√623 at a glance
- Exact value
- √623
- Decimal (10 places)
- 24.9599679487
- Rounded
- 25.0 · 24.96 · 24.960
- Perfect square?
- No — between 24² and 25²
- Rational?
- Irrational
- Both square roots
- ±24.959968
- Prime factorization
- 7 × 89
- Cube root
- 8.540750
How to simplify √623
The prime factorization of 623 is 7 × 89. Every prime appears only once, so there is no pair to bring outside the radical — √623 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 623, 7 and 89 appear an odd number of times, so √623 is irrational and 24.9599679487 is a rounded value.
Where √623 sits between perfect squares
576 = 24² and 625 = 25² are the nearest perfect squares, so √623 lies between 24 and 25. 623 is 47 above 576 and 2 below 625, so the root is closer to 25.
- Straight line between 576 and 625: 24.9592 (0% low)
- Tangent from 24, i.e. 24 + 47 ÷ 48: 24.9792 (0.08% high)
- Tangent from 25, i.e. 25 − 2 ÷ 50: 24.9600 (0% high)
For √623 the tangent at 25 wins, missing by only 0. Tangent estimates shine when the number sits close to a perfect square — here 623 is just 2 below 625.
Finding √623 with the Babylonian method
If a guess is too big, 623 ÷ 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√623) in one step.
Start from the nearest whole number, 25 (25² = 625):
| Step | Guess x | 623 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 25.0000000000 | 24.9200000000 | 24.9600000000 | 4 |
| 2 | 24.9600000000 | 24.9599358974 | 24.9599679487 | all 10 shown |
Because the starting guess was already close, two steps are enough to match √623 = 24.9599679487 to every decimal shown.
√623 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √623 the pattern is [24; 1, 23, 1, 48] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √623 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 |
|---|---|---|
| 24/1 | 24.0000000000 | 9.6 × 10⁻¹ |
| 25/1 | 25.0000000000 | 4.0 × 10⁻² |
| 599/24 | 24.9583333333 | 1.6 × 10⁻³ |
| 624/25 | 24.9600000000 | 3.2 × 10⁻⁵ |
| 30,551/1,224 | 24.9599673203 | 6.3 × 10⁻⁷ |
| 31,175/1,249 | 24.9599679744 | 2.6 × 10⁻⁸ |
The same fractions solve Pell’s equation, x² − 623y² = 1. Its smallest solution in positive whole numbers is x = 624, y = 25.
√623 in geometry and everyday measurements
- A square garage floor of 623 square feet measures about 24.96 ft (25 ft) per side, and its corner-to-corner diagonal is √1246 ≈ 35.3 ft.
- 623 is not a sum of two whole-number squares — the prime factor 7 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √623 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 √623 as its space diagonal.
Square roots near √623 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √620 | 2√155 | 24.8998 | No |
| √621 | 3√69 | 24.9199 | No |
| √622 | √622 | 24.9399 | No |
| √623 | √623 | 24.9600 | No |
| √624 | 4√39 | 24.9800 | No |
| √625 | 25 | 25.0000 | Yes |
| √626 | √626 | 25.0200 | No |
- The cube root of 623 is about 8.540750.
- Squaring undoes the root: (√623)² = 623, while 623² = 388,129 — the number whose square root is 623.
Frequently asked questions
What is the square root of 623?
The square root of 623 is √623, about 24.9599679487. The negative root, −24.959968, also squares to 623.
Is the square root of 623 rational or irrational?
Irrational. 623 is not a perfect square — it falls between 576 and 625 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √623 be simplified?
No. 623 = 7 × 89 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √623 rounded to two decimal places?
√623 ≈ 24.96 to two decimal places (25.0 to one, 24.960 to three). Check: 24.96² = 623.0016, close to 623.