√437 at a glance
- Exact value
- √437
- Decimal (10 places)
- 20.9045449604
- Rounded
- 20.9 · 20.90 · 20.905
- Perfect square?
- No — between 20² and 21²
- Rational?
- Irrational
- Both square roots
- ±20.904545
- Prime factorization
- 19 × 23
- Cube root
- 7.588579
How to simplify √437
The prime factorization of 437 is 19 × 23. Every prime appears only once, so there is no pair to bring outside the radical — √437 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 437, 19 and 23 appear an odd number of times, so √437 is irrational and 20.9045449604 is a rounded value.
Where √437 sits between perfect squares
400 = 20² and 441 = 21² are the nearest perfect squares, so √437 lies between 20 and 21. 437 is 37 above 400 and 4 below 441, so the root is closer to 21.
- Straight line between 400 and 441: 20.9024 (0.01% low)
- Tangent from 20, i.e. 20 + 37 ÷ 40: 20.9250 (0.1% high)
- Tangent from 21, i.e. 21 − 4 ÷ 42: 20.9048 (0% high)
For √437 the tangent at 21 wins, missing by only 0.0002. Tangent estimates shine when the number sits close to a perfect square — here 437 is just 4 below 441.
Finding √437 with the Babylonian method
This is Newton’s method applied to f(x) = x² − 437: following the tangent line down to zero simplifies to averaging x with 437 ÷ x.
Start from the nearest whole number, 21 (21² = 441):
| Step | Guess x | 437 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 21.0000000000 | 20.8095238095 | 20.9047619048 | 3 |
| 2 | 20.9047619048 | 20.9043280182 | 20.9045449615 | 8 |
| 3 | 20.9045449615 | 20.9045449592 | 20.9045449604 | 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 √437 = 20.9045449604 to every decimal shown.
√437 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √437 the pattern is [20; 1, 9, 2, 9, 1, 40] with the block of 6 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √437 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 |
|---|---|---|
| 20/1 | 20.0000000000 | 9.0 × 10⁻¹ |
| 21/1 | 21.0000000000 | 9.5 × 10⁻² |
| 209/10 | 20.9000000000 | 4.5 × 10⁻³ |
| 439/21 | 20.9047619048 | 2.2 × 10⁻⁴ |
| 4,160/199 | 20.9045226131 | 2.2 × 10⁻⁵ |
| 4,599/220 | 20.9045454545 | 4.9 × 10⁻⁷ |
The same fractions solve Pell’s equation, x² − 437y² = 1. Its smallest solution in positive whole numbers is x = 4,599, y = 220.
√437 in geometry and everyday measurements
- A square garage floor of 437 square feet measures about 20.9 ft (20 ft 11 in) per side, and its corner-to-corner diagonal is √874 ≈ 29.6 ft.
- 437 is not a sum of two whole-number squares — the prime factor 19 and 23 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √437 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 6 × 20 box, because 1² + 6² + 20² = 437.
Square roots near √437 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √434 | √434 | 20.8327 | No |
| √435 | √435 | 20.8567 | No |
| √436 | 2√109 | 20.8806 | No |
| √437 | √437 | 20.9045 | No |
| √438 | √438 | 20.9284 | No |
| √439 | √439 | 20.9523 | No |
| √440 | 2√110 | 20.9762 | No |
- The cube root of 437 is about 7.588579.
- Squaring undoes the root: (√437)² = 437, while 437² = 190,969 — the number whose square root is 437.
Frequently asked questions
What is the square root of 437?
The square root of 437 is √437, about 20.9045449604. The negative root, −20.904545, also squares to 437.
Is the square root of 437 rational or irrational?
Irrational. 437 is not a perfect square — it falls between 400 and 441 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √437 be simplified?
No. 437 = 19 × 23 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √437 rounded to two decimal places?
√437 ≈ 20.90 to two decimal places (20.9 to one, 20.905 to three). Check: 20.90² = 436.81, close to 437.