Square root

In mathematics, a square root of a number x is a number y such that y² = x. This means that when you multiply y by itself, you get x. For example, both 4 and -4 are square roots of 16 because 4² = (-4)² = 16.

Every nonnegative real number x has a unique nonnegative square root, called the principal square root, which is denoted by √x. The symbol √ is known as the radical sign or radix. For instance, we write √9 = 3 to show that the principal square root of 9 is 3.

Every positive number x actually has two square roots: √x (which is positive) and -√x (which is negative). These can be written together as ±√x using the ± sign. Square roots of negative numbers are explored in the study of complex numbers, showing how mathematics can expand to include more types of numbers.

History

The idea of square roots has been around for thousands of years. Ancient civilizations like the Babylonians and Egyptians knew how to work with square roots using different methods. For example, a Babylonian clay tablet from around 1800–1600 BC shows an early approximation of the square root of 2.

Later, mathematicians in Ancient India developed methods to find square roots of numbers. The ancient Greeks discovered that the square roots of numbers that are not perfect squares, like the square root of 2, are irrational numbers—numbers that cannot be written as a simple fraction. This discovery was important in the history of mathematics. Over time, new symbols were invented to represent square roots, with the modern symbol "√" first appearing in print in 1525.

Properties and uses

The square root of a number tells us what number, when multiplied by itself, gives the original number. For example, the square root of 16 is 4 because 4 × 4 equals 16. Every positive number has a special nonnegative square root, called the principal square root, which we write using the symbol √.

Square roots are important in geometry. If you know the area of a square, the square root gives you the length of one of its sides. Some square roots are simple whole numbers, like the square root of 4, which is 2. Other square roots are decimals that go on forever without repeating, like the square root of 2, which starts with 1.414….

Computation

Square roots of positive numbers are usually not simple decimals, so we can only get close guesses when we try to calculate them. Most calculators and computer programs have tools to help find these guesses quickly.

One common way to find square roots by hand is called the Babylonian method. It works by repeatedly averaging a guess with the number divided by that guess. Each time you do this, the guess gets closer to the real square root. This method works well and can give very accurate answers after just a few steps.

Square roots of negative and complex numbers

The square of any positive or negative number is positive, and the square of 0 is 0. This means no negative number has a real square root. However, we can find square roots for negative numbers by using complex numbers. Complex numbers include a special number called i, defined so that i² = −1. This lets us find square roots of negative numbers. For example, the principal square root of −1 is i.

For any non-zero complex number, there are exactly two numbers that, when squared, give the original number. One is called the principal square root, and the other is its negative. This helps us work with square roots in more advanced mathematics.