Square Root of 953

√953 at a glance

Exact value

√953

Decimal (10 places)

30.8706980809

Rounded

30.9 · 30.87 · 30.871

Perfect square?

No — between 30² and 31²

Rational?

Irrational

Both square roots

±30.870698

Prime factorization

953

Cube root

9.840813

How to simplify √953

953 is a prime number, so its only factors are 1 and 953. There is no perfect-square factor to pull out, which means √953 is already in its simplest radical form.

The square root of any prime is irrational. If √953 were a fraction a/b in lowest terms, then a² = 953b², so 953 would divide a — and then 953 would divide b too, contradicting “lowest terms.” That is why the decimal 30.8706980809 is only a rounded value.

Where √953 sits between perfect squares

900 = 30² and 961 = 31² are the nearest perfect squares, so √953 lies between 30 and 31. 953 is 53 above 900 and 8 below 961, so the root is closer to 31.

• Straight line between 900 and 961: 30.8689 (0.01% low)
• Tangent from 30, i.e. 30 + 53 ÷ 60: 30.8833 (0.04% high)
• Tangent from 31, i.e. 31 − 8 ÷ 62: 30.8710 (0% high)

For √953 the tangent at 31 wins, missing by only 0.0003. Tangent estimates shine when the number sits close to a perfect square — here 953 is just 8 below 961.

Finding √953 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 953: following the tangent line down to zero simplifies to averaging x with 953 ÷ x.

Start from the nearest whole number, 31 (31² = 961):

The count of correct decimals went 3, 8 and all 10 over 3 steps — roughly doubling each time — until the guess matched √953 = 30.8706980809 to every decimal shown.

√953 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √953 the pattern is [30; 1, 6, 1, 2, 1, 3, 8, 1, 1, 4, 4, 1, …] with the block of 21 terms after the semicolon repeating forever (only the first 12 of the 21 are shown). A pattern that never ends is one more proof that √953 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 − 953y² = 1. Its smallest solution in positive whole numbers is x = 15,090,531,843,660,371,073, y = 488,830,275,367,615,376 — 20 digits for x, even though 953 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: 2,746,864,744² − 953 × 88,979,677² = −1.

√953 in geometry and everyday measurements

• 953 square feet is 88.5 m². Laid out as a square — a small house footprint or a lot — it is about 30.87 ft (30 ft 10 in) on a side.
• 953 = 13² + 28², so by the Pythagorean theorem √953 is the diagonal of a 13 × 28 rectangle — and the distance between the points (0, 0) and (13, 28) on a grid.

Square roots near √953 and related roots

• The cube root of 953 is about 9.840813.
• Squaring undoes the root: (√953)² = 953, while 953² = 908,209 — the number whose square root is 953.

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