Ordering Integers
Every question with a fully worked solution.
- Ordering Integers - Video - Ordering Integers Using a Number Line Watch
Theory
To compare two integers, picture them on a number line: the one further to the right is larger. This Year 7 maths page explains the \(<\) and \(>\) signs, why \(-18\) is smaller than \(-4\), and how to arrange a list of integers in ascending or descending order.
\(>\) means is greater than and \(<\) means is less than. The open side of the sign faces the larger number: \(3 > 2\) and \(-5 < -4\).
On a number line, a number to the right is greater and a number to the left is smaller. Every positive integer is greater than \(0\), and \(0\) is greater than every negative integer.
For two negatives, the one with the bigger size (distance from \(0\)) is further left, so it is smaller: \(-18 < -4\).
Ascending order goes up, from smallest to largest (think of A for a ladder you climb). Descending order goes down, from largest to smallest (D for down).
Reading the signs:
| Expression | Meaning | On a number line |
|---|---|---|
| \(3 > 2\) | \(3\) is greater than \(2\) | \(3\) is to the right of \(2\) |
| \(-5 < -4\) | \(-5\) is less than \(-4\) | \(-5\) is to the left of \(-4\) |
| Means smaller (further left) | Means larger (further right) |
|---|---|
| less than, colder, lower, deeper | greater than, warmer, higher |
| a bigger debt or loss | a bigger balance or profit |
| earlier on a timeline | later on a timeline |
How to compare two integers
- Picture both numbers on a number line.
- The one further right is larger.
- Write the sign with its open side facing the larger number.
How to put integers in ascending order
- Group the negatives, zero and the positives.
- Negatives: the biggest size comes first. Positives: the smallest comes first.
- Join the groups: negatives, zero, positives.
How to put integers in descending order
- Positives first, largest to smallest, then zero.
- Negatives last, starting with the one closest to zero.
(i) \(-6 \;\square\; 1\)
(ii) \(-4 \;\square\; -9\)
(iii) \(0 \;\square\; -5\)
The smaller number is further left.
(i) \(-6 < 1\)
(ii) \(-4 > -9\)
(iii) \(0 > -5\)
Numbers to the right are greater, so reading left to right: \(-3, -1, 2, 3\).
Negatives: \(-15, -8, -3\). Zero: \(0\). Positives: \(2, 7\).
\(-15, -8, -3, 0, 2, 7\). Check: \(-15\) is furthest left and \(7\) furthest right.
Positives first: \(18, 3\). Then negatives, closest to zero first: \(-7, -26, -41, -90\).
\(18, 3, -7, -26, -41, -90\)
(i) List the towns from coldest to warmest.
(ii) How many towns were colder than \(-5^\circ\text{C}\)?
Coldest means smallest, so use ascending order: \(-11 < -6 < -4 < -1 < 2\).
(i) Town D, Town B, Town A, Town C, Town E
(ii) Colder than \(-5\) means left of \(-5\): \(-11\) and \(-6\), so \(2\) towns.
(ii) What is the largest integer less than \(-8\)?
(i) List them: \(-5, -4, -3, -2, -1, 0, 1, 2\). The ends are not included.
(ii) Less than \(-8\) means further left. The first integer to the left is \(-9\).
(i) \(8\) integers
(ii) \(-9\)
Common pitfalls
Frequently asked questions
What does ascending order mean?
From smallest to largest, moving up, like climbing a ladder.
What does descending order mean?
From largest to smallest, moving down.
Which is bigger, -20 or -2?
\(-2\) is bigger. It is closer to zero, so it is further right on the number line.
Is zero bigger than a negative number?
Yes. Zero is greater than every negative number, so \(0 > -100\).
How do you remember which way the sign goes?
The open side of \(<\) or \(>\) always faces the larger number.