What you will learn
- Classify common types of numbers
- Place rational numbers on a number line
- Recognise rational and irrational decimal expansions
- Find rational numbers between two given numbers
The number families
Numbers are organised in nested families. Natural numbers are used for counting. Adding zero gives whole numbers, and adding negative whole numbers gives integers.
A rational number can be written as p/q, where p and q are integers and q is not zero. Every integer is rational because, for example, 5 can be written as 5/1.
Classify −4, 0, 3/5 and √2.
Solution: −4 is an integer and rational number; 0 is whole, integer and rational; 3/5 is rational; √2 is irrational.
Rational and irrational decimals
A rational number has a decimal expansion that either ends or repeats a fixed pattern. For example, 3/8 = 0.375 ends, while 1/3 = 0.333… repeats.
An irrational number has a decimal expansion that neither ends nor repeats. Numbers such as √2 and √3 are irrational. Rational and irrational numbers together form the real numbers.
Is 0.272727… rational?
Solution: Yes. The block 27 repeats, so the number is rational.
Finding numbers between numbers
There are infinitely many rational numbers between any two different rational numbers. One quick method is to take their average. You can repeat the method to find more numbers.
To compare or place numbers on a number line, write them in a common form such as fractions with the same denominator or decimals of sufficient accuracy.
Find one rational number between 1/3 and 1/2.
Solution: Their average is (1/3 + 1/2) ÷ 2 = 5/12. Therefore 5/12 lies between them.
Key terms
- Rational number
- A number expressible as p/q with integers p and q and q ≠ 0.
- Irrational number
- A real number whose decimal expansion is non-terminating and non-repeating.
- Real number
- Any rational or irrational number represented on the number line.
Common mistakes
- Assuming every non-terminating decimal is irrational; repeating decimals are rational.
- Using zero as the denominator of a fraction.
- Thinking there is only one rational number between two rational numbers.
Questions and answers
1. Is −7 a rational number?
Answer: Yes, because −7 = −7/1.
2. Write one rational number between 2/5 and 3/5.
Answer: 1/2
The average is (2/5 + 3/5) ÷ 2 = 1/2.
3. Why is 0.1010010001… irrational?
Answer: Its decimal neither terminates nor repeats a fixed block.
One-minute revision
- Natural ⊂ Whole ⊂ Integers ⊂ Rational ⊂ Real.
- Terminating or repeating decimal means rational.
- Non-terminating and non-repeating decimal means irrational.