√679 at a glance
- Exact value
- √679
- Decimal (10 places)
- 26.0576284416
- Rounded
- 26.1 · 26.06 · 26.058
- Perfect square?
- No — between 26² and 27²
- Rational?
- Irrational
- Both square roots
- ±26.057628
- Prime factorization
- 7 × 97
- Cube root
- 8.789347
How to simplify √679
The prime factorization of 679 is 7 × 97. Every prime appears only once, so there is no pair to bring outside the radical — √679 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 679, 7 and 97 appear an odd number of times, so √679 is irrational and 26.0576284416 is a rounded value.
Where √679 sits between perfect squares
676 = 26² and 729 = 27² are the nearest perfect squares, so √679 lies between 26 and 27. 679 is 3 above 676 and 50 below 729, so the root is closer to 26.
- Straight line between 676 and 729: 26.0566 (0% low)
- Tangent from 26, i.e. 26 + 3 ÷ 52: 26.0577 (0% high)
- Tangent from 27, i.e. 27 − 50 ÷ 54: 26.0741 (0.06% high)
For √679 the tangent at 26 wins, missing by only 0.0001. Tangent estimates shine when the number sits close to a perfect square — here 679 is just 3 above 676.
Finding √679 with the Babylonian method
If a guess is too big, 679 ÷ 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√679) in one step.
Start from the nearest whole number, 26 (26² = 676):
| Step | Guess x | 679 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 26.0000000000 | 26.1153846154 | 26.0576923077 | 4 |
| 2 | 26.0576923077 | 26.0575645756 | 26.0576284417 | 10 |
| 3 | 26.0576284417 | 26.0576284415 | 26.0576284416 | all 10 shown |
The count of correct decimals went 4, 10 and all 10 over 3 steps — roughly doubling each time — until the guess matched √679 = 26.0576284416 to every decimal shown.
√679 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √679 the pattern is [26; 17, 2, 1, 5, 8, 1, 1, 25, 1, 1, 8, 5, …] with the block of 16 terms after the semicolon repeating forever (only the first 12 of the 16 are shown). A pattern that never ends is one more proof that √679 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 | 5.8 × 10⁻² |
| 443/17 | 26.0588235294 | 1.2 × 10⁻³ |
| 912/35 | 26.0571428571 | 4.9 × 10⁻⁴ |
| 1,355/52 | 26.0576923077 | 6.4 × 10⁻⁵ |
| 7,687/295 | 26.0576271186 | 1.3 × 10⁻⁶ |
| 62,851/2,412 | 26.0576285240 | 8.2 × 10⁻⁸ |
The same fractions solve Pell’s equation, x² − 679y² = 1. Its smallest solution in positive whole numbers is x = 17,792,625,320, y = 682,818,291.
√679 in geometry and everyday measurements
- 679 square feet is 63.1 m². Laid out as a square — a small house footprint or a lot — it is about 26.06 ft (26 ft 1 in) on a side.
- 679 is not a sum of two whole-number squares — the prime factor 7 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √679 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 √679 as its space diagonal.
Square roots near √679 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √676 | 26 | 26.0000 | Yes |
| √677 | √677 | 26.0192 | No |
| √678 | √678 | 26.0384 | No |
| √679 | √679 | 26.0576 | No |
| √680 | 2√170 | 26.0768 | No |
| √681 | √681 | 26.0960 | No |
| √682 | √682 | 26.1151 | No |
- The cube root of 679 is about 8.789347.
- Squaring undoes the root: (√679)² = 679, while 679² = 461,041 — the number whose square root is 679.
Frequently asked questions
What is the square root of 679?
The square root of 679 is √679, about 26.0576284416. The negative root, −26.057628, also squares to 679.
Is the square root of 679 rational or irrational?
Irrational. 679 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 √679 be simplified?
No. 679 = 7 × 97 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √679 rounded to two decimal places?
√679 ≈ 26.06 to two decimal places (26.1 to one, 26.058 to three). Check: 26.06² = 679.1236, close to 679.