√969 at a glance
- Exact value
- √969
- Decimal (10 places)
- 31.1287648325
- Rounded
- 31.1 · 31.13 · 31.129
- Perfect square?
- No — between 31² and 32²
- Rational?
- Irrational
- Both square roots
- ±31.128765
- Prime factorization
- 3 × 17 × 19
- Cube root
- 9.895580
How to simplify √969
The prime factorization of 969 is 3 × 17 × 19. Every prime appears only once, so there is no pair to bring outside the radical — √969 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 969, 3, 17 and 19 appear an odd number of times, so √969 is irrational and 31.1287648325 is a rounded value.
Where √969 sits between perfect squares
961 = 31² and 1,024 = 32² are the nearest perfect squares, so √969 lies between 31 and 32. 969 is 8 above 961 and 55 below 1,024, so the root is closer to 31.
- Straight line between 961 and 1,024: 31.1270 (0.01% low)
- Tangent from 31, i.e. 31 + 8 ÷ 62: 31.1290 (0% high)
- Tangent from 32, i.e. 32 − 55 ÷ 64: 31.1406 (0.04% high)
For √969 the tangent at 31 wins, missing by only 0.0003. Tangent estimates shine when the number sits close to a perfect square — here 969 is just 8 above 961.
Finding √969 with the Babylonian method
This is Newton’s method applied to f(x) = x² − 969: following the tangent line down to zero simplifies to averaging x with 969 ÷ x.
Start from the nearest whole number, 31 (31² = 961):
| Step | Guess x | 969 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 31.0000000000 | 31.2580645161 | 31.1290322581 | 3 |
| 2 | 31.1290322581 | 31.1284974093 | 31.1287648337 | 8 |
| 3 | 31.1287648337 | 31.1287648314 | 31.1287648325 | 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 √969 = 31.1287648325 to every decimal shown.
√969 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √969 the pattern is [31; 7, 1, 3, 3, 1, 1, 1, 2, 1, 1, 1, 3, …] 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 √969 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 |
|---|---|---|
| 31/1 | 31.0000000000 | 1.3 × 10⁻¹ |
| 218/7 | 31.1428571429 | 1.4 × 10⁻² |
| 249/8 | 31.1250000000 | 3.8 × 10⁻³ |
| 965/31 | 31.1290322581 | 2.7 × 10⁻⁴ |
| 3,144/101 | 31.1287128713 | 5.2 × 10⁻⁵ |
| 4,109/132 | 31.1287878788 | 2.3 × 10⁻⁵ |
The same fractions solve Pell’s equation, x² − 969y² = 1. Its smallest solution in positive whole numbers is x = 13,588,951, y = 436,540.
√969 in geometry and everyday measurements
- 969 square feet is 90 m². Laid out as a square — a small house footprint or a lot — it is about 31.13 ft (31 ft 2 in) on a side.
- 969 is not a sum of two whole-number squares — the prime factor 3 and 19 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √969 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 22 × 22 box, because 1² + 22² + 22² = 969.
Square roots near √969 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √966 | √966 | 31.0805 | No |
| √967 | √967 | 31.0966 | No |
| √968 | 22√2 | 31.1127 | No |
| √969 | √969 | 31.1288 | No |
| √970 | √970 | 31.1448 | No |
| √971 | √971 | 31.1609 | No |
| √972 | 18√3 | 31.1769 | No |
- The cube root of 969 is about 9.895580.
- Squaring undoes the root: (√969)² = 969, while 969² = 938,961 — the number whose square root is 969.
Frequently asked questions
What is the square root of 969?
The square root of 969 is √969, about 31.1287648325. The negative root, −31.128765, also squares to 969.
Is the square root of 969 rational or irrational?
Irrational. 969 is not a perfect square — it falls between 961 and 1024 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √969 be simplified?
No. 969 = 3 × 17 × 19 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √969 rounded to two decimal places?
√969 ≈ 31.13 to two decimal places (31.1 to one, 31.129 to three). Check: 31.13² = 969.0769, close to 969.