√26 at a glance
- Exact value
- √26
- Decimal (10 places)
- 5.0990195136
- Rounded
- 5.1 · 5.10 · 5.099
- Perfect square?
- No — between 5² and 6²
- Rational?
- Irrational
- Both square roots
- ±5.099020
- Prime factorization
- 2 × 13
- Cube root
- 2.962496
How to simplify √26
The prime factorization of 26 is 2 × 13. Every prime appears only once, so there is no pair to bring outside the radical — √26 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 26, 2 and 13 appear an odd number of times, so √26 is irrational and 5.0990195136 is a rounded value.
Where √26 sits between perfect squares
25 = 5² and 36 = 6² are the nearest perfect squares, so √26 lies between 5 and 6. 26 is 1 above 25 and 10 below 36, so the root is closer to 5.
- Straight line between 25 and 36: 5.0909 (0.16% low)
- Tangent from 5, i.e. 5 + 1 ÷ 10: 5.1000 (0.02% high)
- Tangent from 6, i.e. 6 − 10 ÷ 12: 5.1667 (1.33% high)
For √26 the tangent at 5 wins, missing by only 0.001. Tangent estimates shine when the number sits close to a perfect square — here 26 is just 1 above 25.
Finding √26 with the Babylonian method
The idea is ancient. A Babylonian clay tablet catalogued as YBC 7289, about 3,800 years old, records √2 correct to roughly six decimal places, and Heron of Alexandria described this averaging rule in the first century AD.
Start from the nearest whole number, 5 (5² = 25):
| Step | Guess x | 26 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 5.0000000000 | 5.2000000000 | 5.1000000000 | 3 |
| 2 | 5.1000000000 | 5.0980392157 | 5.0990196078 | 7 |
| 3 | 5.0990196078 | 5.0990194193 | 5.0990195136 | all 10 shown |
The count of correct decimals went 3, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √26 = 5.0990195136 to every decimal shown.
√26 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √26 the pattern is [5; 10] with the block of 1 term after the semicolon repeating forever — the simplest possible pattern, which happens exactly when 26 is one more than a perfect square (5² + 1). A pattern that never ends is one more proof that √26 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 |
|---|---|---|
| 5/1 | 5.0000000000 | 9.9 × 10⁻² |
| 51/10 | 5.1000000000 | 9.8 × 10⁻⁴ |
| 515/101 | 5.0990099010 | 9.6 × 10⁻⁶ |
| 5,201/1,020 | 5.0990196078 | 9.4 × 10⁻⁸ |
The same fractions solve Pell’s equation, x² − 26y² = 1. Its smallest solution in positive whole numbers is x = 51, y = 10. Because the period is odd, the equation with −1 on the right also has a solution: 5² − 26 × 1² = −1.
√26 in geometry and everyday measurements
- A square tile with an area of 26 square inches has sides about 5.099 in long.
- 26 = 1² + 5², so by the Pythagorean theorem √26 is the diagonal of a 1 × 5 rectangle — and the distance between the points (0, 0) and (1, 5) on a grid.
Square roots near √26 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √23 | √23 | 4.7958 | No |
| √24 | 2√6 | 4.8990 | No |
| √25 | 5 | 5.0000 | Yes |
| √26 | √26 | 5.0990 | No |
| √27 | 3√3 | 5.1962 | No |
| √28 | 2√7 | 5.2915 | No |
| √29 | √29 | 5.3852 | No |
- The cube root of 26 is about 2.962496.
- Four times the radicand doubles the root: √104 = 2 × √26 ≈ 10.198039.
Frequently asked questions
What is the square root of 26?
The square root of 26 is √26, about 5.0990195136. The negative root, −5.099020, also squares to 26.
Is the square root of 26 rational or irrational?
Irrational. 26 is not a perfect square — it falls between 25 and 36 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √26 be simplified?
No. 26 = 2 × 13 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √26 rounded to two decimal places?
√26 ≈ 5.10 to two decimal places (5.1 to one, 5.099 to three). Check: 5.10² = 26.01, close to 26.