√59 at a glance
- Exact value
- √59
- Decimal (10 places)
- 7.6811457479
- Rounded
- 7.7 · 7.68 · 7.681
- Perfect square?
- No — between 7² and 8²
- Rational?
- Irrational
- Both square roots
- ±7.681146
- Prime factorization
- 59
- Cube root
- 3.892996
How to simplify √59
59 is a prime number, so its only factors are 1 and 59. There is no perfect-square factor to pull out, which means √59 is already in its simplest radical form.
The square root of any prime is irrational. If √59 were a fraction a/b in lowest terms, then a² = 59b², so 59 would divide a — and then 59 would divide b too, contradicting “lowest terms.” That is why the decimal 7.6811457479 is only a rounded value.
Where √59 sits between perfect squares
49 = 7² and 64 = 8² are the nearest perfect squares, so √59 lies between 7 and 8. 59 is 10 above 49 and 5 below 64, so the root is closer to 8.
- Straight line between 49 and 64: 7.6667 (0.19% low)
- Tangent from 7, i.e. 7 + 10 ÷ 14: 7.7143 (0.43% high)
- Tangent from 8, i.e. 8 − 5 ÷ 16: 7.6875 (0.08% high)
For √59 the tangent at 8 wins, missing by only 0.0064. Tangent estimates shine when the number sits close to a perfect square — here 59 is just 5 below 64.
Finding √59 with the Babylonian method
If a guess is too big, 59 ÷ 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√59) in one step.
Start from the nearest whole number, 8 (8² = 64):
| Step | Guess x | 59 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 8.0000000000 | 7.3750000000 | 7.6875000000 | 2 |
| 2 | 7.6875000000 | 7.6747967480 | 7.6811483740 | 5 |
| 3 | 7.6811483740 | 7.6811431218 | 7.6811457479 | all 10 shown |
The count of correct decimals went 2, 5 and all 10 over 3 steps — roughly doubling each time — until the guess matched √59 = 7.6811457479 to every decimal shown.
√59 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √59 the pattern is [7; 1, 2, 7, 2, 1, 14] with the block of 6 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √59 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 |
|---|---|---|
| 7/1 | 7.0000000000 | 6.8 × 10⁻¹ |
| 8/1 | 8.0000000000 | 3.2 × 10⁻¹ |
| 23/3 | 7.6666666667 | 1.4 × 10⁻² |
| 169/22 | 7.6818181818 | 6.7 × 10⁻⁴ |
| 361/47 | 7.6808510638 | 2.9 × 10⁻⁴ |
| 530/69 | 7.6811594203 | 1.4 × 10⁻⁵ |
The same fractions solve Pell’s equation, x² − 59y² = 1. Its smallest solution in positive whole numbers is x = 530, y = 69.
√59 in geometry and everyday measurements
- A square room or garden bed covering 59 square feet measures about 7.68 ft (7 ft 8 in) along each wall.
- 59 is not a sum of two whole-number squares — 59 is itself a prime that is one less than a multiple of 4, which rules that out — so √59 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 3 × 7 box, because 1² + 3² + 7² = 59.
Square roots near √59 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √56 | 2√14 | 7.4833 | No |
| √57 | √57 | 7.5498 | No |
| √58 | √58 | 7.6158 | No |
| √59 | √59 | 7.6811 | No |
| √60 | 2√15 | 7.7460 | No |
| √61 | √61 | 7.8102 | No |
| √62 | √62 | 7.8740 | No |
- The cube root of 59 is about 3.892996.
- Four times the radicand doubles the root: √236 = 2 × √59 ≈ 15.362291.
Frequently asked questions
What is the square root of 59?
The square root of 59 is √59, about 7.6811457479. The negative root, −7.681146, also squares to 59.
Is the square root of 59 rational or irrational?
Irrational. 59 is not a perfect square — it falls between 49 and 64 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √59 be simplified?
No. 59 is prime, so there is no perfect square to take out of the radical.
What is √59 rounded to two decimal places?
√59 ≈ 7.68 to two decimal places (7.7 to one, 7.681 to three). Check: 7.68² = 58.9824, close to 59.