√190 at a glance
- Exact value
- √190
- Decimal (10 places)
- 13.7840487521
- Rounded
- 13.8 · 13.78 · 13.784
- Perfect square?
- No — between 13² and 14²
- Rational?
- Irrational
- Both square roots
- ±13.784049
- Prime factorization
- 2 × 5 × 19
- Cube root
- 5.748897
How to simplify √190
The prime factorization of 190 is 2 × 5 × 19. Every prime appears only once, so there is no pair to bring outside the radical — √190 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 190, 2, 5 and 19 appear an odd number of times, so √190 is irrational and 13.7840487521 is a rounded value.
Where √190 sits between perfect squares
169 = 13² and 196 = 14² are the nearest perfect squares, so √190 lies between 13 and 14. 190 is 21 above 169 and 6 below 196, so the root is closer to 14.
- Straight line between 169 and 196: 13.7778 (0.05% low)
- Tangent from 13, i.e. 13 + 21 ÷ 26: 13.8077 (0.17% high)
- Tangent from 14, i.e. 14 − 6 ÷ 28: 13.7857 (0.01% high)
For √190 the tangent at 14 wins, missing by only 0.0017. Tangent estimates shine when the number sits close to a perfect square — here 190 is just 6 below 196.
Finding √190 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 | 190 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 14.0000000000 | 13.5714285714 | 13.7857142857 | 2 |
| 2 | 13.7857142857 | 13.7823834197 | 13.7840488527 | 6 |
| 3 | 13.7840488527 | 13.7840486515 | 13.7840487521 | 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 √190 = 13.7840487521 to every decimal shown.
√190 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √190 the pattern is [13; 1, 3, 1, 1, 1, 2, 2, 2, 1, 1, 1, 3, …] with the block of 14 terms after the semicolon repeating forever (only the first 12 of the 14 are shown). A pattern that never ends is one more proof that √190 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 | 7.8 × 10⁻¹ |
| 14/1 | 14.0000000000 | 2.2 × 10⁻¹ |
| 55/4 | 13.7500000000 | 3.4 × 10⁻² |
| 69/5 | 13.8000000000 | 1.6 × 10⁻² |
| 124/9 | 13.7777777778 | 6.3 × 10⁻³ |
| 193/14 | 13.7857142857 | 1.7 × 10⁻³ |
The same fractions solve Pell’s equation, x² − 190y² = 1. Its smallest solution in positive whole numbers is x = 52,021, y = 3,774.
√190 in geometry and everyday measurements
- A square room or garden bed covering 190 square feet measures about 13.78 ft (13 ft 9 in) along each wall.
- 190 is not a sum of two whole-number squares — the prime factor 19 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √190 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 3 × 9 × 10 box, because 3² + 9² + 10² = 190.
Square roots near √190 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √187 | √187 | 13.6748 | No |
| √188 | 2√47 | 13.7113 | No |
| √189 | 3√21 | 13.7477 | No |
| √190 | √190 | 13.7840 | No |
| √191 | √191 | 13.8203 | No |
| √192 | 8√3 | 13.8564 | No |
| √193 | √193 | 13.8924 | No |
- The cube root of 190 is about 5.748897.
- Four times the radicand doubles the root: √760 = 2 × √190 ≈ 27.568098.
Frequently asked questions
What is the square root of 190?
The square root of 190 is √190, about 13.7840487521. The negative root, −13.784049, also squares to 190.
Is the square root of 190 rational or irrational?
Irrational. 190 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 √190 be simplified?
No. 190 = 2 × 5 × 19 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √190 rounded to two decimal places?
√190 ≈ 13.78 to two decimal places (13.8 to one, 13.784 to three). Check: 13.78² = 189.8884, close to 190.