Square root of 2
√2 is about 1.414. It is not a whole number, and here is why.
The answer
The square root of 2 is about 1.414. It is between 1 and 2, because 1 < 2 < 4.
A note about this topic
England’s Year 5 curriculum includes recognising square numbers. The square-root explanations on this page are extension material.
2 drawn as a square
2 counters will not make a square. 1 of them makes a 1 by 1 square, and 1 is left over. A 2 by 2 square needs 2 more. So the side, √2, is more than 1 but less than 2.
Between 1 and 2
1 < 2 < 4, so 1 < √2 < 2
1 × 1 = 1, which is too small, and 2 × 2 = 4, which is too big. So √2 is between 1 and 2. 2 is 1 away from 1 and 2 away from 4, so √2 is nearer 1. It is about 1.414, or 1.41 to 2 decimal places.
Why the decimal never ends
Check the answer by multiplying it by itself: 1.414 × 1.414 = 1.999396. That is very close to 2, but not quite. Adding more decimal places gets closer and closer, but never exactly there.
√2 is an irrational number: its digits go on forever without a repeating pattern, so no fraction is exactly equal to it. That is why we write √2 for the exact value and 1.414 when we need a number to work with.
Where √2 turns up
- A4 paper has nominal dimensions 297 mm by 210 mm. Their ratio, 297 ÷ 210, is about 1.414, close to √2. A-series paper is designed around the √2 ratio so halving a sheet preserves its shape; the stated millimetre dimensions are rounded.
- The diagonal of a square with sides 1 cm long is exactly √2 cm, about 1.41 cm.
Square numbers and their roots
The square numbers up to 144 are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144.
More numbers
See every factor of 2 or check whether 2 is prime.
Questions parents ask
What is √2 to 1 decimal place, and to 5?
1.4 to 1 decimal place, and 1.41421 to 5 decimal places.
A square has an area of 2 cm². How long is each side?
About 1.41 cm. The exact mathematical length is √2 cm. Use an approximation for a measurement.
Can a square root be negative?
Yes. −√2 × −√2 also makes 2, so −√2 is a square root of 2 too. For a positive number, the symbol √ denotes the positive root, √2. The negative root also gives 2 when multiplied by itself.
Continue your number practice
Explore the diagnostic for another set of practice questions.