Square Root of 509

√509 at a glance

Exact value

√509

Decimal (10 places)

22.5610283454

Rounded

22.6 · 22.56 · 22.561

Perfect square?

No — between 22² and 23²

Rational?

Irrational

Both square roots

±22.561028

Prime factorization

509

Cube root

7.984344

How to simplify √509

509 is a prime number, so its only factors are 1 and 509. There is no perfect-square factor to pull out, which means √509 is already in its simplest radical form.

The square root of any prime is irrational. If √509 were a fraction a/b in lowest terms, then a² = 509b², so 509 would divide a — and then 509 would divide b too, contradicting “lowest terms.” That is why the decimal 22.5610283454 is only a rounded value.

Where √509 sits between perfect squares

484 = 22² and 529 = 23² are the nearest perfect squares, so √509 lies between 22 and 23. 509 is 25 above 484 and 20 below 529, so the root is closer to 23.

• Straight line between 484 and 529: 22.5556 (0.02% low)
• Tangent from 22, i.e. 22 + 25 ÷ 44: 22.5682 (0.03% high)
• Tangent from 23, i.e. 23 − 20 ÷ 46: 22.5652 (0.02% high)

For √509 the tangent at 23 wins, missing by only 0.0042. Tangent estimates shine when the number sits close to a perfect square — here 509 is just 20 below 529.

Finding √509 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 509: following the tangent line down to zero simplifies to averaging x with 509 ÷ x.

Start from the nearest whole number, 23 (23² = 529):

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √509 = 22.5610283454 to every decimal shown.

√509 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √509 the pattern is [22; 1, 1, 3, 1, 1, 2, 10, 1, 8, 8, 1, 10, …] with the block of 19 terms after the semicolon repeating forever (only the first 12 of the 19 are shown). A pattern that never ends is one more proof that √509 is irrational.

Cutting the continued fraction off early gives the best fractions for approximating the root — no fraction with a smaller denominator comes closer:

The same fractions solve Pell’s equation, x² − 509y² = 1. Its smallest solution in positive whole numbers is x = 313,201,220,822,405,001, y = 13,882,400,040,814,700 — 18 digits for x, even though 509 is small, which is what makes Pell’s equation famous. Because the period is odd, the equation with −1 on the right also has a solution: 395,727,950² − 509 × 17,540,333² = −1.

√509 in geometry and everyday measurements

• A square garage floor of 509 square feet measures about 22.56 ft (22 ft 7 in) per side, and its corner-to-corner diagonal is √1018 ≈ 31.9 ft.
• 509 = 5² + 22², so by the Pythagorean theorem √509 is the diagonal of a 5 × 22 rectangle — and the distance between the points (0, 0) and (5, 22) on a grid.

Square roots near √509 and related roots

• The cube root of 509 is about 7.984344.
• Squaring undoes the root: (√509)² = 509, while 509² = 259,081 — the number whose square root is 509.

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