Square Root of 883

√883 at a glance

Exact value

√883

Decimal (10 places)

29.7153159162

Rounded

29.7 · 29.72 · 29.715

Perfect square?

No — between 29² and 30²

Rational?

Irrational

Both square roots

±29.715316

Prime factorization

883

Cube root

9.593717

How to simplify √883

883 is a prime number, so its only factors are 1 and 883. There is no perfect-square factor to pull out, which means √883 is already in its simplest radical form.

The square root of any prime is irrational. If √883 were a fraction a/b in lowest terms, then a² = 883b², so 883 would divide a — and then 883 would divide b too, contradicting “lowest terms.” That is why the decimal 29.7153159162 is only a rounded value.

Where √883 sits between perfect squares

841 = 29² and 900 = 30² are the nearest perfect squares, so √883 lies between 29 and 30. 883 is 42 above 841 and 17 below 900, so the root is closer to 30.

• Straight line between 841 and 900: 29.7119 (0.01% low)
• Tangent from 29, i.e. 29 + 42 ÷ 58: 29.7241 (0.03% high)
• Tangent from 30, i.e. 30 − 17 ÷ 60: 29.7167 (0% high)

For √883 the tangent at 30 wins, missing by only 0.0014. Tangent estimates shine when the number sits close to a perfect square — here 883 is just 17 below 900.

Finding √883 with the Babylonian method

If a guess is too big, 883 ÷ 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√883) in one step.

Start from the nearest whole number, 30 (30² = 900):

The count of correct decimals went 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √883 = 29.7153159162 to every decimal shown.

√883 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √883 the pattern is [29; 1, 2, 1, 1, 19, 4, 5, 6, 2, 2, 2, 1, …] with the block of 46 terms after the semicolon repeating forever (only the first 12 of the 46 are shown). A pattern that never ends is one more proof that √883 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 − 883y² = 1. Its smallest solution in positive whole numbers is x = 34,878,475,759,617,272,473,442, y = 1,173,754,162,936,357,802,169 — 23 digits for x, even though 883 is small, which is what makes Pell’s equation famous.

√883 in geometry and everyday measurements

• 883 square feet is 82 m². Laid out as a square — a small house footprint or a lot — it is about 29.72 ft (29 ft 9 in) on a side.
• 883 is not a sum of two whole-number squares — 883 is itself a prime that is one less than a multiple of 4, which rules that out — so √883 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 21 × 21 box, because 1² + 21² + 21² = 883.

Square roots near √883 and related roots

• The cube root of 883 is about 9.593717.
• Squaring undoes the root: (√883)² = 883, while 883² = 779,689 — the number whose square root is 883.

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