Square Root of 269

√269 at a glance

Exact value

√269

Decimal (10 places)

16.4012194669

Rounded

16.4 · 16.40 · 16.401

Perfect square?

No — between 16² and 17²

Rational?

Irrational

Both square roots

±16.401219

Prime factorization

269

Cube root

6.455315

How to simplify √269

269 is a prime number, so its only factors are 1 and 269. There is no perfect-square factor to pull out, which means √269 is already in its simplest radical form.

The square root of any prime is irrational. If √269 were a fraction a/b in lowest terms, then a² = 269b², so 269 would divide a — and then 269 would divide b too, contradicting “lowest terms.” That is why the decimal 16.4012194669 is only a rounded value.

Where √269 sits between perfect squares

256 = 16² and 289 = 17² are the nearest perfect squares, so √269 lies between 16 and 17. 269 is 13 above 256 and 20 below 289, so the root is closer to 16.

• Straight line between 256 and 289: 16.3939 (0.04% low)
• Tangent from 16, i.e. 16 + 13 ÷ 32: 16.4063 (0.03% high)
• Tangent from 17, i.e. 17 − 20 ÷ 34: 16.4118 (0.06% high)

For √269 the tangent at 16 wins, missing by only 0.005. Tangent estimates shine when the number sits close to a perfect square — here 269 is just 13 above 256.

Finding √269 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 269: following the tangent line down to zero simplifies to averaging x with 269 ÷ x.

Start from the nearest whole number, 16 (16² = 256):

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √269 = 16.4012194669 to every decimal shown.

√269 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √269 the pattern is [16; 2, 2, 32] with the block of 3 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √269 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 − 269y² = 1. Its smallest solution in positive whole numbers is x = 13,449, y = 820. Because the period is odd, the equation with −1 on the right also has a solution: 82² − 269 × 5² = −1.

√269 in geometry and everyday measurements

• A square patio or deck of 269 square feet is about 16.4 ft (16 ft 5 in) on each side, so edging all the way around takes 4 × √269 ≈ 65.6 ft.
• 269 = 10² + 13², so by the Pythagorean theorem √269 is the diagonal of a 10 × 13 rectangle — and the distance between the points (0, 0) and (10, 13) on a grid.

Square roots near √269 and related roots

• The cube root of 269 is about 6.455315.
• Squaring undoes the root: (√269)² = 269, while 269² = 72,361 — the number whose square root is 269.

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