√463 at a glance
- Exact value
- √463
- Decimal (10 places)
- 21.5174347914
- Rounded
- 21.5 · 21.52 · 21.517
- Perfect square?
- No — between 21² and 22²
- Rational?
- Irrational
- Both square roots
- ±21.517435
- Prime factorization
- 463
- Cube root
- 7.736188
How to simplify √463
463 is a prime number, so its only factors are 1 and 463. There is no perfect-square factor to pull out, which means √463 is already in its simplest radical form.
The square root of any prime is irrational. If √463 were a fraction a/b in lowest terms, then a² = 463b², so 463 would divide a — and then 463 would divide b too, contradicting “lowest terms.” That is why the decimal 21.5174347914 is only a rounded value.
Where √463 sits between perfect squares
441 = 21² and 484 = 22² are the nearest perfect squares, so √463 lies between 21 and 22. 463 is 22 above 441 and 21 below 484, so the root is closer to 22.
- Straight line between 441 and 484: 21.5116 (0.03% low)
- Tangent from 21, i.e. 21 + 22 ÷ 42: 21.5238 (0.03% high)
- Tangent from 22, i.e. 22 − 21 ÷ 44: 21.5227 (0.02% high)
For √463 the tangent at 22 wins, missing by only 0.0053. Tangent estimates shine when the number sits close to a perfect square — here 463 is just 21 below 484.
Finding √463 with the Babylonian method
If a guess is too big, 463 ÷ x is too small by about the same amount, so their average lands much closer. In fact an error of e shrinks to roughly e² ÷ (2√463) in one step.
Start from the nearest whole number, 22 (22² = 484):
| Step | Guess x | 463 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 22.0000000000 | 21.0454545455 | 21.5227272727 | 2 |
| 2 | 21.5227272727 | 21.5121436114 | 21.5174354421 | 6 |
| 3 | 21.5174354421 | 21.5174341406 | 21.5174347914 | 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 √463 = 21.5174347914 to every decimal shown.
√463 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √463 the pattern is [21; 1, 1, 13, 1, 5, 4, 1, 1, 1, 1, 2, 2, …] with the block of 32 terms after the semicolon repeating forever (only the first 12 of the 32 are shown). A pattern that never ends is one more proof that √463 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 |
|---|---|---|
| 21/1 | 21.0000000000 | 5.2 × 10⁻¹ |
| 22/1 | 22.0000000000 | 4.8 × 10⁻¹ |
| 43/2 | 21.5000000000 | 1.7 × 10⁻² |
| 581/27 | 21.5185185185 | 1.1 × 10⁻³ |
| 624/29 | 21.5172413793 | 1.9 × 10⁻⁴ |
| 3,701/172 | 21.5174418605 | 7.1 × 10⁻⁶ |
The same fractions solve Pell’s equation, x² − 463y² = 1. Its smallest solution in positive whole numbers is x = 247,512,720,456,368, y = 11,502,891,625,161 — 15 digits for x, even though 463 is small, which is what makes Pell’s equation famous.
√463 in geometry and everyday measurements
- A square garage floor of 463 square feet measures about 21.52 ft (21 ft 6 in) per side, and its corner-to-corner diagonal is √926 ≈ 30.4 ft.
- 463 is not a sum of two whole-number squares — 463 is itself a prime that is one less than a multiple of 4, which rules that out — so √463 is never the diagonal of a rectangle with whole-number sides. It is not a sum of three positive squares either, so no whole-number box has √463 as its space diagonal.
Square roots near √463 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √460 | 2√115 | 21.4476 | No |
| √461 | √461 | 21.4709 | No |
| √462 | √462 | 21.4942 | No |
| √463 | √463 | 21.5174 | No |
| √464 | 4√29 | 21.5407 | No |
| √465 | √465 | 21.5639 | No |
| √466 | √466 | 21.5870 | No |
- The cube root of 463 is about 7.736188.
- Squaring undoes the root: (√463)² = 463, while 463² = 214,369 — the number whose square root is 463.
Frequently asked questions
What is the square root of 463?
The square root of 463 is √463, about 21.5174347914. The negative root, −21.517435, also squares to 463.
Is the square root of 463 rational or irrational?
Irrational. 463 is not a perfect square — it falls between 441 and 484 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √463 be simplified?
No. 463 is prime, so there is no perfect square to take out of the radical.
What is √463 rounded to two decimal places?
√463 ≈ 21.52 to two decimal places (21.5 to one, 21.517 to three). Check: 21.52² = 463.1104, close to 463.