√102 at a glance
- Exact value
- √102
- Decimal (10 places)
- 10.0995049384
- Rounded
- 10.1 · 10.10 · 10.100
- Perfect square?
- No — between 10² and 11²
- Rational?
- Irrational
- Both square roots
- ±10.099505
- Prime factorization
- 2 × 3 × 17
- Cube root
- 4.672329
How to simplify √102
The prime factorization of 102 is 2 × 3 × 17. Every prime appears only once, so there is no pair to bring outside the radical — √102 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 102, 2, 3 and 17 appear an odd number of times, so √102 is irrational and 10.0995049384 is a rounded value.
Where √102 sits between perfect squares
100 = 10² and 121 = 11² are the nearest perfect squares, so √102 lies between 10 and 11. 102 is 2 above 100 and 19 below 121, so the root is closer to 10.
- Straight line between 100 and 121: 10.0952 (0.04% low)
- Tangent from 10, i.e. 10 + 2 ÷ 20: 10.1000 (0% high)
- Tangent from 11, i.e. 11 − 19 ÷ 22: 10.1364 (0.36% high)
For √102 the tangent at 10 wins, missing by only 0.0005. Tangent estimates shine when the number sits close to a perfect square — here 102 is just 2 above 100.
Finding √102 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, 10 (10² = 100):
| Step | Guess x | 102 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 10.0000000000 | 10.2000000000 | 10.1000000000 | 3 |
| 2 | 10.1000000000 | 10.0990099010 | 10.0995049505 | 7 |
| 3 | 10.0995049505 | 10.0995049262 | 10.0995049384 | all 10 shown |
The count of correct decimals went 3, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √102 = 10.0995049384 to every decimal shown.
√102 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √102 the pattern is [10; 10, 20] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √102 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 |
|---|---|---|
| 10/1 | 10.0000000000 | 1.0 × 10⁻¹ |
| 101/10 | 10.1000000000 | 5.0 × 10⁻⁴ |
| 2,030/201 | 10.0995024876 | 2.5 × 10⁻⁶ |
| 20,401/2,020 | 10.0995049505 | 1.2 × 10⁻⁸ |
| 410,050/40,601 | 10.0995049383 | 6.0 × 10⁻¹¹ |
| 4,120,901/408,030 | 10.0995049384 | < 10⁻¹⁰ |
The same fractions solve Pell’s equation, x² − 102y² = 1. Its smallest solution in positive whole numbers is x = 101, y = 10.
√102 in geometry and everyday measurements
- A square room or garden bed covering 102 square feet measures about 10.1 ft (10 ft 1 in) along each wall.
- 102 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 √102 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 1 × 10 box, because 1² + 1² + 10² = 102.
Square roots near √102 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √99 | 3√11 | 9.9499 | No |
| √100 | 10 | 10.0000 | Yes |
| √101 | √101 | 10.0499 | No |
| √102 | √102 | 10.0995 | No |
| √103 | √103 | 10.1489 | No |
| √104 | 2√26 | 10.1980 | No |
| √105 | √105 | 10.2470 | No |
- The cube root of 102 is about 4.672329.
- Four times the radicand doubles the root: √408 = 2 × √102 ≈ 20.19901.
Frequently asked questions
What is the square root of 102?
The square root of 102 is √102, about 10.0995049384. The negative root, −10.099505, also squares to 102.
Is the square root of 102 rational or irrational?
Irrational. 102 is not a perfect square — it falls between 100 and 121 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √102 be simplified?
No. 102 = 2 × 3 × 17 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √102 rounded to two decimal places?
√102 ≈ 10.10 to two decimal places (10.1 to one, 10.100 to three). Check: 10.10² = 102.01, close to 102.