Foundations — Integers — operations, number line, absolute value

This page is the toolbox for Integers — operations, number line, absolute value. Before you touch a single rule there, you must be able to read every squiggle it throws at you. We build each symbol from nothing, glue it to a picture, and say why the topic can't live without it.

0. The counting numbers we start from

Before negatives exist, there are the natural numbers: $1,2,3,4,\dots$ — the things you count on your fingers. You already met these in Natural numbers and place value.

Why the topic needs this: integers are built by extending the naturals. If you don't know where we started, you won't see what we added.

1. The symbol "$=$" — sameness

It is not an instruction to "compute". $3+5=8$ says "the point named $3+5$ and the point named $8$ are the same point on the line".

2. The number line — the stage everything happens on

Everything in this topic lives on one horizontal line. Study it before anything else.

Look at the figure:

• the mint dot in the middle is $0$,
• moving right (lavender arrows) grows the number,
• moving left (coral arrows) shrinks it.

Why the topic needs this: the parent note defines addition as "move right", subtraction as "move left", and absolute value as "distance from 0". None of those sentences mean anything without this line.

3. The negative sign "$-$" — the mirror flip

The minus sign wears two hats. You must separate them or you will drown.

Picture a mirror standing at $0$. Reflecting the point $5$ across it lands you on $-5$. Reflecting again brings you back: $-(-5)=5$. The mirror flips direction; two flips return home.

4. The set of integers "$\mathbb{Z}$"

Now we can name the whole collection.

The blackboard-bold letter $\mathbb{Z}$ comes from the German Zahlen, "numbers". When a symbol appears in a curly-brace list $\{\}$, that means "the set/collection of exactly these things".

Why the topic needs this: it gives a one-symbol name for the universe we operate in, so rules can say "for every $a$ in $\mathbb{Z}$..." without re-listing every number.

5. A letter standing for a number: $a,b,c$

Picture a labelled empty crate on the number line; you can slide any dot into it. This is exactly the idea you will lean on hard in Linear equations in one variable.

Why the topic needs this: sign rules like $(-a)+(-b)=-(a+b)$ are general — they must work for all inputs, which requires letters, not specific numbers.

6. Ordering "$<$" and "$>$" — who is further left

The wide-open mouth of the sign always faces the bigger number. Look back at the figure: since $-3$ is left of $1$, we write $-3<1$ — even though $3$ "feels" bigger, the mirror pushed it to the far left.

You will stretch this idea into full chains in Inequalities.

7. Magnitude and the absolute-value bars $|a|$

Now for the star of the parent note. Study its picture:

Read the two-line rule slowly:

• If $a$ is already $0$ or to the right, its distance is $a$. Nothing to do.
• If $a$ is to the left (negative), we apply one mirror flip ($-a$) to turn that negative label into the positive distance. E.g. $|-5|=-(-5)=5$.

In the figure, the coral bracket over $-4$ and the mint bracket over $4$ have the same length — that is the whole point: $|-4|=|4|=4$.

Why the topic needs this: absolute value is how the parent measures "size" of an integer so it can add magnitudes, compare, and prove the triangle inequality $|a+b|\le |a|+|b|$.

8. The identity numbers: $0$ and $1$

Two integers are special because they leave things unchanged — they are the "do-nothing" numbers, and every sign-rule proof secretly leans on them.

Picture them on the line: "$+0$" is standing still; "$\times 1$" is "keep exactly what you have"; "$\times 0$" collapses everything onto the origin.

Why the topic needs this: you cannot prove $(-)\times (-)=(+)$ without $a\times 0=0$ and $a+(-a)=0$. These humble facts are the crowbar in the next section.

9. Why the multiplication signs are forced

Multiplication starts as repeated addition: $3\times 5$ means "take the step $+5$, three times", landing on $15$. But what does $3\times (-5)$ or $(-3)\times (-5)$ mean? We cannot step "$-3$ times". So the signs are not chosen by taste — they are forced by the do-nothing numbers. Watch the picture, then the algebra.

Now the forced algebra (this is the why, using only the identity facts from §8):

Why this step? We never decided the signs; we insisted the answer stay consistent with $a\times 0=0$ and $a+(-a)=0$. Any other choice would break those facts. That is why "negative times negative is positive" — it is the price of a consistent number system, and the double-arrow flip is its picture.

10. Division — the undo, and where it is allowed to live

The sign rules copy multiplication's, because division just runs it backwards: if $(-4)\times (-5)=20$, then $20÷(-4)=-5$.

Why the topic needs this: the parent note computes $20÷(-4)$ freely; you must know when such a division actually produces an integer and why $÷0$ is banned before you trust any answer.

11. Operations as motion — the summary picture

Here is the payoff picture for the additive moves — the ones that are literally walks on the line.

Why $a-b=a+(-b)$ — the formal reason and its picture. "Take away $b$" and "add $-b$" must agree, and here is the airtight reason using §8's inverse fact. Whatever $a-b$ equals, call it $x$; by the meaning of subtraction, $x+b=a$. Now add $-b$ (the mirror of $b$) to both sides: $x+b+(-b)=a+(-b)\Rightarrow x+0=a+(-b)\Rightarrow x=a+(-b).$ So $a-b$ and $a+(-b)$ are the same point — proven, not asserted. In the figure, starting at $3$ and doing $+(-5)$ walks the coral arrow 5 units left to $-2$: that is both "$3-5$" (take away 5) and "$3+(-5)$" (add the mirror), the identical walk.

Why the topic needs this: once motion is the meaning, the sign rules stop being magic. "Two negatives make a positive" in multiplication is just "reverse a reversal" (§9), and in addition it never applies — different motions, different pictures.

12. Grouping and the invisible glue: $()$, $\times$, and juxtaposition

Also: $ab$ (letters written side by side) silently means $a\times b$. The $\times$ is often dropped between letters to save ink.

How the foundations feed the topic

Read it top-down: counting numbers plus a $0$ plus the mirror give you the number line, the number line and the mirror together build $\mathbb{Z}$, and everything after is either motion along the line (operations) or length on the line (absolute value). The identity numbers $0$ and $1$ are the hidden crowbar that forces the multiplication and division sign rules.

Equipment checklist

Cover the right side and answer aloud. If any answer surprises you, re-read that section.