Square Root of 409

√409 at a glance

Exact value

√409

Decimal (10 places)

20.2237484162

Rounded

20.2 · 20.22 · 20.224

Perfect square?

No — between 20² and 21²

Rational?

Irrational

Both square roots

±20.223748

Prime factorization

409

Cube root

7.422914

How to simplify √409

409 is a prime number, so its only factors are 1 and 409. There is no perfect-square factor to pull out, which means √409 is already in its simplest radical form.

The square root of any prime is irrational. If √409 were a fraction a/b in lowest terms, then a² = 409b², so 409 would divide a — and then 409 would divide b too, contradicting “lowest terms.” That is why the decimal 20.2237484162 is only a rounded value.

Where √409 sits between perfect squares

400 = 20² and 441 = 21² are the nearest perfect squares, so √409 lies between 20 and 21. 409 is 9 above 400 and 32 below 441, so the root is closer to 20.

• Straight line between 400 and 441: 20.2195 (0.02% low)
• Tangent from 20, i.e. 20 + 9 ÷ 40: 20.2250 (0.01% high)
• Tangent from 21, i.e. 21 − 32 ÷ 42: 20.2381 (0.07% high)

For √409 the tangent at 20 wins, missing by only 0.0013. Tangent estimates shine when the number sits close to a perfect square — here 409 is just 9 above 400.

Finding √409 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 409: following the tangent line down to zero simplifies to averaging x with 409 ÷ x.

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

The count of correct decimals went 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √409 = 20.2237484162 to every decimal shown.

√409 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √409 the pattern is [20; 4, 2, 7, 1, 1, 1, 4, 2, 2, 13, 13, 2, …] 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 √409 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 − 409y² = 1. Its smallest solution in positive whole numbers is x = 25,052,977,273,092,427,986,049, y = 1,238,789,998,647,218,582,160 — 23 digits for x, even though 409 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: 111,921,796,968² − 409 × 5,534,176,685² = −1.

√409 in geometry and everyday measurements

• A square garage floor of 409 square feet measures about 20.22 ft (20 ft 3 in) per side, and its corner-to-corner diagonal is √818 ≈ 28.6 ft.
• 409 = 3² + 20², so by the Pythagorean theorem √409 is the diagonal of a 3 × 20 rectangle — and the distance between the points (0, 0) and (3, 20) on a grid.

Square roots near √409 and related roots

• The cube root of 409 is about 7.422914.
• Squaring undoes the root: (√409)² = 409, while 409² = 167,281 — the number whose square root is 409.

Need a different value? Use the Square root calculator, or browse the Square roots 1–1,000.