√206 at a glance
- Exact value
- √206
- Decimal (10 places)
- 14.3527000944
- Rounded
- 14.4 · 14.35 · 14.353
- Perfect square?
- No — between 14² and 15²
- Rational?
- Irrational
- Both square roots
- ±14.352700
- Prime factorization
- 2 × 103
- Cube root
- 5.905941
How to simplify √206
The prime factorization of 206 is 2 × 103. Every prime appears only once, so there is no pair to bring outside the radical — √206 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 206, 2 and 103 appear an odd number of times, so √206 is irrational and 14.3527000944 is a rounded value.
Where √206 sits between perfect squares
196 = 14² and 225 = 15² are the nearest perfect squares, so √206 lies between 14 and 15. 206 is 10 above 196 and 19 below 225, so the root is closer to 14.
- Straight line between 196 and 225: 14.3448 (0.05% low)
- Tangent from 14, i.e. 14 + 10 ÷ 28: 14.3571 (0.03% high)
- Tangent from 15, i.e. 15 − 19 ÷ 30: 14.3667 (0.1% high)
For √206 the tangent at 14 wins, missing by only 0.0044. Tangent estimates shine when the number sits close to a perfect square — here 206 is just 10 above 196.
Finding √206 with the Babylonian method
The idea is ancient. A Babylonian clay tablet catalogued as YBC 7289, about 3,800 years old, records √2 correct to roughly six decimal places, and Heron of Alexandria described this averaging rule in the first century AD.
Start from the nearest whole number, 14 (14² = 196):
| Step | Guess x | 206 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 14.0000000000 | 14.7142857143 | 14.3571428571 | 2 |
| 2 | 14.3571428571 | 14.3482587065 | 14.3527007818 | 6 |
| 3 | 14.3527007818 | 14.3526994070 | 14.3527000944 | all 10 shown |
The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √206 = 14.3527000944 to every decimal shown.
√206 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √206 the pattern is [14; 2, 1, 5, 14, 5, 1, 2, 28] with the block of 8 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √206 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 |
|---|---|---|
| 14/1 | 14.0000000000 | 3.5 × 10⁻¹ |
| 29/2 | 14.5000000000 | 1.5 × 10⁻¹ |
| 43/3 | 14.3333333333 | 1.9 × 10⁻² |
| 244/17 | 14.3529411765 | 2.4 × 10⁻⁴ |
| 3,459/241 | 14.3526970954 | 3.0 × 10⁻⁶ |
| 17,539/1,222 | 14.3527004910 | 4.0 × 10⁻⁷ |
The same fractions solve Pell’s equation, x² − 206y² = 1. Its smallest solution in positive whole numbers is x = 59,535, y = 4,148.
√206 in geometry and everyday measurements
- A square patio or deck of 206 square feet is about 14.35 ft (14 ft 4 in) on each side, so edging all the way around takes 4 × √206 ≈ 57.4 ft.
- 206 is not a sum of two whole-number squares — the prime factor 103 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √206 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 3 × 14 box, because 1² + 3² + 14² = 206.
Square roots near √206 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √203 | √203 | 14.2478 | No |
| √204 | 2√51 | 14.2829 | No |
| √205 | √205 | 14.3178 | No |
| √206 | √206 | 14.3527 | No |
| √207 | 3√23 | 14.3875 | No |
| √208 | 4√13 | 14.4222 | No |
| √209 | √209 | 14.4568 | No |
- The cube root of 206 is about 5.905941.
- Four times the radicand doubles the root: √824 = 2 × √206 ≈ 28.7054.
Frequently asked questions
What is the square root of 206?
The square root of 206 is √206, about 14.3527000944. The negative root, −14.352700, also squares to 206.
Is the square root of 206 rational or irrational?
Irrational. 206 is not a perfect square — it falls between 196 and 225 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √206 be simplified?
No. 206 = 2 × 103 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √206 rounded to two decimal places?
√206 ≈ 14.35 to two decimal places (14.4 to one, 14.353 to three). Check: 14.35² = 205.9225, close to 206.