How do we show integers on the number line?

From a number ray to a number line

The number line we drew in earlier classes started at 0 and went on forever to the right. In the language of geometry, that is a number ray. Negative numbers complete it into a true number line, going on forever in both directions.

How many negative numbers are there?

Bela’s Building of Fun had floors from −5 to +6. A mine in the chapter has levels from −200 m to +180 m. But we can imagine bigger buildings and deeper mines. Just as positive numbers +1, +2, +3, … go up without end, negative numbers −1, −2, −3, … go down without end. Positive and negative numbers, together with zero, are called integers:

…, −4, −3, −2, −1, 0, 1, 2, 3, 4, …

The infinite lift becomes a number line

Imagine a lift that goes forever upwards and forever downwards from Level 0. Turn this ‘infinite lift’ through 90° and it becomes the number line: zero in the middle, positive numbers to the right and negative numbers to the left. We usually drop the ‘+’ sign and write positive numbers simply as 1, 2, 3, …

Comparing and ordering integers

On the number line, smaller numbers are to the left of bigger numbers. So 2 < 5, −3 < 2 and −5 < −3. In the building, a lower floor has a smaller number. Two quick facts follow:

• every negative number is less than 0, and 0 is less than every positive number;
• between two negative numbers, the one with the bigger number after the sign is smaller: −10 < −2.

Comparison	Why
−2 < +5	A negative number is less than a positive number
−5 < −3	−5 is to the left of −3
0 > −4	−4 is to the left of 0
−25 < −7	−25 is much farther to the left

Walking along the number line

To the right is the positive (forward) direction, and to the left is the negative (backward) direction. From 5 to 9 you move 4 steps forward, so 5 + 4 = 9 and 9 − 5 = 4. From 9 to 3 you move 6 steps back, a movement of −6, so 9 + (−6) = 3 and 3 − 9 = −6. From 3 to −2 you move 5 steps back: 3 + (−5) = −2 and −2 − 3 = −5.

The unmarked number line

For large numbers, draw a line that shows only 0 and imagine where the other numbers are. For 85 + (−60), start at 85 and move 60 to the left: you land at 25. For (−100) − (+250), ask what must be added to 250 to reach −100: you must go 250 down to 0 and then 100 more, a movement of −350.

Writing integers in order

To list the integers between two numbers, count along the line. Between −4 and 4, in increasing order, are −3, −2, −1, 0, 1, 2, 3. Between −8 and −15 are −14, −13, −12, −11, −10, −9. To see how adding and subtracting work as movement, read adding and subtracting integers.