Numbers and sets

What a set is, how the number systems were built one operation at a time, and where complex numbers come from and how they add and multiply.

August 31, 2026

9 minutes read

1824 words

Contents10

Each kind of number was invented because an operation ran out of answers. Subtraction ran out of the counting numbers, division ran out of the whole numbers, and roots ran out of the fractions. Complex numbers are the last of those widenings, and they arrive the same way the earlier ones did.

Assumed knowledge

Arithmetic, and nothing else. Addition, subtraction, multiplication and division of ordinary numbers, and what a square is.

Ground covered

Sets and the two symbols used to write them. The chain $\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}$ , and what each link had to be added for. Why a negative number has no square root among the reals, and what is done about it. Complex numbers, how they add and multiply, and how they are drawn.

A reader who already knows what $a + bi$ means, and that multiplying two of them expands the brackets and replaces $i^2$ with $-1$ , can skip the page.

Sets and membership

A set is a collection of things. The only question a set answers is whether something is in it or out of it. The things in it are its members.

A small set can be written by listing its members inside braces, as in $\{1, 2, 3\}$ . The symbol $\in$ means is a member of, so $2 \in \{1, 2, 3\}$ is true. A set too large to list is written with a rule instead. The form is braces around a letter standing for a member, a colon, and the rule that member has to satisfy. The even numbers cannot be listed, but the rule deciding them is short, so they are $\{\, n : n \text{ is even} \,\}$ .

One set can sit wholly inside another. The smaller is then a subset of the larger, and the symbol $\subset$ means is a subset of. So $\{1, 2\} \subset \{1, 2, 3\}$ .

Each kind of number below is a set, and each contains the one before it.

Sets of numbers

Counting gives the natural numbers, written $\mathbb{N}$ . They run $0, 1, 2, 3$ and upward, and they answer how many.

Subtraction runs out of them. Taking 5 from 3 has no answer among the natural numbers. Adding the negatives gives the integers, written $\mathbb{Z}$ .

Division runs out of the integers. Dividing 3 by 5 has no integer answer. Adding every ratio of two integers gives the rational numbers, written $\mathbb{Q}$ .

A ratio compares two whole numbers. Five sevenths is the ratio of 5 to 7, written $5/7$ or $5:7$ . It is what each of seven gets when five are shared out equally, and seven such shares make five again. The share is exact even where no knife could cut it so.

The same number turns up under several names in daily use.

Sharing five loaves among seven people gives each one $5/7$ of a loaf.
Dividing 5 by 7 on a calculator shows $0.714285$ and onward, which is that share written as a decimal.
Getting five of the seven answers right is a score of $5/7$ .
A road rising 5 metres over 7 metres of level ground has a gradient of $5/7$ .
Roughly 71.43 per cent is the same quantity counted in hundredths.

The colon form needs one caution. Paint mixed in the ratio $5:7$ takes five parts of one colour to seven of another, and the parts total twelve. The first colour is then $5/12$ of the tin. Five sevenths is the ratio of the two colours to each other, not the share either one takes of the mixture.

Written out as a decimal, a rational number either stops or repeats. $1/4$ is $0.25$ , and it stops. The decimal for $1/3$ is $0.3$ with the 3 repeating forever. Five sevenths repeats the block $714285$ without end.

Some lengths are no ratio at all. The diagonal of a square of side one has length $\sqrt{2}$ , and no two whole numbers stand in that ratio. Its decimal runs $1.41421356$ and onward, never stopping and never repeating. Such a number is called irrational.

An irrational number is exact all the same. $\sqrt{2}$ marks one definite point of the line, and squaring it gives 2 exactly. The length of the decimal does not set rational numbers apart from irrational ones. $1/3$ runs forever as well. Any decimal that can be finished is an approximation, and the symbol $\sqrt{2}$ names the number exactly. What an irrational lacks is a pair of whole numbers to name it, not precision.

Filling those gaps gives the real numbers, written $\mathbb{R}$ . The real numbers are the points of an unbroken line.

Each step widened the set so that one operation always had an answer. Subtraction forced the first, division the second, and roots the third. Every set in the chain contains the one before it as a subset.

\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}

Square roots

Squaring multiplies a number by itself. So $1^2 = 1$ , $2^2 = 4$ , and $3^2 = 9$ . A square root reverses that operation. A square root of 9 is a number whose square is 9.

