Square Root of 433

√433 at a glance

Exact value

√433

Decimal (10 places)

20.8086520467

Rounded

20.8 · 20.81 · 20.809

Perfect square?

No — between 20² and 21²

Rational?

Irrational

Both square roots

±20.808652

Prime factorization

433

Cube root

7.565355

How to simplify √433

433 is a prime number, so its only factors are 1 and 433. There is no perfect-square factor to pull out, which means √433 is already in its simplest radical form.

The square root of any prime is irrational. If √433 were a fraction a/b in lowest terms, then a² = 433b², so 433 would divide a — and then 433 would divide b too, contradicting “lowest terms.” That is why the decimal 20.8086520467 is only a rounded value.

Where √433 sits between perfect squares

400 = 20² and 441 = 21² are the nearest perfect squares, so √433 lies between 20 and 21. 433 is 33 above 400 and 8 below 441, so the root is closer to 21.

• Straight line between 400 and 441: 20.8049 (0.02% low)
• Tangent from 20, i.e. 20 + 33 ÷ 40: 20.8250 (0.08% high)
• Tangent from 21, i.e. 21 − 8 ÷ 42: 20.8095 (0% high)

For √433 the tangent at 21 wins, missing by only 0.0009. Tangent estimates shine when the number sits close to a perfect square — here 433 is just 8 below 441.

Finding √433 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 433: following the tangent line down to zero simplifies to averaging x with 433 ÷ x.

Start from the nearest whole number, 21 (21² = 441):

The count of correct decimals went 3, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √433 = 20.8086520467 to every decimal shown.

√433 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √433 the pattern is [20; 1, 4, 4, 2, 2, 1, 3, 13, 1, 1, 1, 1, …] with the block of 21 terms after the semicolon repeating forever (only the first 12 of the 21 are shown). A pattern that never ends is one more proof that √433 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 − 433y² = 1. Its smallest solution in positive whole numbers is x = 104,564,907,854,286,695,713, y = 5,025,068,784,834,899,736 — 21 digits for x, even though 433 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: 7,230,660,684² − 433 × 347,483,377² = −1.

√433 in geometry and everyday measurements

• A square garage floor of 433 square feet measures about 20.81 ft (20 ft 10 in) per side, and its corner-to-corner diagonal is √866 ≈ 29.4 ft.
• 433 = 12² + 17², so by the Pythagorean theorem √433 is the diagonal of a 12 × 17 rectangle — and the distance between the points (0, 0) and (12, 17) on a grid.

Square roots near √433 and related roots

• The cube root of 433 is about 7.565355.
• Squaring undoes the root: (√433)² = 433, while 433² = 187,489 — the number whose square root is 433.

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