√943 at a glance
- Exact value
- √943
- Decimal (10 places)
- 30.7083050656
- Rounded
- 30.7 · 30.71 · 30.708
- Perfect square?
- No — between 30² and 31²
- Rational?
- Irrational
- Both square roots
- ±30.708305
- Prime factorization
- 23 × 41
- Cube root
- 9.806271
How to simplify √943
The prime factorization of 943 is 23 × 41. Every prime appears only once, so there is no pair to bring outside the radical — √943 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 943, 23 and 41 appear an odd number of times, so √943 is irrational and 30.7083050656 is a rounded value.
Where √943 sits between perfect squares
900 = 30² and 961 = 31² are the nearest perfect squares, so √943 lies between 30 and 31. 943 is 43 above 900 and 18 below 961, so the root is closer to 31.
- Straight line between 900 and 961: 30.7049 (0.01% low)
- Tangent from 30, i.e. 30 + 43 ÷ 60: 30.7167 (0.03% high)
- Tangent from 31, i.e. 31 − 18 ÷ 62: 30.7097 (0% high)
For √943 the tangent at 31 wins, missing by only 0.0014. Tangent estimates shine when the number sits close to a perfect square — here 943 is just 18 below 961.
Finding √943 with the Babylonian method
If a guess is too big, 943 ÷ 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√943) in one step.
Start from the nearest whole number, 31 (31² = 961):
| Step | Guess x | 943 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 31.0000000000 | 30.4193548387 | 30.7096774194 | 2 |
| 2 | 30.7096774194 | 30.7069327731 | 30.7083050962 | 7 |
| 3 | 30.7083050962 | 30.7083050349 | 30.7083050656 | all 10 shown |
The count of correct decimals went 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √943 = 30.7083050656 to every decimal shown.
√943 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √943 the pattern is [30; 1, 2, 2, 2, 1, 60] with the block of 6 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √943 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 |
|---|---|---|
| 30/1 | 30.0000000000 | 7.1 × 10⁻¹ |
| 31/1 | 31.0000000000 | 2.9 × 10⁻¹ |
| 92/3 | 30.6666666667 | 4.2 × 10⁻² |
| 215/7 | 30.7142857143 | 6.0 × 10⁻³ |
| 522/17 | 30.7058823529 | 2.4 × 10⁻³ |
| 737/24 | 30.7083333333 | 2.8 × 10⁻⁵ |
The same fractions solve Pell’s equation, x² − 943y² = 1. Its smallest solution in positive whole numbers is x = 737, y = 24.
√943 in geometry and everyday measurements
- 943 square feet is 87.6 m². Laid out as a square — a small house footprint or a lot — it is about 30.71 ft (30 ft 8 in) on a side.
- 943 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 √943 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 √943 as its space diagonal.
Square roots near √943 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √940 | 2√235 | 30.6594 | No |
| √941 | √941 | 30.6757 | No |
| √942 | √942 | 30.6920 | No |
| √943 | √943 | 30.7083 | No |
| √944 | 4√59 | 30.7246 | No |
| √945 | 3√105 | 30.7409 | No |
| √946 | √946 | 30.7571 | No |
- The cube root of 943 is about 9.806271.
- Squaring undoes the root: (√943)² = 943, while 943² = 889,249 — the number whose square root is 943.
Frequently asked questions
What is the square root of 943?
The square root of 943 is √943, about 30.7083050656. The negative root, −30.708305, also squares to 943.
Is the square root of 943 rational or irrational?
Irrational. 943 is not a perfect square — it falls between 900 and 961 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √943 be simplified?
No. 943 = 23 × 41 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √943 rounded to two decimal places?
√943 ≈ 30.71 to two decimal places (30.7 to one, 30.708 to three). Check: 30.71² = 943.1041, close to 943.