Square Root of 673

√673 at a glance

Exact value

√673

Decimal (10 places)

25.9422435421

Rounded

25.9 · 25.94 · 25.942

Perfect square?

No — between 25² and 26²

Rational?

Irrational

Both square roots

±25.942244

Prime factorization

673

Cube root

8.763381

How to simplify √673

673 is a prime number, so its only factors are 1 and 673. There is no perfect-square factor to pull out, which means √673 is already in its simplest radical form.

The square root of any prime is irrational. If √673 were a fraction a/b in lowest terms, then a² = 673b², so 673 would divide a — and then 673 would divide b too, contradicting “lowest terms.” That is why the decimal 25.9422435421 is only a rounded value.

Where √673 sits between perfect squares

625 = 25² and 676 = 26² are the nearest perfect squares, so √673 lies between 25 and 26. 673 is 48 above 625 and 3 below 676, so the root is closer to 26.

• Straight line between 625 and 676: 25.9412 (0% low)
• Tangent from 25, i.e. 25 + 48 ÷ 50: 25.9600 (0.07% high)
• Tangent from 26, i.e. 26 − 3 ÷ 52: 25.9423 (0% high)

For √673 the tangent at 26 wins, missing by only 0.0001. Tangent estimates shine when the number sits close to a perfect square — here 673 is just 3 below 676.

Finding √673 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 673: following the tangent line down to zero simplifies to averaging x with 673 ÷ x.

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

The count of correct decimals went 4, 10 and all 10 over 3 steps — roughly doubling each time — until the guess matched √673 = 25.9422435421 to every decimal shown.

√673 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √673 the pattern is [25; 1, 16, 3, 5, 2, 3, 1, 1, 6, 1, 5, 1, …] with the block of 31 terms after the semicolon repeating forever (only the first 12 of the 31 are shown). A pattern that never ends is one more proof that √673 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 − 673y² = 1. Its smallest solution in positive whole numbers is x = 4,765,506,835,465,395,993,032,041,249, y = 183,696,788,896,587,421,699,032,600 — 28 digits for x, even though 673 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: 48,813,455,293,932² − 673 × 1,881,620,424,025² = −1.

√673 in geometry and everyday measurements

• 673 square feet is 62.5 m². Laid out as a square — a small house footprint or a lot — it is about 25.94 ft (25 ft 11 in) on a side.
• 673 = 12² + 23², so by the Pythagorean theorem √673 is the diagonal of a 12 × 23 rectangle — and the distance between the points (0, 0) and (12, 23) on a grid.

Square roots near √673 and related roots

• The cube root of 673 is about 8.763381.
• Squaring undoes the root: (√673)² = 673, while 673² = 452,929 — the number whose square root is 673.

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