Integers & Absolute Value

Below zero is a direction, and how far from zero is a separate question.

Monday morning was -2°. Tuesday was -5°. Which was colder?

Tuesday. Everyone knows that from standing outside, even before any arithmetic.

which morning was colder? 10 5 0 −5 −10 Monday −5° Tuesday −2° −5 is further down, so it is the smaller number

But 5 is bigger than 2, so on the number line -5 sits further left — and further left means smaller.

So -5 < -2, even though 5 > 2. The minus sign reverses the order.

Two different questions hide here: which is smaller, and which is further from zero. Tuesday is colder and also further from zero, and keeping those apart is what this unit is about.

Opposites, and what absolute value measures -5 -4 -3 -2 -1 0 1 2 3 4 5 opposites 3 steps 3 steps |−3| = |3| = 3

Reading the number line

Numbers get bigger as you move right and smaller as you move left. That rule never changes.

So -5 is less than -2, because it sits further left, and -1 is less than 0.

Any two numbers can be compared just by seeing which is further right.

The minus reverses the order

5 > 2 but -5 < -2. Comparing the digits and ignoring the sign is the commonest mistake here, and a quick sketch of the line prevents it.

Opposites

Every number has an opposite the same distance from zero on the other side.

The opposite of 7 is -7; the opposite of -7 is 7. Zero is its own opposite.

Adding a number to its opposite always gives zero, which is why they cancel.

Two minuses

-(-4) means the opposite of -4, which is 4. The second minus reverses the first.

Absolute value is a distance

|-7| asks only how far from zero, not which side. The answer is 7.

Distance is never negative, so an absolute value is never negative either.

That is why |-7| and |7| are equal: the two numbers are the same distance out.

Two questions, kept apart

"Which is smaller?" and "which is further from zero?" have different answers for negatives. -9 is smaller than -2 but further from zero.

Where negatives turn up

Temperature below zero, floors below ground, money owed, depths below sea level, golf under par.

In each case the negative is a direction, not a smaller amount.

A debt of $50 is written -50, and it is a larger debt than -20 even though the number is smaller.

Read what zero means

Zero is sea level, or ground floor, or breaking even. Knowing what zero stands for tells you what the negatives mean.

Ordering a mixed list

Put them on a line rather than trying to sort them in your head.

Mark zero first, then place the negatives to the left and the positives to the right.

Reading left to right gives the order, smallest first, with no guesswork.

Fractions too

-12 sits between -1 and 0. The line handles fractions and decimals just as happily as whole numbers.

Worked examples

Comparing

Which is smaller, -8 or -3?

  1. Sketch the line and mark both. -8 is further left.
  2. Further left means smaller. The digits alone would mislead here.
  3. So -8 < -3. Though |-8| = 8 is the larger distance from zero.

-8

Absolute value

Work out |-12| + |5|.

  1. |-12| = 12, since it is 12 steps from zero. Distance, not direction.
  2. |5| = 5. Already positive, so unchanged.
  3. 12 + 5 = 17. Both bars had to be resolved before adding.

17

Ordering

Put -4, 2, -7, 0 and -1 in order, smallest first.

  1. Mark zero, then place the negatives to its left. -7, -4, -1 in that order going right.
  2. Place 2 to the right of zero. The only positive.
  3. Read off left to right: -7, -4, -1, 0, 2. The line does the sorting.

-7, -4, -1, 0, 2

Words to know

Integer
A whole number, positive, negative or zero. No fractions.
Opposite
The same distance from zero, on the other side. The opposite of -7 is 7.
Absolute value
Distance from zero, written |-7| = 7. Never negative.
Number line
A line with numbers increasing to the right.
Inequality
A statement that one number is less than or greater than another.

Mistakes to watch for

Comparing negatives by their digits. -8 is smaller than -3, even though 8 is bigger than 3. The minus sign reverses the order, and a sketch of the number line settles it every time.

Giving a negative absolute value. |-7| = 7, never -7 — it measures distance from zero, and a distance is never negative.

Confusing "smaller" with "closer to zero". -9 is smaller than -2 but further from zero. The two questions have opposite answers for negatives.

Reading -(-4) as -4. The outer minus takes the opposite, so the answer is 4. Two minuses undo each other.

Thinking a debt of -50 is smaller than a debt of -20 in ordinary language. As a number it is smaller; as a debt it is larger, and word problems rely on that distinction.

Negative numbers were resisted for centuries. European mathematicians called them "absurd" as late as the 1700s, though Chinese accountants had used them since about 200 BC.

Bank statements, lifts, golf scores and sea level all use them, and each one has its own word for below zero.

The absolute value bars were introduced by Weierstrass in 1841 — remarkably recent for something so basic.

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