Nine has two of them. $3^2 = 9$ , and $(-3)^2 = 9$ as well, because a negative times a negative is positive. Every positive real number has two square roots, one positive and one negative. Zero has one, which is zero itself.

Negative numbers have none at all. Multiplying two positive numbers gives a positive result. Multiplying two negative numbers also gives a positive result. Every real number $x$ therefore satisfies $x^2 \ge 0$ , and no real number squares to $-1$ .

An operation with no answer forced each earlier widening. Subtraction had no answer in $\mathbb{N}$ , and division had none in $\mathbb{Z}$ . The square root of a negative number has none in $\mathbb{R}$ .

Imaginary unit

Every earlier step widened the system by adding what was missing. The same is done here.

The imaginary unit, written $i$ , is defined by a single property.

i^2 = -1 \tag{1}

Nothing else about $i$ is assumed. It is not a real number and it has no place on the number line. It is a new object whose square is $-1$ .

The square root of $-1$ is not unique either, exactly as nine has two. Since $(-i)^2 = (-1)^2 i^2$ , and $(-1)^2 = 1$ , the value $-i$ squares to $-1$ as well.

Multiplying $i$ by a real number $b$ gives $bi$ . Every number with that form is an imaginary number. For instance $3i$ squares to $-9$ , because $3i \times 3i = 9 i^2$ .

Complex numbers

Adding a real number to an imaginary one gives a complex number. It is written $a + bi$ , where $a$ and $b$ are real numbers. The number $a$ is the real part, and $b$ is the imaginary part. The set of all of them is written $\mathbb{C}$ .

Nothing is lost by the widening. Every real number is already in $\mathbb{C}$ , as the case $b = 0$ . Every imaginary number is there too, as the case $a = 0$ . So $\mathbb{R} \subset \mathbb{C}$ , and the chain of sets gains one more link. The letter $z$ stands for a complex number by convention.

Addition works term by term. The real parts add, and the imaginary parts add.

(a + bi) + (p + qi) = (a + p) + (b + q)\,i \tag{2}

Multiplication works by expanding the brackets in the usual way. Every $i^2$ that appears is then replaced by $-1$ , which is what equation (1) allows.

\begin{aligned} (a + bi)(p + qi) &= ap + aqi + bpi + bq\,i^2 \\ &= ap + aqi + bpi - bq \\ &= (ap - bq) + (aq + bp)\,i \end{aligned} \tag{3}

For example, $(1 + 2i)(3 + i)$ gives $3 + i + 6i + 2i^2$ . The last term is $-2$ , so the result is $1 + 7i$ .

Squaring is the case where both factors are the same number. Setting $p = a$ and $q = b$ in equation (3) gives its rule.

(a + bi)^2 = (a^2 - b^2) + 2ab\,i \tag{4}

So $(1 + 2i)^2$ is $(1 - 4) + 4i$ , which is $-3 + 4i$ .

The complex plane

A complex number carries two independent ordinary numbers, $a$ and $b$ . Drawing one therefore needs two axes rather than a single line. The number $a + bi$ is drawn as the point with horizontal coordinate $a$ and vertical coordinate $b$ . The horizontal axis holds the ordinary numbers. The vertical axis holds the multiples of $i$ . The plane of all these points is the complex plane.

The number 3 + 2i sits three squares along and two up. A real number sits on the horizontal axis, and a multiple of i on the vertical one.

Addition slides points across that plane. Equation (2) adds the two coordinates separately, which is what sliding a point does to its coordinates. Adding one fixed number therefore slides every point by the same distance in the same direction. That displacement runs from the origin to the fixed number. Distances and angles between points are unchanged.

Adding two numbers. The second arrow is carried to the tip of the first, and its tip lands on the sum.

Both figures name a point by two coordinates, one distance along and one distance up. The same point can be named another way. Say how far it lies from the origin, and in which direction, and that fixes it just as exactly. Those two numbers are one distance and one angle, and they are the point’s polar coordinates .

The second naming is what multiplication needs. Equation (3) changes both coordinates at once, so multiplication has no reading as simple as sliding. In polar coordinates it has one. Multiplying two complex numbers multiplies their distances and adds their angles.

Figure sources

Both drawings come from one script kept beside this page.

planes.py draws the plane and the addition on one ruled grid, so the paper a reader learns in the first is the paper the second argues on. One unit per square, with five fine squares to each ruled one, the way squared paper is printed. It needs NumPy and Matplotlib.

planes.py

uv run --with numpy --with matplotlib python3 planes.py