√183 at a glance
- Exact value
- √183
- Decimal (10 places)
- 13.5277492585
- Rounded
- 13.5 · 13.53 · 13.528
- Perfect square?
- No — between 13² and 14²
- Rational?
- Irrational
- Both square roots
- ±13.527749
- Prime factorization
- 3 × 61
- Cube root
- 5.677411
How to simplify √183
The prime factorization of 183 is 3 × 61. Every prime appears only once, so there is no pair to bring outside the radical — √183 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 183, 3 and 61 appear an odd number of times, so √183 is irrational and 13.5277492585 is a rounded value.
Where √183 sits between perfect squares
169 = 13² and 196 = 14² are the nearest perfect squares, so √183 lies between 13 and 14. 183 is 14 above 169 and 13 below 196, so the root is closer to 14.
- Straight line between 169 and 196: 13.5185 (0.07% low)
- Tangent from 13, i.e. 13 + 14 ÷ 26: 13.5385 (0.08% high)
- Tangent from 14, i.e. 14 − 13 ÷ 28: 13.5357 (0.06% high)
For √183 the tangent at 14 wins, missing by only 0.008. Tangent estimates shine when the number sits close to a perfect square — here 183 is just 13 below 196.
Finding √183 with the Babylonian method
If a guess is too big, 183 ÷ 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√183) in one step.
Start from the nearest whole number, 14 (14² = 196):
| Step | Guess x | 183 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 14.0000000000 | 13.0714285714 | 13.5357142857 | 2 |
| 2 | 13.5357142857 | 13.5197889182 | 13.5277516020 | 5 |
| 3 | 13.5277516020 | 13.5277469150 | 13.5277492585 | all 10 shown |
The count of correct decimals went 2, 5 and all 10 over 3 steps — roughly doubling each time — until the guess matched √183 = 13.5277492585 to every decimal shown.
√183 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √183 the pattern is [13; 1, 1, 8, 1, 1, 26] with the block of 6 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √183 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 |
|---|---|---|
| 13/1 | 13.0000000000 | 5.3 × 10⁻¹ |
| 14/1 | 14.0000000000 | 4.7 × 10⁻¹ |
| 27/2 | 13.5000000000 | 2.8 × 10⁻² |
| 230/17 | 13.5294117647 | 1.7 × 10⁻³ |
| 257/19 | 13.5263157895 | 1.4 × 10⁻³ |
| 487/36 | 13.5277777778 | 2.9 × 10⁻⁵ |
The same fractions solve Pell’s equation, x² − 183y² = 1. Its smallest solution in positive whole numbers is x = 487, y = 36.
√183 in geometry and everyday measurements
- A square room or garden bed covering 183 square feet measures about 13.53 ft (13 ft 6 in) along each wall.
- 183 is not a sum of two whole-number squares — the prime factor 3 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √183 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 √183 as its space diagonal.
Square roots near √183 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √180 | 6√5 | 13.4164 | No |
| √181 | √181 | 13.4536 | No |
| √182 | √182 | 13.4907 | No |
| √183 | √183 | 13.5277 | No |
| √184 | 2√46 | 13.5647 | No |
| √185 | √185 | 13.6015 | No |
| √186 | √186 | 13.6382 | No |
- The cube root of 183 is about 5.677411.
- Four times the radicand doubles the root: √732 = 2 × √183 ≈ 27.055499.
Frequently asked questions
What is the square root of 183?
The square root of 183 is √183, about 13.5277492585. The negative root, −13.527749, also squares to 183.
Is the square root of 183 rational or irrational?
Irrational. 183 is not a perfect square — it falls between 169 and 196 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √183 be simplified?
No. 183 = 3 × 61 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √183 rounded to two decimal places?
√183 ≈ 13.53 to two decimal places (13.5 to one, 13.528 to three). Check: 13.53² = 183.0609, close to 183.