√667 at a glance
- Exact value
- √667
- Decimal (10 places)
- 25.8263431403
- Rounded
- 25.8 · 25.83 · 25.826
- Perfect square?
- No — between 25² and 26²
- Rational?
- Irrational
- Both square roots
- ±25.826343
- Prime factorization
- 23 × 29
- Cube root
- 8.737260
How to simplify √667
The prime factorization of 667 is 23 × 29. Every prime appears only once, so there is no pair to bring outside the radical — √667 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 667, 23 and 29 appear an odd number of times, so √667 is irrational and 25.8263431403 is a rounded value.
Where √667 sits between perfect squares
625 = 25² and 676 = 26² are the nearest perfect squares, so √667 lies between 25 and 26. 667 is 42 above 625 and 9 below 676, so the root is closer to 26.
- Straight line between 625 and 676: 25.8235 (0.01% low)
- Tangent from 25, i.e. 25 + 42 ÷ 50: 25.8400 (0.05% high)
- Tangent from 26, i.e. 26 − 9 ÷ 52: 25.8269 (0% high)
For √667 the tangent at 26 wins, missing by only 0.0006. Tangent estimates shine when the number sits close to a perfect square — here 667 is just 9 below 676.
Finding √667 with the Babylonian method
If a guess is too big, 667 ÷ 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√667) in one step.
Start from the nearest whole number, 26 (26² = 676):
| Step | Guess x | 667 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 26.0000000000 | 25.6538461538 | 25.8269230769 | 3 |
| 2 | 25.8269230769 | 25.8257632167 | 25.8263431468 | 8 |
| 3 | 25.8263431468 | 25.8263431338 | 25.8263431403 | all 10 shown |
The count of correct decimals went 3, 8 and all 10 over 3 steps — roughly doubling each time — until the guess matched √667 = 25.8263431403 to every decimal shown.
√667 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √667 the pattern is [25; 1, 4, 1, 3, 7, 8, 2, 8, 7, 3, 1, 4, …] with the block of 14 terms after the semicolon repeating forever (only the first 12 of the 14 are shown). A pattern that never ends is one more proof that √667 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 |
|---|---|---|
| 25/1 | 25.0000000000 | 8.3 × 10⁻¹ |
| 26/1 | 26.0000000000 | 1.7 × 10⁻¹ |
| 129/5 | 25.8000000000 | 2.6 × 10⁻² |
| 155/6 | 25.8333333333 | 7.0 × 10⁻³ |
| 594/23 | 25.8260869565 | 2.6 × 10⁻⁴ |
| 4,313/167 | 25.8263473054 | 4.2 × 10⁻⁶ |
The same fractions solve Pell’s equation, x² − 667y² = 1. Its smallest solution in positive whole numbers is x = 107,119,097, y = 4,147,668.
√667 in geometry and everyday measurements
- 667 square feet is 62 m². Laid out as a square — a small house footprint or a lot — it is about 25.83 ft (25 ft 10 in) on a side.
- 667 is not a sum of two whole-number squares — the prime factor 23 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √667 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 15 × 21 box, because 1² + 15² + 21² = 667.
Square roots near √667 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √664 | 2√166 | 25.7682 | No |
| √665 | √665 | 25.7876 | No |
| √666 | 3√74 | 25.8070 | No |
| √667 | √667 | 25.8263 | No |
| √668 | 2√167 | 25.8457 | No |
| √669 | √669 | 25.8650 | No |
| √670 | √670 | 25.8844 | No |
- The cube root of 667 is about 8.737260.
- Squaring undoes the root: (√667)² = 667, while 667² = 444,889 — the number whose square root is 667.
Frequently asked questions
What is the square root of 667?
The square root of 667 is √667, about 25.8263431403. The negative root, −25.826343, also squares to 667.
Is the square root of 667 rational or irrational?
Irrational. 667 is not a perfect square — it falls between 625 and 676 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √667 be simplified?
No. 667 = 23 × 29 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √667 rounded to two decimal places?
√667 ≈ 25.83 to two decimal places (25.8 to one, 25.826 to three). Check: 25.83² = 667.1889, close to 667.