Square Root of 503

√503 at a glance

Exact value

√503

Decimal (10 places)

22.4276614920

Rounded

22.4 · 22.43 · 22.428

Perfect square?

No — between 22² and 23²

Rational?

Irrational

Both square roots

±22.427661

Prime factorization

503

Cube root

7.952848

How to simplify √503

503 is a prime number, so its only factors are 1 and 503. There is no perfect-square factor to pull out, which means √503 is already in its simplest radical form.

The square root of any prime is irrational. If √503 were a fraction a/b in lowest terms, then a² = 503b², so 503 would divide a — and then 503 would divide b too, contradicting “lowest terms.” That is why the decimal 22.4276614920 is only a rounded value.

Where √503 sits between perfect squares

484 = 22² and 529 = 23² are the nearest perfect squares, so √503 lies between 22 and 23. 503 is 19 above 484 and 26 below 529, so the root is closer to 22.

• Straight line between 484 and 529: 22.4222 (0.02% low)
• Tangent from 22, i.e. 22 + 19 ÷ 44: 22.4318 (0.02% high)
• Tangent from 23, i.e. 23 − 26 ÷ 46: 22.4348 (0.03% high)

For √503 the tangent at 22 wins, missing by only 0.0042. Tangent estimates shine when the number sits close to a perfect square — here 503 is just 19 above 484.

Finding √503 with the Babylonian method

If a guess is too big, 503 ÷ 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√503) in one step.

Start from the nearest whole number, 22 (22² = 484):

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √503 = 22.4276614920 to every decimal shown.

√503 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √503 the pattern is [22; 2, 2, 1, 21, 1, 2, 2, 44] with the block of 8 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √503 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 − 503y² = 1. Its smallest solution in positive whole numbers is x = 24,648, y = 1,099.

√503 in geometry and everyday measurements

• A square garage floor of 503 square feet measures about 22.43 ft (22 ft 5 in) per side, and its corner-to-corner diagonal is √1006 ≈ 31.7 ft.
• 503 is not a sum of two whole-number squares — 503 is itself a prime that is one less than a multiple of 4, which rules that out — so √503 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 √503 as its space diagonal.

Square roots near √503 and related roots

• The cube root of 503 is about 7.952848.
• Squaring undoes the root: (√503)² = 503, while 503² = 253,009 — the number whose square root is 503.

Need a different value? Use the Square root calculator, or browse the Square roots 1–1,000.