√687 at a glance
- Exact value
- √687
- Decimal (10 places)
- 26.2106848442
- Rounded
- 26.2 · 26.21 · 26.211
- Perfect square?
- No — between 26² and 27²
- Rational?
- Irrational
- Both square roots
- ±26.210685
- Prime factorization
- 3 × 229
- Cube root
- 8.823731
How to simplify √687
The prime factorization of 687 is 3 × 229. Every prime appears only once, so there is no pair to bring outside the radical — √687 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 687, 3 and 229 appear an odd number of times, so √687 is irrational and 26.2106848442 is a rounded value.
Where √687 sits between perfect squares
676 = 26² and 729 = 27² are the nearest perfect squares, so √687 lies between 26 and 27. 687 is 11 above 676 and 42 below 729, so the root is closer to 26.
- Straight line between 676 and 729: 26.2075 (0.01% low)
- Tangent from 26, i.e. 26 + 11 ÷ 52: 26.2115 (0% high)
- Tangent from 27, i.e. 27 − 42 ÷ 54: 26.2222 (0.04% high)
For √687 the tangent at 26 wins, missing by only 0.0009. Tangent estimates shine when the number sits close to a perfect square — here 687 is just 11 above 676.
Finding √687 with the Babylonian method
If a guess is too big, 687 ÷ 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√687) in one step.
Start from the nearest whole number, 26 (26² = 676):
| Step | Guess x | 687 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 26.0000000000 | 26.4230769231 | 26.2115384615 | 3 |
| 2 | 26.2115384615 | 26.2098312546 | 26.2106848581 | 7 |
| 3 | 26.2106848581 | 26.2106848303 | 26.2106848442 | 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 √687 = 26.2106848442 to every decimal shown.
√687 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √687 the pattern is [26; 4, 1, 2, 1, 16, 1, 2, 1, 4, 52] with the block of 10 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √687 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 | 2.1 × 10⁻¹ |
| 105/4 | 26.2500000000 | 3.9 × 10⁻² |
| 131/5 | 26.2000000000 | 1.1 × 10⁻² |
| 367/14 | 26.2142857143 | 3.6 × 10⁻³ |
| 498/19 | 26.2105263158 | 1.6 × 10⁻⁴ |
| 8,335/318 | 26.2106918239 | 7.0 × 10⁻⁶ |
The same fractions solve Pell’s equation, x² − 687y² = 1. Its smallest solution in positive whole numbers is x = 165,337, y = 6,308.
√687 in geometry and everyday measurements
- 687 square feet is 63.8 m². Laid out as a square — a small house footprint or a lot — it is about 26.21 ft (26 ft 3 in) on a side.
- 687 is not a sum of two whole-number squares — the prime factor 3 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √687 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 √687 as its space diagonal.
Square roots near √687 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √684 | 6√19 | 26.1534 | No |
| √685 | √685 | 26.1725 | No |
| √686 | 7√14 | 26.1916 | No |
| √687 | √687 | 26.2107 | No |
| √688 | 4√43 | 26.2298 | No |
| √689 | √689 | 26.2488 | No |
| √690 | √690 | 26.2679 | No |
- The cube root of 687 is about 8.823731.
- Squaring undoes the root: (√687)² = 687, while 687² = 471,969 — the number whose square root is 687.
Frequently asked questions
What is the square root of 687?
The square root of 687 is √687, about 26.2106848442. The negative root, −26.210685, also squares to 687.
Is the square root of 687 rational or irrational?
Irrational. 687 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 √687 be simplified?
No. 687 = 3 × 229 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √687 rounded to two decimal places?
√687 ≈ 26.21 to two decimal places (26.2 to one, 26.211 to three). Check: 26.21² = 686.9641, close to 687.