√890 at a glance
- Exact value
- √890
- Decimal (10 places)
- 29.8328677804
- Rounded
- 29.8 · 29.83 · 29.833
- Perfect square?
- No — between 29² and 30²
- Rational?
- Irrational
- Both square roots
- ±29.832868
- Prime factorization
- 2 × 5 × 89
- Cube root
- 9.619002
How to simplify √890
The prime factorization of 890 is 2 × 5 × 89. Every prime appears only once, so there is no pair to bring outside the radical — √890 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 890, 2, 5 and 89 appear an odd number of times, so √890 is irrational and 29.8328677804 is a rounded value.
Where √890 sits between perfect squares
841 = 29² and 900 = 30² are the nearest perfect squares, so √890 lies between 29 and 30. 890 is 49 above 841 and 10 below 900, so the root is closer to 30.
- Straight line between 841 and 900: 29.8305 (0.01% low)
- Tangent from 29, i.e. 29 + 49 ÷ 58: 29.8448 (0.04% high)
- Tangent from 30, i.e. 30 − 10 ÷ 60: 29.8333 (0% high)
For √890 the tangent at 30 wins, missing by only 0.0005. Tangent estimates shine when the number sits close to a perfect square — here 890 is just 10 below 900.
Finding √890 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, 30 (30² = 900):
| Step | Guess x | 890 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 30.0000000000 | 29.6666666667 | 29.8333333333 | 3 |
| 2 | 29.8333333333 | 29.8324022346 | 29.8328677840 | 8 |
| 3 | 29.8328677840 | 29.8328677767 | 29.8328677804 | all 10 shown |
The count of correct decimals went 3, 8 and all 10 over 3 steps — roughly doubling each time — until the guess matched √890 = 29.8328677804 to every decimal shown.
√890 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √890 the pattern is [29; 1, 4, 1, 58] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √890 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 |
|---|---|---|
| 29/1 | 29.0000000000 | 8.3 × 10⁻¹ |
| 30/1 | 30.0000000000 | 1.7 × 10⁻¹ |
| 149/5 | 29.8000000000 | 3.3 × 10⁻² |
| 179/6 | 29.8333333333 | 4.7 × 10⁻⁴ |
| 10,531/353 | 29.8328611898 | 6.6 × 10⁻⁶ |
| 10,710/359 | 29.8328690808 | 1.3 × 10⁻⁶ |
The same fractions solve Pell’s equation, x² − 890y² = 1. Its smallest solution in positive whole numbers is x = 179, y = 6.
√890 in geometry and everyday measurements
- 890 square feet is 82.7 m². Laid out as a square — a small house footprint or a lot — it is about 29.83 ft (29 ft 10 in) on a side.
- 890 = 7² + 29² = 19² + 23², so by the Pythagorean theorem √890 is the diagonal of rectangles measuring 7 × 29 and 19 × 23 — and the distance between the points (0, 0) and (7, 29) on a grid.
Square roots near √890 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √887 | √887 | 29.7825 | No |
| √888 | 2√222 | 29.7993 | No |
| √889 | √889 | 29.8161 | No |
| √890 | √890 | 29.8329 | No |
| √891 | 9√11 | 29.8496 | No |
| √892 | 2√223 | 29.8664 | No |
| √893 | √893 | 29.8831 | No |
- The cube root of 890 is about 9.619002.
- Squaring undoes the root: (√890)² = 890, while 890² = 792,100 — the number whose square root is 890.
Frequently asked questions
What is the square root of 890?
The square root of 890 is √890, about 29.8328677804. The negative root, −29.832868, also squares to 890.
Is the square root of 890 rational or irrational?
Irrational. 890 is not a perfect square — it falls between 841 and 900 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √890 be simplified?
No. 890 = 2 × 5 × 89 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √890 rounded to two decimal places?
√890 ≈ 29.83 to two decimal places (29.8 to one, 29.833 to three). Check: 29.83² = 889.8289, close to 890.