Math101 · Independent Ontario learning library
Math101learn.math101.caReal Numbers
Real numbers fill the number line and include natural numbers, whole numbers, integers, rational numbers, and irrational numbers.
The real numbers are all numbers that can be located on an ordinary number line.
The number-family map
Number sets are nested. A natural number such as $5$ is also whole, integer, rational, and real.
| Set | Typical description | Examples |
|---|---|---|
| natural | counting numbers | $1,2,3,\ldots$ |
| whole | natural numbers and zero | $0,1,2,\ldots$ |
| integers | whole numbers and their negatives | $-4,0,12$ |
| rational | can be written as $a/b$ | $3/5,-2,0.7$ |
| irrational | cannot be written as $a/b$ | $\sqrt2,\pi$ |
| real | rational or irrational | every entry above |
Some courses include $0$ among natural numbers and others do not. State the convention when it matters.
Rational versus irrational
A rational number can be expressed as $a/b$ where $a$ and $b$ are integers and $b\ne0$. Its decimal terminates or repeats. An irrational number has a nonterminating, nonrepeating decimal.
The number $\sqrt{49}=7$ is rational even though it is written with a radical. The number $\sqrt{10}$ is irrational because $10$ is not a perfect square. Classification depends on value, not appearance.
Classifying efficiently
To classify a number, simplify it first. Then choose the most specific set and remember the larger sets that contain it.
Ordering on the number line
Numbers farther right are greater. For negative numbers, the one closer to zero is greater: $-2>-7$. Decimal approximations help compare irrational values. Since $\sqrt5\approx2.236$ and $7/3\approx2.333$, we have
Use enough decimal places to make the comparison reliable; rounding both values too early can hide a difference.
Absolute value and distance
Absolute value measures distance from zero:
Therefore $|-6|=6$ and $|4|=4$. Distance cannot be negative. The distance between two real numbers $a$ and $b$ is $|a-b|$.
Operations and closure
Adding, subtracting, or multiplying real numbers produces another real number. Division also stays real when the divisor is not zero. Some operations can leave the real-number system: $\sqrt{-1}$ is not real, and division by zero is undefined rather than a new number.
Rational and irrational behaviour can be surprising. A rational plus an irrational is irrational, but two irrational numbers may add to a rational number: $\sqrt2+(-\sqrt2)=0$.
Intervals
Intervals describe continuous pieces of the real line. The inequality $-1\le x<4$ becomes $[-1,4)$: a square bracket includes an endpoint and a round bracket excludes it. Infinity always uses a round bracket because it is a direction, not a reachable real endpoint.
Common mistakes
Assuming every radical is irrational. Simplify first; $\sqrt{81}=9$.
Calling a repeating decimal irrational. Repeating decimals are rational.
Forgetting containment. An integer is also rational and real.
Thinking “negative” means “not real.” Negative integers, fractions, and decimals are real; only certain operations such as even roots of negative numbers leave the real system.
Quick self-check
- Have I simplified the expression before classifying it?
- Does its decimal terminate, repeat, or do neither?
- Which is the smallest named set containing it?
- Where would it lie relative to nearby integers?
Related topics
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
Which statement best classifies √10?
- 10 is not a perfect square, so √10 cannot be written as a ratio of integers.
- It is irrational.
- Every irrational number is real.
End of lesson
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