In Pre-Algebra you met irrational numbers like $\sqrt{2}$ and $\pi$. Algebra uses every kind of number at once, so it helps to know how the kinds fit together. Each group sits inside a bigger one, like nesting boxes.
- Natural numbers: the counting numbers $1, 2, 3, \ldots$
- Whole numbers: the natural numbers and $0$.
- Integers: the whole numbers and their opposites: $\ldots, -2, -1, 0, 1, 2, \ldots$
- Rational numbers: every number that can be written as $\frac{a}{b}$, where $a$ and $b$ are integers and $b \ne 0$. As decimals, they end ($0.75$) or repeat forever ($0.\overline{3}$).
- Irrational numbers: real numbers that are not rational. Their decimals never end and never repeat, like $\sqrt{2} = 1.41421\ldots$ and $\pi$.
- Real numbers: all of these together, every point on the number line.
Why it works
Why is every integer rational? Because any integer $n$ equals $\frac{n}{1}$, so $-4 = \frac{-4}{1}$. Why is a repeating decimal rational? Take $x = 0.\overline{3}$. Then $10x = 3.\overline{3}$, and subtracting gives $9x = 3$, so $x = \frac{1}{3}$. And why is $\sqrt{2}$ not rational? If $\sqrt{2} = \frac{a}{b}$ in lowest terms, then $a^2 = 2b^2$, so $a$ is even. Writing $a = 2k$ gives $b^2 = 2k^2$, so $b$ is even too. Both even contradicts lowest terms, so no such fraction exists.
Worked examples
Worked example: A square root that simplifies
Which groups does $-\sqrt{16}$ belong to?
- Simplify first: $\sqrt{16} = 4$, so $-\sqrt{16} = -4$.
- $-4$ is an integer. It is not natural or whole, because it is negative.
- Every integer is rational: $-4 = \frac{-4}{1}$.
- Every rational number is real.
Answer: Integers, rational numbers, real numbers.
Worked example: A square root that does not
Which groups does $\sqrt{20}$ belong to?
- $20$ is not a perfect square: $4^2 = 16$ and $5^2 = 25$.
- So $\sqrt{20} = 4.4721\ldots$ never ends or repeats: it is irrational.
- Every irrational number is real.
Answer: Irrational numbers, real numbers.
Common mistake
Two classic slips: thinking every square root is irrational, and thinking a negative integer like $-4$ is not rational. $\sqrt{25} = 5$ is a natural number, and $-4 = \frac{-4}{1}$ is rational. Quick check: simplify the number first, then ask, “Can I write it as a fraction of two integers?”
Key idea
Natural $\subset$ whole $\subset$ integers $\subset$ rational, and rational plus irrational make up the real numbers. A number is rational exactly when it can be written as $\frac{a}{b}$ with integers $a$ and $b \ne 0$.