Square Root of 2
The square root of 2 is approximately 1.414214, or exactly √2 in simplified radical form. No, 2 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
Where √2 sits
1² = 1 and 2² = 4, and 2 falls between them, so √2 is between 1 and 2. It's closer to 1 (1 above 1, 2 below 4), which matches the decimal 1.414214.
| k | k² | Compared to 2 |
|---|---|---|
| 1 | 1 | 1 below |
| 2 | 4 | 2 above |
| 3 | 9 | 7 above |
| 4 | 16 | 14 above |
| 5 | 25 | 23 above |
| 6 | 36 | 34 above |
Working it out by hand: Newton's method
Start with a guess (the nearest whole root, 1), divide 2 by it, and average the guess with the result. Each pass roughly doubles the number of correct digits:
| Pass | Guess x | 2 ÷ x | New guess (x + 2÷x) ÷ 2 |
|---|---|---|---|
| 1 | 1 | 2 | 1.5 |
| 2 | 1.5 | 1.333333 | 1.416667 |
| 3 | 1.416667 | 1.411765 | 1.414216 |
| 4 | 1.414216 | 1.414211 | 1.414214 |
After 4 passes the guess is 1.414214, already matching √2 = 1.414214 to six decimal places.
Exact form from the prime factors
2 = 2. No prime appears twice, so nothing can be pulled out of the root -- √2 is already in simplest form.
Frequently Asked Questions
What is the square root of 2?
The square root of 2 is approximately 1.414214, or exactly √2 in simplified radical form.
Is 2 a perfect square?
No, 2 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
What two whole numbers is √2 between?
Between 1 and 2, because 1² = 1 and 2² = 4.
What are the square roots of the numbers right next to 2?
√1 is exactly 1, and √3 is about 1.7321. Square roots increase more slowly as the number gets bigger, which is why consecutive whole numbers have roots that get closer together the higher you go.
What is √2 squared?
Exactly 2 -- squaring undoes the square root. Squaring the rounded decimal 1.414214 gives 2, which is why the radical form √2 is the exact answer and the decimal is an approximation.
What is the negative square root of 2?
−1.414214. Every positive number has two square roots, one positive and one negative, because (−1.414214)² is also 2. The radical sign √ by convention means the positive one.
More Number Theory
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