Square Root of 937

√937 at a glance

Exact value

√937

Decimal (10 places)

30.6104557300

Rounded

30.6 · 30.61 · 30.610

Perfect square?

No — between 30² and 31²

Rational?

Irrational

Both square roots

±30.610456

Prime factorization

937

Cube root

9.785429

How to simplify √937

937 is a prime number, so its only factors are 1 and 937. There is no perfect-square factor to pull out, which means √937 is already in its simplest radical form.

The square root of any prime is irrational. If √937 were a fraction a/b in lowest terms, then a² = 937b², so 937 would divide a — and then 937 would divide b too, contradicting “lowest terms.” That is why the decimal 30.6104557300 is only a rounded value.

Where √937 sits between perfect squares

900 = 30² and 961 = 31² are the nearest perfect squares, so √937 lies between 30 and 31. 937 is 37 above 900 and 24 below 961, so the root is closer to 31.

• Straight line between 900 and 961: 30.6066 (0.01% low)
• Tangent from 30, i.e. 30 + 37 ÷ 60: 30.6167 (0.02% high)
• Tangent from 31, i.e. 31 − 24 ÷ 62: 30.6129 (0.01% high)

For √937 the tangent at 31 wins, missing by only 0.0024. Tangent estimates shine when the number sits close to a perfect square — here 937 is just 24 below 961.

Finding √937 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 937: following the tangent line down to zero simplifies to averaging x with 937 ÷ x.

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

The count of correct decimals went 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √937 = 30.6104557300 to every decimal shown.

√937 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √937 the pattern is [30; 1, 1, 1, 1, 3, 4, 2, 3, 6, 1, 1, 19, …] with the block of 33 terms after the semicolon repeating forever (only the first 12 of the 33 are shown). A pattern that never ends is one more proof that √937 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 − 937y² = 1. Its smallest solution in positive whole numbers is x = 480,644,425,002,415,999,597,113,107,233, y = 15,701,968,936,415,353,889,062,192,632 — 30 digits for x, even though 937 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: 490,226,695,010,796² − 937 × 16,015,008,052,621² = −1.

√937 in geometry and everyday measurements

• 937 square feet is 87.1 m². Laid out as a square — a small house footprint or a lot — it is about 30.61 ft (30 ft 7 in) on a side.
• 937 = 19² + 24², so by the Pythagorean theorem √937 is the diagonal of a 19 × 24 rectangle — and the distance between the points (0, 0) and (19, 24) on a grid.

Square roots near √937 and related roots

• The cube root of 937 is about 9.785429.
• Squaring undoes the root: (√937)² = 937, while 937² = 877,969 — the number whose square root is 937.

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