How to Calculate Square Roots
This calculator determines the square root (sqrt) and cube root (cbrt) of a number, also checking whether it is a perfect square or perfect cube.
The Square Root
The square root of a number n is the value x such that x² = n. Geometrically, if n is the area of a square, sqrt(n) is the length of its side. The square root is defined only for non-negative numbers in the real number system.
Perfect squares are numbers whose root is an integer: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100... All other positive numbers have irrational roots -- numbers with infinitely many non-repeating decimal digits.
The Cube Root
The cube root of n is the value x such that x³ = n. Geometrically, if n is the volume of a cube, cbrt(n) is the length of its edge. Unlike the square root, the cube root is also defined for negative numbers: cbrt(-8) = -2.
The Discovery of Irrationals
The Pythagorean school (5th century BC) discovered with dismay that sqrt(2) cannot be expressed as a ratio of integers. This discovery, attributed to Hippasus of Metapontum, was revolutionary: it demonstrated that rational numbers are not sufficient to describe reality. The proof by contradiction that sqrt(2) is irrational is one of the jewels of mathematics.
Historical Calculation Methods
Before calculators, the Babylonian method (or Heron's method) was used: start from an initial estimate x0 and iteratively improve with x_{n+1} = (x_n + n/x_n)/2. This method converges rapidly: for sqrt(2), starting from x0 = 1, after only 4 iterations you get an approximation correct to 8 decimal places.
Useful Properties of Roots
- sqrt(a x b) = sqrt(a) x sqrt(b) -- the root of a product is the product of roots
- sqrt(a / b) = sqrt(a) / sqrt(b) -- the root of a quotient is the quotient of roots
- (sqrt(a))² = a -- the square root is the inverse of squaring
- sqrt(a²) = |a| -- the result is always non-negative