Integers & Rational Numbers

Integers include zero, the natural numbers, and their negatives, so they stretch out on both sides of the number line. We need negatives to describe things like five below zero or three floors underground, where direction matters as much as size. The distance from zero is a number's absolute value, letting us compare magnitudes even when the signs differ. Drag the marker along the number line to watch each integer's position and absolute value.

Expanding the World of Numbers

👀 Why Do We Need Negative Numbers?

①Temperature below 0°C: -5°C

②Owing money: -₩30,000

③Many situations can't be expressed with natural numbers alone

Integers on the Number Line

Select number3

🔍 What is Absolute Value?

①Distance to 0 on the number line

②|3| = 3, |-3| = 3 → magnitude ignoring sign

③Always ≥ 0

④|0| = 0

Ordering Numbers

Order of Integers

(negative) < 0 < (positive)

Further right on the number line = larger

Comparing Negatives

A negative with a larger absolute value is smaller

-5 < -3 (since |-5| > |-3|, -5 is smaller)

💡 Comparison Rules

①Positives are always > 0

②Negatives are always < 0

③Among negatives, larger absolute value = smaller number

④A positive is always greater than a negative

Work It Out

Example 1

Find the absolute values of −5 and +3.

Absolute value is the distance from 0 on the number line, so drop the sign.

|−5| = 5

Find the distance for +3 as well.

|+3| = 3

▸ |−5| = 5, |+3| = 3

Absolute value is a distance, so it is always at least 0.

Example 2

Arrange −3, +2, 0, −1/2 from smallest to largest.

Among negatives, a larger absolute value is smaller, so −3 < −1/2.

−3 < −1/2 < 0 < +2

List from the smallest.

−3, −1/2, 0, +2

▸ −3, −1/2, 0, +2

(negative) < 0 < (positive); among negatives, larger absolute value is smaller.

Exam Key Points

Number Classification

Rationals = integers ∪ non-integer rationals

Every number expressible as a (reduced) fraction

Grade-7 school exam type

Find all numbers whose absolute value is 4.

①4

②−4

③0 and 4

④−4 and 4

⑤−4, 0, 4

▸ ④ −4 and 4

A number with absolute value 4 is at distance 4 from 0.

|x| = 4

There are two, one on each side of 0.

x = +4 or x = −4

🎯 Exam Key Points

①Natural ⊂ Integer ⊂ Rational

②0 is neither positive nor negative

③Same |·| and opposite signs → sum = 0

④Rational = a/b (b≠0, a,b ∈ Z)

⑤To the right on the number line → larger

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GCD & LCM

Operations on Integers & Rationals

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