The number line , going both ways

The number line is one of the cleanest mental pictures in mathematics. A straight line, a chosen zero, and equal markings going both ways , that is all there is to it. But on this line live all the integers, ordered neatly from smallest to largest.

Concept

Draw a horizontal line. Mark a point on it and call it $0$. Choose a unit of distance (say $1$ cm). Step to the right by that unit and mark $1$. Step again and mark $2$, and so on. Now step to the left of $0$ , that point is $-1$. Step again to the left for $-2$, and so on.

The result is the integer number line:

$\dots, -4, -3, -2, -1, 0, 1, 2, 3, 4, \dots$

The number line has an important rule: moving right increases the number, moving left decreases it. So:

• $5$ is to the right of $0$, so $5 > 0$.
• $-3$ is to the left of $0$, so $-3 < 0$.
• $-7$ is to the left of $-2$, so $-7 < -2$.

Distance from zero

The distance from any integer $n$ to $0$ on the number line is just the "size" of $n$, ignoring the sign. We sometimes call this the absolute value of $n$, written $|n|$.

• $|5| = 5$.
• $|-7| = 7$.
• $|0| = 0$.

Two opposite numbers (like $5$ and $-5$) have the same distance from $0$ , they are mirror reflections across the zero.

Addition as movement on the number line

If you start at $n$ and add a positive number $p$, you move $p$ steps to the right. If you add a negative number (or subtract a positive), you move steps to the left.

• $3 + 4$: start at $3$, move $4$ right, land at $7$. ✓
• $3 + (-5)$: start at $3$, move $5$ left, land at $-2$.
• $(-2) + (-3)$: start at $-2$, move $3$ left, land at $-5$.

This visual picture is the most reliable way to understand integer arithmetic before memorising any rules.

Comparison rules

For any two integers $a$ and $b$:

• $a > b$ if $a$ is to the right of $b$ on the number line.
• $a < b$ if $a$ is to the left.
• $a = b$ if they coincide.

So even though "$-100$ is a big negative", it is less than $-1$ , because $-100$ sits far to the left.

The number line goes on forever

In both directions, the integers continue without end. This is one of the great realisations: there is no biggest integer, and there is no smallest integer. The number line is infinite.

Worked examples

Example 1. On a number line, where is $-4$ relative to $-1$?

• $-4$ is to the left of $-1$. So $-4 < -1$.

Example 2. What is the distance between $-3$ and $4$ on the number line?

• $|4 - (-3)| = |4 + 3| = 7$ units.

Example 3. Start at $-5$, move $8$ steps to the right. Where do you end up?

• $-5 + 8 = 3$.

Example 4. Order from smallest to largest: $-12, 4, -3, 0, 8, -1, -8$.

• $-12 < -8 < -3 < -1 < 0 < 4 < 8$.

Example 5. Find $|-9|$ and $|6|$.

• $|-9| = 9$, $|6| = 6$.

Example 6. Two friends are at $-3$ and $5$ on the number line. How many steps apart are they?

• $5 - (-3) = 8$ steps.

Try it yourself

1. Draw a number line from $-10$ to $+10$ and mark all the integers.
2. Mark the locations of: $-7, 3, -2, 0, 9, -5$.
3. Find the distance between $-6$ and $4$.
4. What is $|-15|$?
5. Order from smallest to largest: $-9, 2, -3, 0, -1, 5$.
6. Find $a$ if $a$ is the same distance from $0$ as $-7$ but on the opposite side.
7. Start at $-3$ and move $9$ steps left. Where are you?
8. Investigate: if $a < 0$ and $b > 0$, can you say which is greater, $a$ or $b$, without knowing their values?

Activity

Floor-line walk. Mark the floor with chalk or tape: a long line with $0$ in the middle and integers $-5$ to $+5$ at regular intervals. Stand at $0$. A friend calls out instructions like "add $3$", "subtract $5$", "add $-2$". For each, walk the right number of steps (right for positive add, left for negative add or for subtraction). Try $5$ instructions and see where you end up. This is integer arithmetic with your feet.