Square Root of 859

√859 at a glance

Exact value

√859

Decimal (10 places)

29.3087017795

Rounded

29.3 · 29.31 · 29.309

Perfect square?

No — between 29² and 30²

Rational?

Irrational

Both square roots

±29.308702

Prime factorization

859

Cube root

9.505998

How to simplify √859

859 is a prime number, so its only factors are 1 and 859. There is no perfect-square factor to pull out, which means √859 is already in its simplest radical form.

The square root of any prime is irrational. If √859 were a fraction a/b in lowest terms, then a² = 859b², so 859 would divide a — and then 859 would divide b too, contradicting “lowest terms.” That is why the decimal 29.3087017795 is only a rounded value.

Where √859 sits between perfect squares

841 = 29² and 900 = 30² are the nearest perfect squares, so √859 lies between 29 and 30. 859 is 18 above 841 and 41 below 900, so the root is closer to 29.

• Straight line between 841 and 900: 29.3051 (0.01% low)
• Tangent from 29, i.e. 29 + 18 ÷ 58: 29.3103 (0.01% high)
• Tangent from 30, i.e. 30 − 41 ÷ 60: 29.3167 (0.03% high)

For √859 the tangent at 29 wins, missing by only 0.0016. Tangent estimates shine when the number sits close to a perfect square — here 859 is just 18 above 841.

Finding √859 with the Babylonian method

If a guess is too big, 859 ÷ 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√859) in one step.

Start from the nearest whole number, 29 (29² = 841):

The count of correct decimals went 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √859 = 29.3087017795 to every decimal shown.

√859 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √859 the pattern is [29; 3, 4, 5, 1, 1, 1, 2, 2, 3, 2, 19, 9, …] with the block of 46 terms after the semicolon repeating forever (only the first 12 of the 46 are shown). A pattern that never ends is one more proof that √859 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 − 859y² = 1. Its smallest solution in positive whole numbers is x = 2,058,844,771,979,643,060,124,010, y = 70,246,877,103,894,937,291,269 — 25 digits for x, even though 859 is small, which is what makes Pell’s equation famous.

√859 in geometry and everyday measurements

• 859 square feet is 79.8 m². Laid out as a square — a small house footprint or a lot — it is about 29.31 ft (29 ft 4 in) on a side.
• 859 is not a sum of two whole-number squares — 859 is itself a prime that is one less than a multiple of 4, which rules that out — so √859 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 3 × 3 × 29 box, because 3² + 3² + 29² = 859.

Square roots near √859 and related roots

• The cube root of 859 is about 9.505998.
• Squaring undoes the root: (√859)² = 859, while 859² = 737,881 — the number whose square root is 859.

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