Coaching Centre
Real numbers and intervals
Natural numbers, integers, rationals, irrationals and reals, plus intervals on the number line in inequality and bracket notation.
The number system
| Set | Symbol | What it contains | Examples |
|---|---|---|---|
| Natural numbers | $\mathbb{N}$ | Counting numbers | $1, 2, 3, \ldots$ |
| Integers | $\mathbb{Z}$ | Whole numbers and their negatives | $-2, 0, 5$ |
| Rational numbers | $\mathbb{Q}$ | Any fraction $\tfrac{a}{b}$ of integers, $b \ne 0$ | $\tfrac34, -2, 0.\overline{3}$ |
| Irrational numbers | Real numbers that are not fractions | $\sqrt{2}, \pi, e$ | |
| Real numbers | $\mathbb{R}$ | Every point on the number line | all of the above |
Write $0.\overline{27} = 0.272727\ldots$ as a fraction.
Intervals
An interval is a connected piece of the number line. A filled dot (or square bracket) means the endpoint is included; an open dot (or round bracket) means it is not.
| Inequality | Interval notation | Type |
|---|---|---|
| $-1 \le x \le 3$ | $[-1, 3]$ | closed |
| $-1 < x < 3$ | $(-1, 3)$ | open |
| $-1 \le x < 3$ | $[-1, 3)$ | half-closed |
| $x > 2$ | $(2, \infty)$ | unbounded |
| $x \le 0$ | $(-\infty, 0]$ | unbounded |
The green band is the interval. Use the 0/1 sliders to include (1) or exclude (0) each endpoint, and read the inequality.
Write the interval in bracket notation before you read the answer: square bracket when an endpoint is included, round when it is not.
Which number is irrational?
$7$ is not a perfect square, so $\sqrt{7}$ is irrational. $\sqrt{16} = 4$ is an integer.
$x > -3$ in interval notation is:
$-3$ is not included, and infinity always gets a round bracket.
Write $0.\overline{4}$ as a fraction.
$x = 0.444\ldots$, $10x = 4.444\ldots$, so $9x = 4$ and $x = \tfrac49$.
Which set does $-5$ not belong to?
The natural numbers are only the counting numbers $1, 2, 3, \ldots$
Write $0.\overline{12} = 0.1212\ldots$ as a fraction in lowest terms.
$100x - x = 12$, so $x = \tfrac{12}{99} = \tfrac{4}{33}$.
$[-2, 5)$ as an inequality is:
The square bracket includes $-2$; the round bracket excludes $5$.
Which interval is bounded?
A bounded interval has two endpoints.
Is $0.1010010001\ldots$ (one more zero each time) rational?
A rational decimal must stop or repeat the same block forever. This one does neither.
New numbers every time. Keep going until you get five in a row.