√415 at a glance
- Exact value
- √415
- Decimal (10 places)
- 20.3715487875
- Rounded
- 20.4 · 20.37 · 20.372
- Perfect square?
- No — between 20² and 21²
- Rational?
- Irrational
- Both square roots
- ±20.371549
- Prime factorization
- 5 × 83
- Cube root
- 7.459036
How to simplify √415
The prime factorization of 415 is 5 × 83. Every prime appears only once, so there is no pair to bring outside the radical — √415 is already in simplest form (a "square-free" radicand).
A whole number has a rational square root only when every prime in its factorization appears an even number of times. In 415, 5 and 83 appear an odd number of times, so √415 is irrational and 20.3715487875 is a rounded value.
Where √415 sits between perfect squares
400 = 20² and 441 = 21² are the nearest perfect squares, so √415 lies between 20 and 21. 415 is 15 above 400 and 26 below 441, so the root is closer to 20.
- Straight line between 400 and 441: 20.3659 (0.03% low)
- Tangent from 20, i.e. 20 + 15 ÷ 40: 20.3750 (0.02% high)
- Tangent from 21, i.e. 21 − 26 ÷ 42: 20.3810 (0.05% high)
For √415 the tangent at 20 wins, missing by only 0.0035. Tangent estimates shine when the number sits close to a perfect square — here 415 is just 15 above 400.
Finding √415 with the Babylonian method
If a guess is too big, 415 ÷ x is too small by about the same amount, so their average lands much closer. In fact an error of e shrinks to roughly e² ÷ (2√415) in one step.
Start from the nearest whole number, 20 (20² = 400):
| Step | Guess x | 415 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 20.0000000000 | 20.7500000000 | 20.3750000000 | 2 |
| 2 | 20.3750000000 | 20.3680981595 | 20.3715490798 | 6 |
| 3 | 20.3715490798 | 20.3715484952 | 20.3715487875 | all 10 shown |
The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √415 = 20.3715487875 to every decimal shown.
√415 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √415 the pattern is [20; 2, 1, 2, 4, 6, 1, 1, 3, 1, 1, 6, 4, …] with the block of 16 terms after the semicolon repeating forever (only the first 12 of the 16 are shown). A pattern that never ends is one more proof that √415 is irrational.
Cutting the continued fraction off early gives the best fractions for approximating the root — no fraction with a smaller denominator comes closer:
| Fraction | Decimal | Error |
|---|---|---|
| 20/1 | 20.0000000000 | 3.7 × 10⁻¹ |
| 41/2 | 20.5000000000 | 1.3 × 10⁻¹ |
| 61/3 | 20.3333333333 | 3.8 × 10⁻² |
| 163/8 | 20.3750000000 | 3.5 × 10⁻³ |
| 713/35 | 20.3714285714 | 1.2 × 10⁻⁴ |
| 4,441/218 | 20.3715596330 | 1.1 × 10⁻⁵ |
The same fractions solve Pell’s equation, x² − 415y² = 1. Its smallest solution in positive whole numbers is x = 18,412,804, y = 903,849.
√415 in geometry and everyday measurements
- A square garage floor of 415 square feet measures about 20.37 ft (20 ft 4 in) per side, and its corner-to-corner diagonal is √830 ≈ 28.8 ft.
- 415 is not a sum of two whole-number squares — the prime factor 83 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √415 is never the diagonal of a rectangle with whole-number sides. It is not a sum of three positive squares either, so no whole-number box has √415 as its space diagonal.
Square roots near √415 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √412 | 2√103 | 20.2978 | No |
| √413 | √413 | 20.3224 | No |
| √414 | 3√46 | 20.3470 | No |
| √415 | √415 | 20.3715 | No |
| √416 | 4√26 | 20.3961 | No |
| √417 | √417 | 20.4206 | No |
| √418 | √418 | 20.4450 | No |
- The cube root of 415 is about 7.459036.
- Squaring undoes the root: (√415)² = 415, while 415² = 172,225 — the number whose square root is 415.
Frequently asked questions
What is the square root of 415?
The square root of 415 is √415, about 20.3715487875. The negative root, −20.371549, also squares to 415.
Is the square root of 415 rational or irrational?
Irrational. 415 is not a perfect square — it falls between 400 and 441 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √415 be simplified?
No. 415 = 5 × 83 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √415 rounded to two decimal places?
√415 ≈ 20.37 to two decimal places (20.4 to one, 20.372 to three). Check: 20.37² = 414.9369, close to 415.