Complex Numbers

Description

Complex numbers are numbers that can be written in the form $a+bi$, where $a$ and $b$ are real numbers and $i$ is $\sqrt{-1}$, the imaginary unit. The steps of adding, subtracting, and multiplying complex numbers are much the same as those used to perform these operations with algebraic expressions. Division of complex numbers, however, is similar to the process of rationalizing the denominator of a fraction. The fact $i^2=-1$ is useful for simplifying products and quotients of complex numbers.

A quadratic equation with no real roots has two roots that are complex numbers. An understanding of complex numbers is necessary for solving this type of quadratic equation. Quadratic equations with complex roots can be solved using some of the same methods used for quadratic equations with real roots, such as completing the square or using the quadratic formula.

At A Glance

• A solution of the equation $x^2 = -1$ is defined as the imaginary unit, written as $i$.
• Multiples of $i$ are called pure imaginary numbers, written in the form $bi$, where $b$ is a real number.
• A complex number is the sum of a real number and a pure imaginary number. Complex numbers can be written in the form $a + bi$, where $a$ and $b$ are real numbers. Complex numbers can be graphed in a coordinate plane with a real axis and an imaginary axis.
• Complex numbers are added and subtracted by combining like terms.
• Powers of i are computed using the property $i^2 = -1$. Complex numbers are multiplied by using the distributive property and powers of $i$.
• Division by a complex number $a + bi$ is performed by multiplying the numerator and denominator by the complex conjugate, $a - bi$.
• Quadratic equations with no linear term can be solved by isolating $x^2$ and taking the square root of both sides of the equation. If $x^2$ is equal to a negative number, then the solutions of the equation are pure imaginary numbers.
• Quadratic equations with no real roots can be solved by completing the square.
• Quadratic equations with no real roots can be solved by using the quadratic formula.