Cube root

In mathematics, a cube root of a number x is a number y that, when multiplied by itself three times, gives the original number x. This means y times y times y equals x, or y³ = x. For example, the cube root of 8 is 2 because 2 × 2 × 2 = 8.

Every real number has exactly one real cube root. This is the main cube root we usually think about. For positive numbers, this real cube root is also the main, or principal, cube root. But for negative numbers, the real cube root is different from the principal cube root, which can be a complex number.

Complex numbers, which include numbers with an imaginary part, have three cube roots each. For example, the number 8 has one real cube root, which is 2, and two other cube roots that are complex numbers. These complex cube roots help us understand more about how numbers behave in advanced math.

The cube root is a special kind of math operation, called a multivalued function, because a number can have more than one cube root depending on the type of numbers we are looking at. In some special number systems, a number might even have infinitely many cube roots!

Definition

The cube roots of a number x are the numbers y which satisfy the equation y³ = x. This means that when you multiply y by itself three times, you get x.

Properties

For any real number, there is exactly one real cube root. This means if you multiply this number by itself three times, you get the original number. For example, the cube root of 8 is 2, because 2 × 2 × 2 = 8.

When we think about numbers that include imaginary parts (complex numbers), there are three cube roots for each number. For real numbers, one of these roots is real, and the other two are complex numbers that are opposites of each other. For example, the cube roots of 1 are 1, and two complex numbers that are mirrors of each other.

Impossibility of compass-and-straightedge construction

Cube roots help us solve two famous geometry puzzles. The first puzzle is about dividing an angle into three equal parts, called angle trisection. The second puzzle is about finding the side length of a cube that has twice the volume of another cube with a known side length, called doubling the cube.

In 1837, a mathematician named Pierre Wantzel proved that neither of these puzzles can be solved using just a compass and straightedge.

Numerical methods

Newton's method is a step-by-step way to find the cube root of a number. It starts with a guess and gets better each time you repeat the steps.

Halley's method is another step-by-step way that can find the answer faster, but it needs a bit more work each time.

Both methods need a good starting guess to work well. Sometimes people change parts of the number to get a better starting point. There is also a special way to keep getting closer to the answer using parts of the number itself.

Appearance in solutions of third and fourth degree equations

Cubic equations are special math problems where the highest power of the unknown number is 3. These can always be solved using cube roots and square roots. Sometimes, the answers can be simpler if one of the solutions is a simple whole number, called a rational number.

Quartic equations are another type of math problem that can also be solved using cube roots and square roots.

History

The idea of finding cube roots dates back thousands of years. Ancient Babylonian mathematicians around 1800 BCE were already thinking about it. Later, around the fourth century BCE, the famous thinker Plato asked a tricky question: how to make a cube’s volume double using only a compass and straightedge. This would need a special length that turned out to be impossible to create with those tools.

Different cultures developed ways to calculate cube roots. A Chinese book called The Nine Chapters on the Mathematical Art, written around the second century BCE, included a method for this. Around the first century CE, Hero of Alexandria, a Greek mathematician, also created a way to find cube roots. Much later, in 499 CE, the Indian mathematician Aryabhata shared a method for finding cube roots of large numbers in his work, the Aryabhatiya.