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Square Roots: Babylonian Clay Tablets to Your Calculator

By Tally Calculators · Published Oct 27, 2026 · Updated Oct 27, 2026 · 3 min read

The square root of a number is the value that, multiplied by itself, gives that number. The square root of 144 is 12 because 12 × 12 = 144. Humans have been calculating square roots for nearly four thousand years, and the methods have changed less than you might expect.

What a square root means

Geometrically, the square root of an area is the side length of a square with that area. Algebraically, √x is the same as x raised to the power ½. Every positive number has two square roots, a positive and a negative one (both 12 and −12 square to 144), and the √ sign by convention means the positive one. Roots of perfect squares such as 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121 and 144 are whole numbers. Most other roots are irrational: their decimals never end or repeat.

YBC 7289: a 3,700-year-old calculation

A small clay tablet held at Yale University, catalogued as YBC 7289 and dated to roughly 1800–1600 BCE, shows a square with its two diagonals drawn. Along one side is the number 30, and on the diagonal appears the sexagesimal number 1;24,51,10. Converted to decimal, that is 1.41421296…, an approximation of √2 that is correct to about six decimal places. Multiplying by 30 gives the diagonal, about 42.426.

Heron’s (the Babylonian) method

The method is simple: guess a root, divide the number by your guess, and take the average of the two. Repeat. To find √2, start with 1.5:

The number of correct digits roughly doubles each step. The technique is named after Hero of Alexandria, who described it in the first century CE, and it is a special case of the Newton–Raphson method developed in the 17th century.

The irrational √2

Ancient Greek mathematicians proved that √2 cannot be written as a fraction of whole numbers, a discovery that shocked those who believed all quantities were ratios. The proof is by contradiction and fits in a few lines, which makes it a favourite introduction to mathematical proof.

The radical sign and negative roots

The √ symbol appeared in print in the 16th century. It is usually credited to the German mathematician Christoff Rudolff, who used it in his 1525 algebra text Die Coss. Roots of negative numbers were long considered impossible until Italian mathematicians such as Rafael Bombelli, in 1572, showed how to work with them, laying the groundwork for imaginary and complex numbers. Today √−1 is written as i.

How calculators and computers do it

Modern processors use fast iterative algorithms from the same family as Heron’s method, often with a table-based first guess so only a few steps are needed. The IEEE 754 floating-point standard requires square root to be correctly rounded, so you get the closest possible answer. A famous shortcut known as the “fast inverse square root”, used in the source code of the 1999 game Quake III Arena, used a magic constant to get an approximation quickly for 3D graphics.

Handy tips

Frequently asked questions

How do you find a square root by hand?

Use a guess-and-average method like Heron’s: average your guess with the number divided by your guess, and repeat until it settles.

Is the square root of 2 rational?

No. It is irrational, so its decimal expansion never ends or repeats.

Can you take the square root of a negative number?

Not within the real numbers. In complex numbers the square root of −1 is defined as i.

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This article is for general information only and is not financial, medical or professional advice.