Square root

The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3, because 3 multiplied by 3 is 9. The square root operation is fundamental in mathematics, especially in algebra, geometry, and calculus. The symbol for the square root is √, known as the radical sign.

Definition

The square root of a number x is denoted as √x. For any positive real number x, there are two square roots: the positive square root, denoted as √x, and the negative square root, denoted as -√x. For example, the square roots of 25 are 5 and -5, because both 5² (5*5) and (-5)² (-5*-5) equal 25. The square root of zero is zero.

Properties

Square roots have several important properties. One of the most fundamental is that the square root of a product is equal to the product of the square roots of the factors: √(ab) = √a * √b, where a and b are positive real numbers. Another important property is that the square root of a fraction is equal to the square root of the numerator divided by the square root of the denominator: √(a/b) = √a / √b.

Complex Numbers

For negative numbers, the square root is not a real number but a complex number. The square root of a negative number a is defined as √a = i√|a|, where i is the imaginary unit, satisfying i² = -1. For example, the square root of -9 is 3i, because (3i)² = -9.

Calculating Square Roots

There are various methods for calculating square roots, ranging from simple estimation to complex algorithms like the Newton-Raphson method. Before the advent of calculators, methods such as the Babylonian method were commonly used for manual calculations.

Applications

Square roots are used in many areas of mathematics and applied sciences. They are essential in solving quadratic equations, calculating distances in geometry using the Pythagorean theorem, and in statistical formulas such as the standard deviation. In engineering, square roots are used in calculations involving areas and volumes.

See Also

• Exponentiation
• Radical expression
• Complex number
• Pythagorean theorem
• Quadratic equation