Square Root of 283

√283 at a glance

Exact value

√283

Decimal (10 places)

16.8226038413

Rounded

16.8 · 16.82 · 16.823

Perfect square?

No — between 16² and 17²

Rational?

Irrational

Both square roots

±16.822604

Prime factorization

283

Cube root

6.565414

How to simplify √283

283 is a prime number, so its only factors are 1 and 283. There is no perfect-square factor to pull out, which means √283 is already in its simplest radical form.

The square root of any prime is irrational. If √283 were a fraction a/b in lowest terms, then a² = 283b², so 283 would divide a — and then 283 would divide b too, contradicting “lowest terms.” That is why the decimal 16.8226038413 is only a rounded value.

Where √283 sits between perfect squares

256 = 16² and 289 = 17² are the nearest perfect squares, so √283 lies between 16 and 17. 283 is 27 above 256 and 6 below 289, so the root is closer to 17.

• Straight line between 256 and 289: 16.8182 (0.03% low)
• Tangent from 16, i.e. 16 + 27 ÷ 32: 16.8438 (0.13% high)
• Tangent from 17, i.e. 17 − 6 ÷ 34: 16.8235 (0.01% high)

For √283 the tangent at 17 wins, missing by only 0.0009. Tangent estimates shine when the number sits close to a perfect square — here 283 is just 6 below 289.

Finding √283 with the Babylonian method

If a guess is too big, 283 ÷ 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√283) in one step.

Start from the nearest whole number, 17 (17² = 289):

The count of correct decimals went 3, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √283 = 16.8226038413 to every decimal shown.

√283 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √283 the pattern is [16; 1, 4, 1, 1, 1, 3, 10, 1, 15, 1, 10, 3, …] with the block of 18 terms after the semicolon repeating forever (only the first 12 of the 18 are shown). A pattern that never ends is one more proof that √283 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 − 283y² = 1. Its smallest solution in positive whole numbers is x = 138,274,082, y = 8,219,541.

√283 in geometry and everyday measurements

• A square patio or deck of 283 square feet is about 16.82 ft (16 ft 10 in) on each side, so edging all the way around takes 4 × √283 ≈ 67.3 ft.
• 283 is not a sum of two whole-number squares — 283 is itself a prime that is one less than a multiple of 4, which rules that out — so √283 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 3 × 7 × 15 box, because 3² + 7² + 15² = 283.

Square roots near √283 and related roots

• The cube root of 283 is about 6.565414.
• Squaring undoes the root: (√283)² = 283, while 283² = 80,089 — the number whose square root is 283.

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