The integers are the positive integers like 1, 2, 3, zero, and the negative integers like −1, −2, −3, all together.
We need negative numbers to describe the opposite side of a zero point, like 5 degrees below zero or three floors below ground; add fractions, and together with the integers you get the rational numbers.
The distance of a number from 0 on the number line is its absolute value: +3 and −3 point in opposite directions, but both have absolute value 3.
On this page you see how the sets of numbers fit inside each other, then pick numbers on a number line to check absolute values and compare sizes.
From natural numbers to rational numbers
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How the sets fit: natural numbers ⊂ integers ⊂ rational numbers
👀 Why we need negative numbers
①If 5°C above zero is +5°C, then 5°C below zero can be written −5°C
②If the third floor above ground is +3, then the third floor below ground is −3
③When quantities go in opposite directions from a zero point, we mark them with a positive sign + or a negative sign −
④Adding non-integers such as −1/2 and 2/3 gives the rational numbers. As the picture shows, the natural numbers sit inside the integers, and the integers inside the rationals
Integers and absolute value on the number line
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Number line: positive integers in red, negative in blue, with the absolute value of the chosen number and its opposite
3
🔍 Absolute value
①The absolute value of a number is the distance between its point on the number line and the origin 0
②|+3| = 3 and |−3| = 3. In the picture, +3 and −3 are the same distance from 0, on opposite sides
③An absolute value is a distance, so it is 0 or positive: |0| = 0
④Apart from 0, there are always two numbers with the same absolute value, one positive and one negative
Comparing numbers
Ordering integers
(negative) < 0 < (positive)
The further right a number is on the number line, the larger it is
Comparing negatives
The larger its absolute value, the smaller the negative number
|−5| > |−3|, so −5 < −3
💡 Comparison rules
①Positive numbers are greater than 0 and negative numbers are less than 0
②If the signs differ, the positive number is always greater
③Between positives, the one with the larger absolute value is greater
④Between negatives, the one with the larger absolute value is smaller, because it lies further left on the number line
Check with examples
Example 1
Find the absolute values of −5 and +3.
1
Absolute value is the distance to the origin, so drop the sign and look only at the size.
|−5| = 5
2
+3 is also 3 away from the origin.
|+3| = 3
▸ |−5| = 5, |+3| = 3
−5 is smaller than +3 but has the larger absolute value. Comparing numbers and comparing absolute values are different questions.
Example 2
Put the four numbers −3, +2, 0, −1/2 in order from smallest to largest.
1
Between negatives, the larger absolute value is smaller, so −3 < −1/2.
−3 < −1/2 < 0 < +2
2
Write them from smallest to largest.
−3, −1/2, 0, +2
▸ −3, −1/2, 0, +2
Mark the four points on a number line and they appear in this order from left to right.
Exam checklist
Rational numbers
a/b (a and b integers, b ≠ 0)
A number that can be written as a fraction with integer numerator and denominator, and a denominator that is not 0
Grade 7 school test style
Which are all the numbers whose absolute value is 4?
①4
②−4
③0 and 4
④−4 and 4
⑤−4, 0, 4
▸ ④ −4 and 4
1
A number with absolute value 4 is 4 away from the origin.
|x| = 4
2
There is one on each side of the origin.
x = +4 or x = −4
🎯 Exam checklist
①Remember the inclusion natural numbers ⊂ integers ⊂ rational numbers
②0 is neither positive nor negative
③Two numbers with the same absolute value and opposite signs face each other across 0, and their sum is 0
④The integer 3 can be written 3/1, so it is a rational number
⑤When comparing negatives, the larger absolute value means the smaller number