What is a Rational Number?
Key Definition
A rational number is any number that can be written as a fraction a/b, where:
- a and b are integers (whole numbers, positive or negative)
- b ≠ 0 (the denominator cannot be zero)
This includes:
Integers
5, −3, 0
5 = 5/1, −3 = −3/1Fractions
2/3, −7/4
Already in a/b formDecimals
0.5, 0.333...
Terminate or repeatWhy "Rational"?
The word "rational" comes from "ratio", not from "reasonable"!
Every rational number is a ratio of two integers. Think of it as comparing two whole numbers.
Remember: Rational = Ratio-nal (can be written as a ratio)
The Symbol for Rational Numbers
ℚ
From the word "Quotient"
ℤ ⊂ ℚ means "integers are inside rationals"
Integers ARE Rational Numbers
Every integer is automatically a rational number! Here's why:
| Integer | As a Fraction | Explanation |
|---|---|---|
| 5 | 5/1 | 5 divided by 1 equals 5 |
| −3 | −3/1 | −3 divided by 1 equals −3 |
| 0 | 0/1 | 0 divided by 1 equals 0 |
| 7 | 7/1 | 7 divided by 1 equals 7 |
| −5 | −5/1 | −5 divided by 1 equals −5 |
Interactive visualization — available in the full lesson
The set of integers (ℤ) is completely inside the set of rational numbers (ℚ)