Introduction
When we learn about numbers, we start with the natural numbers (1, 2, 3, …) and before long we meet negative numbers, fractions, irrational numbers, and even imaginary numbers. And a question can creep in.
"Why do new numbers keep getting invented? Weren't the naturals enough? Who decided 'from now on, let's count fractions as numbers too'?"
Here's the bottom line up front — numbers expanded out of necessity, not taste. Every time some operation posed a problem that "the existing numbers couldn't answer," people built one more room onto the house of numbers to fill that gap. That is the history of our number system. And the event that forced that construction most dramatically was the discovery that \(\sqrt{2}\) can never be written as a fraction.
The first place the naturals get stuck — subtraction
The natural numbers are "the numbers for counting how many." Addition and multiplication always land back inside the naturals. \(2 + 3 = 5\), \(2 \times 3 = 6\) — the result is again a natural number.
Subtraction, though, is different.
$$3 - 5 = \,?$$"Take 5 apples away from 3 apples" makes no sense in the world of the naturals. The answer falls outside the natural-number room. To fill that gap we bring in negative numbers, and the naturals together with 0 and the negatives are called the integers.
- Naturals \(\mathbb{N}\): \(1, 2, 3, \dots\)
- Integers \(\mathbb{Z}\): \(\dots, -2, -1, 0, 1, 2, \dots\)
Here \(\mathbb{Z}\) is just the symbol for the integers (it comes from the German Zahlen); there's nothing to be intimidated by. The only point is that "to subtract freely, we needed negative numbers."
The second snag — division
Once we have the integers, subtraction is solved. But now division steps outside the room.
$$1 \div 3 = \,?$$Dividing \(1\) by \(3\) doesn't come out evenly in integers. What fills this gap is the fraction, and every number that can be written as a ratio of integers is called a rational number \(\mathbb{Q}\).
$$\mathbb{Q} = \left\{ \frac{a}{b} \;\middle|\; a, b \text{ are integers}, \; b \neq 0 \right\}$$The word "rational" here doesn't mean "sensible" — it's the translation of ratio. So a rational number = "a number that can be written as a ratio of integers." Decimals that terminate or repeat, like \(0.5 = \tfrac{1}{2}\) or \(0.333\dots = \tfrac{1}{3}\), are all rational numbers.
By this point it feels like "surely now every number can be written as a fraction." And that is exactly where one of the most shocking events in the history of mathematics breaks out.
√2 cannot be written as a fraction
By the Pythagorean theorem, the diagonal of a square with side length \(1\) has length \(\sqrt{2}\). It is clearly "a length that exists," yet this number cannot be written as any fraction. Let's follow why.
Intuition: If \(\sqrt{2}\) were a fraction, it would have to be writable as a fully reduced fraction \(\dfrac{a}{b}\) that can't be reduced any further. But once you set it up that way, you're forced to conclude that \(a\) and \(b\) are both even — which collides head-on with the premise that "it can't be reduced any further."
Let's confirm it with the algebra. Set \(\sqrt{2} = \dfrac{a}{b}\) and square both sides:
$$2 = \frac{a^2}{b^2} \quad\Longrightarrow\quad a^2 = 2b^2$$- The right-hand side is \(2 \times (\text{integer})\), so \(a^2\) is even. But squaring an odd number gives an odd number, so for \(a^2\) to be even, \(a\) must be even as well. Hence we can write \(a = 2k\).
- Substituting this gives \((2k)^2 = 2b^2\), that is, \(4k^2 = 2b^2 \Rightarrow b^2 = 2k^2\). By the same reasoning, \(b\) is also even.
In the end \(a\) and \(b\) are both even — meaning the fraction can be reduced by \(2\) once more, which tears down the starting assumption that it was "fully reduced." A contradiction. Therefore it is impossible to write \(\sqrt{2}\) as a fraction.
The explanation you often hear — "it's irrational because its decimal goes on forever" — is the consequence, not the reason. The real reason is "it can't be written as a ratio of integers," and that is precisely why, when you expand it as a decimal, it has no choice but to run on endlessly and irregularly.
A number that can't be written as a fraction this way is called an irrational number, and the rationals and irrationals taken all together are called the real numbers \(\mathbb{R}\). The irrationals fill the gaps along the number line, so that at last an "unbroken straight line" is completed.
The last room — √(−1)
Once we reach the reals, every point on the number line corresponds to a number. But yet another operation steps outside the room.
$$x^2 = -1 \quad\Longrightarrow\quad x = \sqrt{-1} = \,?$$For real numbers, squaring any number gives something \(0\) or greater (even a negative squared becomes positive). So "a real number whose square is \(-1\)" is nowhere on the number line. To fill this final gap, we bring in, by convention, a new number whose square is \(-1\), written \(i\).
$$i^2 = -1$$Numbers of the form \(a + bi\), built by adding this \(i\) to the reals, are the complex numbers \(\mathbb{C}\). \(i\) is not a "nonexistent number" forced into being — just like the earlier negatives, fractions, and irrationals, it is simply one more room built onto the house to fill a spot where the existing numbers had no answer.
Why \(i\) even comes to carry the geometric meaning of "a 90° rotation" is continued in Why is multiplying by i a 90° rotation?.
The pattern that threads it all together
If we boil the whole flow so far down to a single sentence, it's this.
When an operation's answer can't be produced by the numbers we currently have, we create a new room to hold that answer, and the numbers widen.
| Stuck operation | An example that fails | The new number that filled it | The widened house |
|---|---|---|---|
| Subtraction | \(3 - 5\) | negative numbers | integers \(\mathbb{Z}\) |
| Division | \(1 \div 3\) | fractions | rationals \(\mathbb{Q}\) |
| Square root | \(\sqrt{2}\) | irrational numbers | reals \(\mathbb{R}\) |
| Square root of a negative | \(\sqrt{-1}\) | the imaginary number \(i\) | complex numbers \(\mathbb{C}\) |
And so the house of numbers stacks up neatly as a chain of containments.
$$\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{C}$$The inner numbers are still right there inside the outer rooms (the natural number \(3\) is also an integer, also a rational, also a real, also a complex number). Widening doesn't make the old numbers disappear; it only increases the kinds of answers that can be held.
Try it yourself
In the diagram below, the concentric circles run from the inside out as \(\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{C}\). Press an operation button and the dot shoots outward to the first room where that answer can live. And in the \(\sqrt{2}\) chase at the bottom, you can see for yourself that no matter how large you make the denominator \(b\), \(a^2\) and \(2b^2\) never line up — that is, \(\sqrt{2}\) can't be matched by any ratio of integers.
Closing
That numbers widened from the naturals all the way to the complex numbers was not anyone's matter of taste — it was a necessary construction, built to solve the problems operations posed. Subtraction summoned the negatives, division summoned the fractions, the square root summoned the irrationals, and the square root of a negative summoned the imaginary numbers. Right at the center of it stood the discovery that "\(\sqrt{2}\) is not a fraction," and this single event shattered the world of the rationals and opened the door to the reals.
Next time you meet a new number or symbol, try asking — "which stuck operation was this number made to solve?" Follow that answer and you'll find that the number system, which looked so abstract, was really a very natural "record of problem-solving."
Related reading — Up next: Why is multiplying by i a 90° rotation? (what kind of structure this post's last room \(\mathbb{C}\) actually is) · Why do logarithms turn multiplication into addition?