Square Root of 20
The square root of 20 is approximately 4.472136, or exactly 2√5 in simplified radical form. No, 20 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
Where √20 sits
4² = 16 and 5² = 25, and 20 falls between them, so √20 is between 4 and 5. It's closer to 4 (4 above 16, 5 below 25), which matches the decimal 4.472136.
| k | k² | Compared to 20 |
|---|---|---|
| 2 | 4 | 16 below |
| 3 | 9 | 11 below |
| 4 | 16 | 4 below |
| 5 | 25 | 5 above |
| 6 | 36 | 16 above |
| 7 | 49 | 29 above |
Working it out by hand: Newton's method
Start with a guess (the nearest whole root, 4), divide 20 by it, and average the guess with the result. Each pass roughly doubles the number of correct digits:
| Pass | Guess x | 20 ÷ x | New guess (x + 20÷x) ÷ 2 |
|---|---|---|---|
| 1 | 4 | 5 | 4.5 |
| 2 | 4.5 | 4.444444 | 4.472222 |
| 3 | 4.472222 | 4.47205 | 4.472136 |
| 4 | 4.472136 | 4.472136 | 4.472136 |
After 4 passes the guess is 4.472136, already matching √20 = 4.472136 to six decimal places.
Exact form from the prime factors
20 = 2^2 × 5. Each pair of matching primes comes out of the root as one factor, and what's left stays inside: √20 = 2√5. That's the exact value; the decimal 4.472136 is a rounding of it.
Frequently Asked Questions
What is the square root of 20?
The square root of 20 is approximately 4.472136, or exactly 2√5 in simplified radical form.
Is 20 a perfect square?
No, 20 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
What two whole numbers is √20 between?
Between 4 and 5, because 4² = 16 and 5² = 25.
What are the square roots of the numbers right next to 20?
√19 is about 4.3589, and √21 is about 4.5826. 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 √20 squared?
Exactly 20 -- squaring undoes the square root. Squaring the rounded decimal 4.472136 gives 20, which is why the radical form 2√5 is the exact answer and the decimal is an approximation.
What is the negative square root of 20?
−4.472136. Every positive number has two square roots, one positive and one negative, because (−4.472136)² is also 20. The radical sign √ by convention means the positive one.
More Number Theory
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