√690 at a glance
- Exact value
- √690
- Decimal (10 places)
- 26.2678510731
- Rounded
- 26.3 · 26.27 · 26.268
- Perfect square?
- No — between 26² and 27²
- Rational?
- Irrational
- Both square roots
- ±26.267851
- Prime factorization
- 2 × 3 × 5 × 23
- Cube root
- 8.836556
How to simplify √690
The prime factorization of 690 is 2 × 3 × 5 × 23. Every prime appears only once, so there is no pair to bring outside the radical — √690 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 690, 2, 3, 5 and 23 appear an odd number of times, so √690 is irrational and 26.2678510731 is a rounded value.
Where √690 sits between perfect squares
676 = 26² and 729 = 27² are the nearest perfect squares, so √690 lies between 26 and 27. 690 is 14 above 676 and 39 below 729, so the root is closer to 26.
- Straight line between 676 and 729: 26.2642 (0.01% low)
- Tangent from 26, i.e. 26 + 14 ÷ 52: 26.2692 (0.01% high)
- Tangent from 27, i.e. 27 − 39 ÷ 54: 26.2778 (0.04% high)
For √690 the tangent at 26 wins, missing by only 0.0014. Tangent estimates shine when the number sits close to a perfect square — here 690 is just 14 above 676.
Finding √690 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, 26 (26² = 676):
| Step | Guess x | 690 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 26.0000000000 | 26.5384615385 | 26.2692307692 | 2 |
| 2 | 26.2692307692 | 26.2664714495 | 26.2678511094 | 7 |
| 3 | 26.2678511094 | 26.2678510369 | 26.2678510731 | all 10 shown |
The count of correct decimals went 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √690 = 26.2678510731 to every decimal shown.
√690 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √690 the pattern is [26; 3, 1, 2, 1, 3, 52] with the block of 6 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √690 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 |
|---|---|---|
| 26/1 | 26.0000000000 | 2.7 × 10⁻¹ |
| 79/3 | 26.3333333333 | 6.5 × 10⁻² |
| 105/4 | 26.2500000000 | 1.8 × 10⁻² |
| 289/11 | 26.2727272727 | 4.9 × 10⁻³ |
| 394/15 | 26.2666666667 | 1.2 × 10⁻³ |
| 1,471/56 | 26.2678571429 | 6.1 × 10⁻⁶ |
The same fractions solve Pell’s equation, x² − 690y² = 1. Its smallest solution in positive whole numbers is x = 1,471, y = 56.
√690 in geometry and everyday measurements
- 690 square feet is 64.1 m². Laid out as a square — a small house footprint or a lot — it is about 26.27 ft (26 ft 3 in) on a side.
- 690 is not a sum of two whole-number squares — the prime factor 3 and 23 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √690 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 8 × 25 box, because 1² + 8² + 25² = 690.
Square roots near √690 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √687 | √687 | 26.2107 | No |
| √688 | 4√43 | 26.2298 | No |
| √689 | √689 | 26.2488 | No |
| √690 | √690 | 26.2679 | No |
| √691 | √691 | 26.2869 | No |
| √692 | 2√173 | 26.3059 | No |
| √693 | 3√77 | 26.3249 | No |
- The cube root of 690 is about 8.836556.
- Squaring undoes the root: (√690)² = 690, while 690² = 476,100 — the number whose square root is 690.
Frequently asked questions
What is the square root of 690?
The square root of 690 is √690, about 26.2678510731. The negative root, −26.267851, also squares to 690.
Is the square root of 690 rational or irrational?
Irrational. 690 is not a perfect square — it falls between 676 and 729 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √690 be simplified?
No. 690 = 2 × 3 × 5 × 23 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √690 rounded to two decimal places?
√690 ≈ 26.27 to two decimal places (26.3 to one, 26.268 to three). Check: 26.27² = 690.1129, close to 690.