√13 at a glance
- Exact value
- √13
- Decimal (10 places)
- 3.6055512755
- Rounded
- 3.6 · 3.61 · 3.606
- Perfect square?
- No — between 3² and 4²
- Rational?
- Irrational
- Both square roots
- ±3.605551
- Prime factorization
- 13
- Cube root
- 2.351335
How to simplify √13
13 is a prime number, so its only factors are 1 and 13. There is no perfect-square factor to pull out, which means √13 is already in its simplest radical form.
The square root of any prime is irrational. If √13 were a fraction a/b in lowest terms, then a² = 13b², so 13 would divide a — and then 13 would divide b too, contradicting “lowest terms.” That is why the decimal 3.6055512755 is only a rounded value.
Where √13 sits between perfect squares
9 = 3² and 16 = 4² are the nearest perfect squares, so √13 lies between 3 and 4. 13 is 4 above 9 and 3 below 16, so the root is closer to 4.
- Straight line between 9 and 16: 3.5714 (0.95% low)
- Tangent from 3, i.e. 3 + 4 ÷ 6: 3.6667 (1.7% high)
- Tangent from 4, i.e. 4 − 3 ÷ 8: 3.6250 (0.54% high)
For √13 the tangent at 4 wins, missing by only 0.0194. Tangent estimates shine when the number sits close to a perfect square — here 13 is just 3 below 16.
Finding √13 with the Babylonian method
This is Newton’s method applied to f(x) = x² − 13: following the tangent line down to zero simplifies to averaging x with 13 ÷ x.
Start from the nearest whole number, 4 (4² = 16):
| Step | Guess x | 13 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 4.0000000000 | 3.2500000000 | 3.6250000000 | 1 |
| 2 | 3.6250000000 | 3.5862068966 | 3.6056034483 | 4 |
| 3 | 3.6056034483 | 3.6054991034 | 3.6055512758 | 9 |
| 4 | 3.6055512758 | 3.6055512751 | 3.6055512755 | all 10 shown |
The count of correct decimals went 1, 4, 9 and all 10 over 4 steps — roughly doubling each time — until the guess matched √13 = 3.6055512755 to every decimal shown.
√13 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √13 the pattern is [3; 1, 1, 1, 1, 6] with the block of 5 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √13 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 |
|---|---|---|
| 3/1 | 3.0000000000 | 6.1 × 10⁻¹ |
| 4/1 | 4.0000000000 | 3.9 × 10⁻¹ |
| 7/2 | 3.5000000000 | 1.1 × 10⁻¹ |
| 11/3 | 3.6666666667 | 6.1 × 10⁻² |
| 18/5 | 3.6000000000 | 5.6 × 10⁻³ |
| 119/33 | 3.6060606061 | 5.1 × 10⁻⁴ |
The same fractions solve Pell’s equation, x² − 13y² = 1. Its smallest solution in positive whole numbers is x = 649, y = 180. Because the period is odd, the equation with −1 on the right also has a solution: 18² − 13 × 5² = −1.
√13 in geometry and everyday measurements
- A square tile with an area of 13 square inches has sides about 3.606 in long.
- 13 = 2² + 3², so by the Pythagorean theorem √13 is the diagonal of a 2 × 3 rectangle — and the distance between the points (0, 0) and (2, 3) on a grid.
Square roots near √13 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √10 | √10 | 3.1623 | No |
| √11 | √11 | 3.3166 | No |
| √12 | 2√3 | 3.4641 | No |
| √13 | √13 | 3.6056 | No |
| √14 | √14 | 3.7417 | No |
| √15 | √15 | 3.8730 | No |
| √16 | 4 | 4.0000 | Yes |
- The cube root of 13 is about 2.351335.
- Four times the radicand doubles the root: √52 = 2 × √13 ≈ 7.211103.
Frequently asked questions
What is the square root of 13?
The square root of 13 is √13, about 3.6055512755. The negative root, −3.605551, also squares to 13.
Is the square root of 13 rational or irrational?
Irrational. 13 is not a perfect square — it falls between 9 and 16 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √13 be simplified?
No. 13 is prime, so there is no perfect square to take out of the radical.
What is √13 rounded to two decimal places?
√13 ≈ 3.61 to two decimal places (3.6 to one, 3.606 to three). Check: 3.61² = 13.0321, close to 13.