Estimating square roots; irrational numbers on the line
Trap a square root between whole numbers, zoom in to tenths and hundredths by squaring, and sort rational from irrational numbers on the real line.
- UK: Year 8
- France: grade 8
- Germany: grade 8
- Iran: grade 8
- Afghanistan: grade 8
- Syria: grade 9
- Palestine: grade 8
- Ukraine: grade 8
- Russia: grade 8
Guide
The square root of 20, √20, is the number that makes 20 when you multiply it by itself. 20 is not a square number, so √20 is not a whole number.
To estimate it, find the square numbers either side: 16 < 20 < 25, so √20 is between 4 and 5. Then zoom in and square tenths: 4.4 × 4.4 = 19.36 and 4.5 × 4.5 = 20.25.
√2 is a real point on the line (it is the length of the diagonal of a square with side 1), but its decimal never ends and never repeats. A number like that is an irrational number. π is irrational too.
A decimal that repeats for ever is still rational: 0.333… is 13. Rational and irrational numbers together make the real numbers, and every one of them has its own place on the line.