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Mathematics · Chapter 2: Real Numbers
Natural Numbers, Integers, Rational and Irrational Numbers
The family of numbers: natural, whole, integers, rational and irrational numbers, and changing recurring decimals to fractions.
Key termRememberExampleWatch out
01Concept
Numbers did not all arrive at once. People first needed numbers to count sheep and coins, then a number for nothing, then numbers below zero, then parts of things. Each time, the family of numbers grew. Let us meet each member of the family.
Natural and whole numbers
| Set | Symbol | Members | Used for |
|---|---|---|---|
| Natural numbers | ℕ | 1, 2, 3, 4, … | Counting students in a class |
| Whole numbers | W | 0, 1, 2, 3, … (natural numbers and zero) | Number of goals in a match (can be 0) |
| Integers | ℤ | …, −3, −2, −1, 0, 1, 2, 3, … | Temperature in Skardu in winter: −15 °C |
Rational numbers
A rational number is any number that can be written as p/q, where p and q are integers and q ≠ 0. The word comes from ratio.
- ½, −¾, 7/3 are rational.
- Every integer is rational: 5 = 5/1, −2 = −2/1.
- Terminating decimals are rational: 0.25 = 1/4, 1.6 = 8/5.
- Recurring decimals are rational: 0.333… = 1/3, 0.1666… = 1/6.
Irrational numbers
An irrational number is a number that cannot be written as p/q. Its decimal form never ends and never repeats.
- √2 = 1.41421356…
- √3, √5, √7 (square roots of numbers that are not perfect squares)
- π = 3.14159265…
- e = 2.71828… (you will meet it in Logarithms)
Real numbers
Put the rational and irrational numbers together and you get the real numbers (ℝ): every number that has a place on the number line.
Turning a recurring decimal into a fraction
Example: write 0.454545… as a fraction.
- Let x = 0.454545…
- Two digits repeat, so multiply by 100: 100x = 45.454545…
- Subtract: 100x − x = 45, so 99x = 45
- x = 45/99 = 5/11
02Virtual Lab
Try this topic for yourself in the virtual lab. It opens on its own page with steps, tasks and questions.
03Real World Examples
Temperatures below zero
In January the temperature in Skardu and Quetta can fall to −10 °C or lower. Negative integers let us record it.
Bank balance
If you spend Rs 500 more than you have in a mobile wallet that allows overdraft, your balance is −500: a negative integer.
π in every circle
The wheel of a rickshaw, a roti or a cricket ball: the distance round any circle is π times its diameter, and π is irrational.
04Practice Questions
Try to answer in your own words first, then open the model answer.
Short questions
Q1.Define a rational number and give three examples.Show answer
Answer
A rational number can be written as p/q, where p and q are integers and q ≠ 0.
Examples: 3/4, −5 (= −5/1), 0.2 (= 1/5).
Q2.Which of these are irrational: √9, √10, 0.75, π, 1/7?Show answer
Answer
√10 and π are irrational.
- √9 = 3 (rational)
- 0.75 = 3/4 (rational)
- 1/7 = 0.142857142857… repeats, so it is rational
Q3.Is 0 a natural number? Is it a whole number? An integer?Show answer
Answer
0 is not a natural number, but it is a whole number, an integer, a rational number and a real number.
Q4.Write 0.777… as a fraction.Show answer
Answer
- Let x = 0.777…
- 10x = 7.777…
- 10x − x = 7, so 9x = 7
- x = 7/9
Q5.Write 0.1̇2̇ (that is, 0.121212…) as a fraction.Show answer
Answer
- x = 0.121212…
- 100x = 12.121212…
- 99x = 12
- x = 12/99 = 4/33
Q6.Is every integer a rational number? Is every rational number an integer?Show answer
Answer
Yes, every integer n can be written as n/1. No, ½ is rational but not an integer.
Long questions
Q7.Explain the difference between rational and irrational numbers, with two examples of each, and say how their decimal forms differ.Show answer
Answer
A rational number can be written as a fraction p/q (q ≠ 0). Its decimal either terminates (0.25) or recurs (0.333…).
An irrational number cannot be written as a fraction. Its decimal goes on for ever without repeating (√2 = 1.41421…, π = 3.14159…).
Together they make up the real numbers.