- Maths
- Classes 9-10
- 5 min read
Rational and irrational numbers
Tell rational numbers from irrational ones, turn recurring decimals into fractions and find where √2 sits on the number line.
Let us begin
The big idea
Every number you have met so far is a real number, and real numbers come in two families. Rational numbers can be written as fractions. Irrational numbers, like √2 and π, cannot, however hard you try.
A rational number is any number that can be written as p/q, where p and q are integers and q is not zero. So 3 (which is 3/1), −5/8, 0.75 (which is 3/4) and 0.333... (which is 1/3) are all rational.
When you write a rational number as a decimal, one of two things happens: the decimal stops (0.75, 2.5) or it repeats a block forever (0.333..., 0.272727...). A repeating decimal is written with a bar over the repeating block.
An irrational number is a real number that is not rational. Its decimal goes on forever without ever repeating a block. √2 = 1.41421356..., √3, √5 and π = 3.14159265... are irrational. Note that √4 = 2 and √9 = 3 are rational, because they are whole numbers.
Every real number has a place on the number line. √2 is the length of the diagonal of a square of side 1, so it can be drawn exactly with a compass, even though its decimal never ends.
Between any two rational numbers there are infinitely many more. One quick way to find a number between a and b is to take their average (a + b)/2.
Worked examples
See how it works
Is 0.75 rational?
Yes. 0.75 = 75/100 = 3/4. Any decimal that stops can be written over a power of ten, so it is rational.
Turn 0.333... into a fraction
Let x = 0.333... Multiply by 10: 10x = 3.333... Subtract the first line from the second: 10x − x = 3.333... − 0.333..., so 9x = 3 and x = 3/9 = 1/3.
Turn 0.272727... into a fraction
The repeating block has two digits, so multiply by 100. Let x = 0.272727..., then 100x = 27.272727... Subtract: 99x = 27, so x = 27/99 = 3/11.
A rational number between 1/3 and 1/2
Average them: (1/3 + 1/2)/2 = (2/6 + 3/6)/2 = (5/6)/2 = 5/12. Check: 1/3 = 4/12 and 1/2 = 6/12, and 5/12 sits between them.
Try it at home
Draw √2 with a square
You will need
- A sheet of paper
- A pencil and ruler
- A length of string or a strip of paper
- A scale marked in centimetres
What to do
- Draw a number line along the bottom of the page. Mark 0 and then 1, 2 and 3, using 4 cm for each unit.
- On the segment from 0 to 1, draw a square of side 4 cm standing on the line.
- Draw the diagonal from the point 0 to the opposite top corner of the square. This diagonal has length √2 units.
- Hold one end of the string at 0, stretch it along the diagonal and pinch it at the corner. Swing the pinched end down to the number line and mark the spot.
- Measure the mark with the scale: it should be about 5.66 cm from 0, which is 4 × 1.414. That point is √2.
Practice sheet
Check what you learned
Practice sheet: Rational and irrational numbers (Classes 9-10, Maths)
- Is 7 a rational number? Why?
- Write 0.6 as a fraction in its lowest terms.
- Is √16 rational or irrational?
- Is √7 rational or irrational?
- Write 0.444... as a fraction.
- Write 0.181818... as a fraction.
- Find a rational number between 1/4 and 1/2.
- Which of these decimals is irrational: 0.125, 0.121212..., 0.1010010001...?
- Is the product √2 × √2 rational or irrational?
- Is π equal to 22/7?
Answer key
Answers: Rational and irrational numbers
- Yes, because 7 = 7/1, a fraction of integers.
- 0.6 = 6/10 = 3/5.
- Rational, because √16 = 4.
- Irrational, because 7 is not a perfect square and the decimal of √7 never repeats.
- Let x = 0.444..., then 10x = 4.444..., so 9x = 4 and x = 4/9.
- Let x = 0.181818..., then 100x = 18.1818..., so 99x = 18 and x = 18/99 = 2/11.
- 3/8, the average: (1/4 + 1/2)/2 = (3/4)/2 = 3/8. Any number between them is fine.
- 0.1010010001..., because its pattern never repeats a fixed block.
- Rational: √2 × √2 = 2.
- No. 22/7 is only a rational approximation of π. π itself is irrational.