Square Root of 683

√683 at a glance

Exact value

√683

Decimal (10 places)

26.1342686907

Rounded

26.1 · 26.13 · 26.134

Perfect square?

No — between 26² and 27²

Rational?

Irrational

Both square roots

±26.134269

Prime factorization

683

Cube root

8.806572

How to simplify √683

683 is a prime number, so its only factors are 1 and 683. There is no perfect-square factor to pull out, which means √683 is already in its simplest radical form.

The square root of any prime is irrational. If √683 were a fraction a/b in lowest terms, then a² = 683b², so 683 would divide a — and then 683 would divide b too, contradicting “lowest terms.” That is why the decimal 26.1342686907 is only a rounded value.

Where √683 sits between perfect squares

676 = 26² and 729 = 27² are the nearest perfect squares, so √683 lies between 26 and 27. 683 is 7 above 676 and 46 below 729, so the root is closer to 26.

• Straight line between 676 and 729: 26.1321 (0.01% low)
• Tangent from 26, i.e. 26 + 7 ÷ 52: 26.1346 (0% high)
• Tangent from 27, i.e. 27 − 46 ÷ 54: 26.1481 (0.05% high)

For √683 the tangent at 26 wins, missing by only 0.0003. Tangent estimates shine when the number sits close to a perfect square — here 683 is just 7 above 676.

Finding √683 with the Babylonian method

If a guess is too big, 683 ÷ 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√683) in one step.

Start from the nearest whole number, 26 (26² = 676):

The count of correct decimals went 3, 8 and all 10 over 3 steps — roughly doubling each time — until the guess matched √683 = 26.1342686907 to every decimal shown.

√683 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √683 the pattern is [26; 7, 2, 4, 3, 1, 1, 25, 1, 1, 3, 4, 2, …] with the block of 14 terms after the semicolon repeating forever (only the first 12 of the 14 are shown). A pattern that never ends is one more proof that √683 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 − 683y² = 1. Its smallest solution in positive whole numbers is x = 170,067,682, y = 6,507,459.

√683 in geometry and everyday measurements

• 683 square feet is 63.5 m². Laid out as a square — a small house footprint or a lot — it is about 26.13 ft (26 ft 2 in) on a side.
• 683 is not a sum of two whole-number squares — 683 is itself a prime that is one less than a multiple of 4, which rules that out — so √683 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 3 × 7 × 25 box, because 3² + 7² + 25² = 683.

Square roots near √683 and related roots

• The cube root of 683 is about 8.806572.
• Squaring undoes the root: (√683)² = 683, while 683² = 466,489 — the number whose square root is 683.

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