√669 at a glance
- Exact value
- √669
- Decimal (10 places)
- 25.8650343128
- Rounded
- 25.9 · 25.87 · 25.865
- Perfect square?
- No — between 25² and 26²
- Rational?
- Irrational
- Both square roots
- ±25.865034
- Prime factorization
- 3 × 223
- Cube root
- 8.745985
How to simplify √669
The prime factorization of 669 is 3 × 223. Every prime appears only once, so there is no pair to bring outside the radical — √669 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 669, 3 and 223 appear an odd number of times, so √669 is irrational and 25.8650343128 is a rounded value.
Where √669 sits between perfect squares
625 = 25² and 676 = 26² are the nearest perfect squares, so √669 lies between 25 and 26. 669 is 44 above 625 and 7 below 676, so the root is closer to 26.
- Straight line between 625 and 676: 25.8627 (0.01% low)
- Tangent from 25, i.e. 25 + 44 ÷ 50: 25.8800 (0.06% high)
- Tangent from 26, i.e. 26 − 7 ÷ 52: 25.8654 (0% high)
For √669 the tangent at 26 wins, missing by only 0.0004. Tangent estimates shine when the number sits close to a perfect square — here 669 is just 7 below 676.
Finding √669 with the Babylonian method
This is Newton’s method applied to f(x) = x² − 669: following the tangent line down to zero simplifies to averaging x with 669 ÷ x.
Start from the nearest whole number, 26 (26² = 676):
| Step | Guess x | 669 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 26.0000000000 | 25.7307692308 | 25.8653846154 | 3 |
| 2 | 25.8653846154 | 25.8646840149 | 25.8650343151 | 8 |
| 3 | 25.8650343151 | 25.8650343104 | 25.8650343128 | 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 √669 = 25.8650343128 to every decimal shown.
√669 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √669 the pattern is [25; 1, 6, 2, 2, 3, 1, 9, 1, 1, 2, 1, 12, …] with the block of 38 terms after the semicolon repeating forever (only the first 12 of the 38 are shown). A pattern that never ends is one more proof that √669 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 | 8.7 × 10⁻¹ |
| 26/1 | 26.0000000000 | 1.3 × 10⁻¹ |
| 181/7 | 25.8571428571 | 7.9 × 10⁻³ |
| 388/15 | 25.8666666667 | 1.6 × 10⁻³ |
| 957/37 | 25.8648648649 | 1.7 × 10⁻⁴ |
| 3,259/126 | 25.8650793651 | 4.5 × 10⁻⁵ |
The same fractions solve Pell’s equation, x² − 669y² = 1. Its smallest solution in positive whole numbers is x = 14,226,117,859,054,135, y = 550,013,492,618,436 — 17 digits for x, even though 669 is small, which is what makes Pell’s equation famous.
√669 in geometry and everyday measurements
- 669 square feet is 62.2 m². Laid out as a square — a small house footprint or a lot — it is about 25.87 ft (25 ft 10 in) on a side.
- 669 is not a sum of two whole-number squares — the prime factor 3 and 223 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √669 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 4 × 13 × 22 box, because 4² + 13² + 22² = 669.
Square roots near √669 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √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 |
| √671 | √671 | 25.9037 | No |
| √672 | 4√42 | 25.9230 | No |
- The cube root of 669 is about 8.745985.
- Squaring undoes the root: (√669)² = 669, while 669² = 447,561 — the number whose square root is 669.
Frequently asked questions
What is the square root of 669?
The square root of 669 is √669, about 25.8650343128. The negative root, −25.865034, also squares to 669.
Is the square root of 669 rational or irrational?
Irrational. 669 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 √669 be simplified?
No. 669 = 3 × 223 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √669 rounded to two decimal places?
√669 ≈ 25.87 to two decimal places (25.9 to one, 25.865 to three). Check: 25.87² = 669.2569, close to 669.