Square Root of 83

√83 at a glance

Exact value

√83

Decimal (10 places)

9.1104335791

Rounded

9.1 · 9.11 · 9.110

Perfect square?

No — between 9² and 10²

Rational?

Irrational

Both square roots

±9.110434

Prime factorization

83

Cube root

4.362071

How to simplify √83

83 is a prime number, so its only factors are 1 and 83. There is no perfect-square factor to pull out, which means √83 is already in its simplest radical form.

The square root of any prime is irrational. If √83 were a fraction a/b in lowest terms, then a² = 83b², so 83 would divide a — and then 83 would divide b too, contradicting “lowest terms.” That is why the decimal 9.1104335791 is only a rounded value.

Where √83 sits between perfect squares

81 = 9² and 100 = 10² are the nearest perfect squares, so √83 lies between 9 and 10. 83 is 2 above 81 and 17 below 100, so the root is closer to 9.

• Straight line between 81 and 100: 9.1053 (0.06% low)
• Tangent from 9, i.e. 9 + 2 ÷ 18: 9.1111 (0.01% high)
• Tangent from 10, i.e. 10 − 17 ÷ 20: 9.1500 (0.43% high)

For √83 the tangent at 9 wins, missing by only 0.0007. Tangent estimates shine when the number sits close to a perfect square — here 83 is just 2 above 81.

Finding √83 with the Babylonian method

If a guess is too big, 83 ÷ 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√83) in one step.

Start from the nearest whole number, 9 (9² = 81):

The count of correct decimals went 3, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √83 = 9.1104335791 to every decimal shown.

√83 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √83 the pattern is [9; 9, 18] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √83 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 − 83y² = 1. Its smallest solution in positive whole numbers is x = 82, y = 9.

√83 in geometry and everyday measurements

• A square room or garden bed covering 83 square feet measures about 9.11 ft (9 ft 1 in) along each wall.
• 83 is not a sum of two whole-number squares — 83 is itself a prime that is one less than a multiple of 4, which rules that out — so √83 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 1 × 9 box, because 1² + 1² + 9² = 83.

Square roots near √83 and related roots

• The cube root of 83 is about 4.362071.
• Four times the radicand doubles the root: √332 = 2 × √83 ≈ 18.220867.

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