Square Root of 383

√383 at a glance

Exact value

√383

Decimal (10 places)

19.5703857908

Rounded

19.6 · 19.57 · 19.570

Perfect square?

No — between 19² and 20²

Rational?

Irrational

Both square roots

±19.570386

Prime factorization

383

Cube root

7.262167

How to simplify √383

383 is a prime number, so its only factors are 1 and 383. There is no perfect-square factor to pull out, which means √383 is already in its simplest radical form.

The square root of any prime is irrational. If √383 were a fraction a/b in lowest terms, then a² = 383b², so 383 would divide a — and then 383 would divide b too, contradicting “lowest terms.” That is why the decimal 19.5703857908 is only a rounded value.

Where √383 sits between perfect squares

361 = 19² and 400 = 20² are the nearest perfect squares, so √383 lies between 19 and 20. 383 is 22 above 361 and 17 below 400, so the root is closer to 20.

• Straight line between 361 and 400: 19.5641 (0.03% low)
• Tangent from 19, i.e. 19 + 22 ÷ 38: 19.5789 (0.04% high)
• Tangent from 20, i.e. 20 − 17 ÷ 40: 19.5750 (0.02% high)

For √383 the tangent at 20 wins, missing by only 0.0046. Tangent estimates shine when the number sits close to a perfect square — here 383 is just 17 below 400.

Finding √383 with the Babylonian method

If a guess is too big, 383 ÷ 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√383) in one step.

Start from the nearest whole number, 20 (20² = 400):

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √383 = 19.5703857908 to every decimal shown.

√383 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √383 the pattern is [19; 1, 1, 3, 19, 3, 1, 1, 38] with the block of 8 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √383 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 − 383y² = 1. Its smallest solution in positive whole numbers is x = 18,768, y = 959.

√383 in geometry and everyday measurements

• A square patio or deck of 383 square feet is about 19.57 ft (19 ft 7 in) on each side, so edging all the way around takes 4 × √383 ≈ 78.3 ft.
• 383 is not a sum of two whole-number squares — 383 is itself a prime that is one less than a multiple of 4, which rules that out — so √383 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 √383 as its space diagonal.

Square roots near √383 and related roots

• The cube root of 383 is about 7.262167.
• Squaring undoes the root: (√383)² = 383, while 383² = 146,689 — the number whose square root is 383.

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