√359 at a glance
- Exact value
- √359
- Decimal (10 places)
- 18.9472953215
- Rounded
- 18.9 · 18.95 · 18.947
- Perfect square?
- No — between 18² and 19²
- Rational?
- Irrational
- Both square roots
- ±18.947295
- Prime factorization
- 359
- Cube root
- 7.107194
How to simplify √359
359 is a prime number, so its only factors are 1 and 359. There is no perfect-square factor to pull out, which means √359 is already in its simplest radical form.
The square root of any prime is irrational. If √359 were a fraction a/b in lowest terms, then a² = 359b², so 359 would divide a — and then 359 would divide b too, contradicting “lowest terms.” That is why the decimal 18.9472953215 is only a rounded value.
Where √359 sits between perfect squares
324 = 18² and 361 = 19² are the nearest perfect squares, so √359 lies between 18 and 19. 359 is 35 above 324 and 2 below 361, so the root is closer to 19.
- Straight line between 324 and 361: 18.9459 (0.01% low)
- Tangent from 18, i.e. 18 + 35 ÷ 36: 18.9722 (0.13% high)
- Tangent from 19, i.e. 19 − 2 ÷ 38: 18.9474 (0% high)
For √359 the tangent at 19 wins, missing by only 0.0001. Tangent estimates shine when the number sits close to a perfect square — here 359 is just 2 below 361.
Finding √359 with the Babylonian method
If a guess is too big, 359 ÷ 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√359) in one step.
Start from the nearest whole number, 19 (19² = 361):
| Step | Guess x | 359 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 19.0000000000 | 18.8947368421 | 18.9473684211 | 4 |
| 2 | 18.9473684211 | 18.9472222222 | 18.9472953216 | 9 |
| 3 | 18.9472953216 | 18.9472953214 | 18.9472953215 | all 10 shown |
The count of correct decimals went 4, 9 and all 10 over 3 steps — roughly doubling each time — until the guess matched √359 = 18.9472953215 to every decimal shown.
√359 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √359 the pattern is [18; 1, 17, 1, 36] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √359 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 |
|---|---|---|
| 18/1 | 18.0000000000 | 9.5 × 10⁻¹ |
| 19/1 | 19.0000000000 | 5.3 × 10⁻² |
| 341/18 | 18.9444444444 | 2.9 × 10⁻³ |
| 360/19 | 18.9473684211 | 7.3 × 10⁻⁵ |
| 13,301/702 | 18.9472934473 | 1.9 × 10⁻⁶ |
| 13,661/721 | 18.9472954230 | 1.0 × 10⁻⁷ |
The same fractions solve Pell’s equation, x² − 359y² = 1. Its smallest solution in positive whole numbers is x = 360, y = 19.
√359 in geometry and everyday measurements
- A square patio or deck of 359 square feet is about 18.95 ft (18 ft 11 in) on each side, so edging all the way around takes 4 × √359 ≈ 75.8 ft.
- 359 is not a sum of two whole-number squares — 359 is itself a prime that is one less than a multiple of 4, which rules that out — so √359 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 √359 as its space diagonal.
Square roots near √359 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √356 | 2√89 | 18.8680 | No |
| √357 | √357 | 18.8944 | No |
| √358 | √358 | 18.9209 | No |
| √359 | √359 | 18.9473 | No |
| √360 | 6√10 | 18.9737 | No |
| √361 | 19 | 19.0000 | Yes |
| √362 | √362 | 19.0263 | No |
- The cube root of 359 is about 7.107194.
- Squaring undoes the root: (√359)² = 359, while 359² = 128,881 — the number whose square root is 359.
Frequently asked questions
What is the square root of 359?
The square root of 359 is √359, about 18.9472953215. The negative root, −18.947295, also squares to 359.
Is the square root of 359 rational or irrational?
Irrational. 359 is not a perfect square — it falls between 324 and 361 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √359 be simplified?
No. 359 is prime, so there is no perfect square to take out of the radical.
What is √359 rounded to two decimal places?
√359 ≈ 18.95 to two decimal places (18.9 to one, 18.947 to three). Check: 18.95² = 359.1025, close to 359.