Square Root of 3
The square root of 3 is approximately 1.732051, or exactly √3 in simplified radical form. No, 3 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
Where √3 sits
1² = 1 and 2² = 4, and 3 falls between them, so √3 is between 1 and 2. It's closer to 2 (2 above 1, 1 below 4), which matches the decimal 1.732051.
| k | k² | Compared to 3 |
|---|---|---|
| 1 | 1 | 2 below |
| 2 | 4 | 1 above |
| 3 | 9 | 6 above |
| 4 | 16 | 13 above |
| 5 | 25 | 22 above |
| 6 | 36 | 33 above |
Working it out by hand: Newton's method
Start with a guess (the nearest whole root, 1), divide 3 by it, and average the guess with the result. Each pass roughly doubles the number of correct digits:
| Pass | Guess x | 3 ÷ x | New guess (x + 3÷x) ÷ 2 |
|---|---|---|---|
| 1 | 1 | 3 | 2 |
| 2 | 2 | 1.5 | 1.75 |
| 3 | 1.75 | 1.714286 | 1.732143 |
| 4 | 1.732143 | 1.731959 | 1.732051 |
After 4 passes the guess is 1.732051, already matching √3 = 1.732051 to six decimal places.
Exact form from the prime factors
3 = 3. No prime appears twice, so nothing can be pulled out of the root -- √3 is already in simplest form.
Frequently Asked Questions
What is the square root of 3?
The square root of 3 is approximately 1.732051, or exactly √3 in simplified radical form.
Is 3 a perfect square?
No, 3 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
What two whole numbers is √3 between?
Between 1 and 2, because 1² = 1 and 2² = 4.
What are the square roots of the numbers right next to 3?
√2 is about 1.4142, and √4 is exactly 2. 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 √3 squared?
Exactly 3 -- squaring undoes the square root. Squaring the rounded decimal 1.732051 gives 3, which is why the radical form √3 is the exact answer and the decimal is an approximation.
What is the negative square root of 3?
−1.732051. Every positive number has two square roots, one positive and one negative, because (−1.732051)² is also 3. The radical sign √ by convention means the positive one.
More Number Theory
- Square Root Calculator (look up any number)
- Factors of 3
- GCF Calculator
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