The Integers

Subtraction has a problem. Inside the counting numbers (the natural numbers) — $1, 2, 3, \ldots$ — you can add two of them and stay inside the set; you can multiply them and stay inside; but the moment you try $3 - 5$, you fall off the edge. No counting number answers it. The integers exist to plug that hole.

Building the set

To make subtraction always work, mathematicians extended the counting numbers in both directions: zero in the middle, and for every positive integer, a negative twin. The full set is written $\mathbb{Z}$ — from the German Zahlen, "numbers" — and looks like:

The dotted set is infinite in both directions. The arrows mean "keep going forever."

The number line, geometrically

Every integer corresponds to exactly one point on a horizontal line, equally spaced, zero at the middle. Right is positive; left is negative. The line itself is your most reliable intuition tool — when in doubt, draw it.

(Drag the marker in the widget above. Watch its opposite light up on the far side of zero. Try $-3$.)

Opposites and zero

For every integer $a$ there is a unique opposite, written $-a$, sitting at the same distance from zero on the other side. The defining property is $a + (-a) = 0$ — they cancel. Zero is its own opposite, because zero is already at zero distance from itself.

Absolute value — distance, not sign

The absolute value $|a|$ asks one question only: how far is $a$ from zero? Distance carries no direction, so $|5| = 5$ and $|{-5}| = 5$. Formally:

The second branch looks strange — negative of a negative? — but it is just the formal way of saying "flip the sign on negatives so the result becomes positive."

Closure: which operations stay inside $\mathbb{Z}$?

You can add two integers and get an integer. Multiply two integers, integer. Subtract, integer. But divide — and the integers betray you. $7 \div 2 = 3.5$, which is NOT in $\mathbb{Z}$. We say $\mathbb{Z}$ is closed under addition, subtraction, and multiplication, but not under division. This single gap motivates the rational numbers, three sections from now.

Try it

Use the number-line widget above to:

• Set $a =$ in the widget. Before pressing it: where do you expect the opposite to land? Now confirm it lights up at the matching point and try a few more integers.
• Compute $|{-7}| - |3|$ visually: count from zero to $-7$ (7 steps), then subtract 3.
• Place $a =$ and $b =$. Before placing them: predict $|a - b|$ in your head. Now place the points and check (hint: the answer is NOT $|a| - |b|$).

Pause: if $a$ and $b$ are both negative, is $a - b$ positive or negative? Try a small example to test your intuition before scrolling on.

Try it in code

A trap to watch for

Beginners routinely write $|a - b| = |a| - |b|$. This is wrong, and the reason it feels right is that absolute value LOOKS like it should distribute over subtraction the way negation does. Try $a = 3, b = -2$: $|3 - (-2)| = |5| = 5$, but $|3| - |{-2}| = 3 - 2 = 1$. The first answer is the distance between $3$ and $-2$ on the number line; the second is meaningless arithmetic. The right rule: the absolute value of a difference is the distance between two numbers — always non-negative, always geometrically interpretable.

What you now know

You can now place any integer on the number line, find its opposite, compute its absolute value as a distance from zero, and reason about which arithmetic operations keep you inside $\mathbb{Z}$. The next section uses these tools to define addition and subtraction of integers — which is where the negative-sign rules ("subtracting a negative is adding") stop feeling like magic and start feeling like geometry.

Quick check

References

• Lang, S. (1971). Basic Mathematics. Springer. Chapter 1, §1 — the canonical motivation-first treatment of integers.
• Stewart, I. (2017). Infinity: A Very Short Introduction. Oxford. Accessible discussion of why infinite sets in both directions are well-defined.
• Devlin, K. (1994). Mathematics: The Science of Patterns. Scientific American Library. Chapter 2 connects the geometric and algebraic views of number systems.