Square Root of 203

√203 at a glance

Exact value

√203

Decimal (10 places)

14.2478068488

Rounded

14.2 · 14.25 · 14.248

Perfect square?

No — between 14² and 15²

Rational?

Irrational

Both square roots

±14.247807

Prime factorization

7 × 29

Cube root

5.877131

How to simplify √203

The prime factorization of 203 is 7 × 29. Every prime appears only once, so there is no pair to bring outside the radical — √203 is already in simplest form (a "square-free" radicand).

A whole number has a rational square root only when every prime in its factorization appears an even number of times. In 203, 7 and 29 appear an odd number of times, so √203 is irrational and 14.2478068488 is a rounded value.

Where √203 sits between perfect squares

196 = 14² and 225 = 15² are the nearest perfect squares, so √203 lies between 14 and 15. 203 is 7 above 196 and 22 below 225, so the root is closer to 14.

• Straight line between 196 and 225: 14.2414 (0.05% low)
• Tangent from 14, i.e. 14 + 7 ÷ 28: 14.2500 (0.02% high)
• Tangent from 15, i.e. 15 − 22 ÷ 30: 14.2667 (0.13% high)

For √203 the tangent at 14 wins, missing by only 0.0022. Tangent estimates shine when the number sits close to a perfect square — here 203 is just 7 above 196.

Finding √203 with the Babylonian method

If a guess is too big, 203 ÷ 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√203) in one step.

Start from the nearest whole number, 14 (14² = 196):

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √203 = 14.2478068488 to every decimal shown.

√203 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √203 the pattern is [14; 4, 28] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √203 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 − 203y² = 1. Its smallest solution in positive whole numbers is x = 57, y = 4.

√203 in geometry and everyday measurements

• A square patio or deck of 203 square feet is about 14.25 ft (14 ft 3 in) on each side, so edging all the way around takes 4 × √203 ≈ 57 ft.
• 203 is not a sum of two whole-number squares — the prime factor 7 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √203 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 9 × 11 box, because 1² + 9² + 11² = 203.

Square roots near √203 and related roots

• The cube root of 203 is about 5.877131.
• Four times the radicand doubles the root: √812 = 2 × √203 ≈ 28.495614.

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