Number
Number systems
Tell a rational number from an irrational one, and place any real number on the number line.
Until now every number you have met could be written as one whole number over another: 4 is 4/1, 0.25 is 1/4, and even 0.333… is 1/3. Numbers of that kind are called rational. The surprise of this year is that they do not fill the number line. There are gaps, and the numbers sitting in them — √2, √3, π — cannot be written as a fraction however hard you try. Those are the irrational numbers, and rational and irrational together are called the real numbers.
How to spot which is which
Write the number as a decimal. If it stops, it is rational. If it goes on for ever but repeats a block of digits, it is still rational. Only a decimal that goes on for ever and never settles into a repeat is irrational.
Worked example
Is √9 rational or irrational?
Work out √9 first. It is 3.
A square root only stays awkward when the number under it is not a perfect square.
Have a play
Every tick on this line is a rational number. Move along it and watch where √2 (about 1.41), √3 (about 1.73) and π (about 3.14) would have to squeeze in.
Tap anywhere on the line.
Try it together
Between which two whole numbers does √30 sit?
You do not need a calculator. Squaring whole numbers until you pass 30 is enough.
1.What is 5 × 5?
Watch out
A square root sign does not make a number irrational. √49 is 7, as rational as any number you have met. It is the number underneath that decides.
Have a go
Have a go on your own: which of these is rational?
Ready to practise?
Eight questions on what you have just read. Nothing is timed, and you can play as many times as you like.