√618 at a glance
- Exact value
- √618
- Decimal (10 places)
- 24.8596057893
- Rounded
- 24.9 · 24.86 · 24.860
- Perfect square?
- No — between 24² and 25²
- Rational?
- Irrational
- Both square roots
- ±24.859606
- Prime factorization
- 2 × 3 × 103
- Cube root
- 8.517840
How to simplify √618
The prime factorization of 618 is 2 × 3 × 103. Every prime appears only once, so there is no pair to bring outside the radical — √618 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 618, 2, 3 and 103 appear an odd number of times, so √618 is irrational and 24.8596057893 is a rounded value.
Where √618 sits between perfect squares
576 = 24² and 625 = 25² are the nearest perfect squares, so √618 lies between 24 and 25. 618 is 42 above 576 and 7 below 625, so the root is closer to 25.
- Straight line between 576 and 625: 24.8571 (0.01% low)
- Tangent from 24, i.e. 24 + 42 ÷ 48: 24.8750 (0.06% high)
- Tangent from 25, i.e. 25 − 7 ÷ 50: 24.8600 (0% high)
For √618 the tangent at 25 wins, missing by only 0.0004. Tangent estimates shine when the number sits close to a perfect square — here 618 is just 7 below 625.
Finding √618 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, 25 (25² = 625):
| Step | Guess x | 618 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 25.0000000000 | 24.7200000000 | 24.8600000000 | 3 |
| 2 | 24.8600000000 | 24.8592115849 | 24.8596057924 | 8 |
| 3 | 24.8596057924 | 24.8596057862 | 24.8596057893 | 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 √618 = 24.8596057893 to every decimal shown.
√618 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √618 the pattern is [24; 1, 6, 8, 6, 1, 48] with the block of 6 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √618 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 | 8.6 × 10⁻¹ |
| 25/1 | 25.0000000000 | 1.4 × 10⁻¹ |
| 174/7 | 24.8571428571 | 2.5 × 10⁻³ |
| 1,417/57 | 24.8596491228 | 4.3 × 10⁻⁵ |
| 8,676/349 | 24.8595988539 | 6.9 × 10⁻⁶ |
| 10,093/406 | 24.8596059113 | 1.2 × 10⁻⁷ |
The same fractions solve Pell’s equation, x² − 618y² = 1. Its smallest solution in positive whole numbers is x = 10,093, y = 406.
√618 in geometry and everyday measurements
- A square garage floor of 618 square feet measures about 24.86 ft (24 ft 10 in) per side, and its corner-to-corner diagonal is √1236 ≈ 35.2 ft.
- 618 is not a sum of two whole-number squares — the prime factor 3 and 103 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √618 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 16 × 19 box, because 1² + 16² + 19² = 618.
Square roots near √618 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √615 | √615 | 24.7992 | No |
| √616 | 2√154 | 24.8193 | No |
| √617 | √617 | 24.8395 | No |
| √618 | √618 | 24.8596 | No |
| √619 | √619 | 24.8797 | No |
| √620 | 2√155 | 24.8998 | No |
| √621 | 3√69 | 24.9199 | No |
- The cube root of 618 is about 8.517840.
- Squaring undoes the root: (√618)² = 618, while 618² = 381,924 — the number whose square root is 618.
Frequently asked questions
What is the square root of 618?
The square root of 618 is √618, about 24.8596057893. The negative root, −24.859606, also squares to 618.
Is the square root of 618 rational or irrational?
Irrational. 618 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 √618 be simplified?
No. 618 = 2 × 3 × 103 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √618 rounded to two decimal places?
√618 ≈ 24.86 to two decimal places (24.9 to one, 24.860 to three). Check: 24.86² = 618.0196, close to 618.