That picture works perfectly for counting numbers and then breaks the moment you ask what \frac{1}{2} \times \frac{1}{3} means, since you cannot add something half a time. The better picture, which survives every extension, is scaling: 3 \times 4 is the number four stretched to three times its size, and \frac{1}{2} \times 4 is four shrunk to half its size. Hold onto scaling; it is the picture that will still work in Chapter 4.2 when we multiply matrices.
Multiplication is also commutative and associative, its identity is 1, and it connects to addition through the property that makes all of algebra work:
a \times (b + c) = a \times b + a \times c
This is the distributive law. It is why 7 \times 12 can be computed as 7 \times 10 + 7 \times 2 = 70 + 14 = 84, which is what you actually do in your head. Chapter 2.1 shows that the entire technique of expanding brackets is this one law applied repeatedly.
Subtraction is the undoing of addition. "What must I add to 3 to get 5?" is the real meaning of 5 - 3. It is not commutative (5-3 \neq 3-5) and not associative, which is why 8 - 3 - 2 needs a convention (left to right) to be unambiguous.
Division is the undoing of multiplication. "What must I multiply 3 by to get 12?" is the real meaning of 12 \div 3. Also not commutative, also not associative.
Seeing subtraction and division as inverse questions rather than as separate operations is the single most useful reframing in elementary arithmetic, because it immediately explains things that otherwise have to be memorised — including why dividing by zero is not merely forbidden but meaningless, which we settle in the next chapter.
9. Why the order of operations exists
2 + 3 \times 4 is 14, not 20. Every schoolchild is told this and almost none are told why.
It is not a law of nature. It is a convention about notation, chosen for a reason. Multiplication is a shorthand for repeated addition, so an expression like 2 + 3 \times 4 naturally means "two, plus three lots of four". Giving multiplication the tighter grip lets us write polynomials — the expressions of Chapter 2.2 — without drowning in brackets. Compare:
3x^2 + 2x + 1 \qquad\text{versus}\qquad ((3 \times (x^2)) + (2 \times x)) + 1
The convention buys readability, nothing more. A different convention would need different brackets and would describe the same mathematics. This matters because programming languages make exactly the same kind of choice, and occasionally a different one — which is why Volume I, 3.1 talks about operator precedence when it builds a parser.
10. Where this shows up in your life
Every price tag and every clock. Place value is so completely absorbed that reading 2,499 as two thousand four hundred and ninety-nine feels like perception rather than calculation. It is calculation; you were trained into it for years.
Every computer. Volume I, 1.3 is this chapter in base two. The digits are 0 and 1, each step left multiplies by two instead of ten, and everything else — carrying, column arithmetic, the growth of numeral length with the logarithm of the number — is identical. A computer is a place-value machine.
Barcodes and card numbers. The last digit of your bank card is a check digit, computed from the others so that a mistyped digit produces an invalid number. It works by arithmetic on the positions, which is only possible because position carries meaning. Chapter 1.6 shows the actual calculation.
The word "algorithm" in every technology conversation you have. It came from a man explaining place-value arithmetic, and it originally meant nothing more than "the step-by-step method for calculating with these Indian digits".
Numbers now exist, and they can be written and combined. But the system has a hole in it: subtraction runs off the end. The next chapter fills it, and in doing so answers the question every schoolchild asks and almost no teacher answers — why two negatives make a positive.