Basics of complex numbers and how to manipulate them in equations

Complex numbers are numbers that have both a real and imaginary component. In this tutorial, we will introduce the basics of complex numbers and how to manipulate them in equations.

A complex number is typically written in the form a + bi, where a is the real part and b is the imaginary part. The real part of a complex number is a regular number that we are familiar with, such as 1, 2, or 3. The imaginary part of a complex number is a multiple of the square root of -1, which is denoted by the symbol i. For example, the complex number 2 + 3i has a real part of 2 and an imaginary part of 3i.

To manipulate complex numbers in equations, we can use the same rules and operations that we use for regular numbers. For example, we can add and subtract complex numbers by adding or subtracting the real and imaginary parts separately. So, to add the complex numbers 2 + 3i and 4 + 5i, we would add the real parts to get 6 and add the imaginary parts to get 8i.

We can also multiply and divide complex numbers using the rules of algebra. For example, to multiply the complex numbers 2 + 3i and 4 + 5i, we would use the distributive property to get 8 + 12i + 12i + 15i^2. Since i^2 is equal to -1, this simplifies to 8 + 12i – 15, which is equal to -7 + 12i.

In this tutorial, we will go over these concepts in more detail and practice manipulating complex numbers in equations. We will also learn about the conjugate of a complex number and how to use it to simplify complex equations. By the end of this tutorial, you should have a good understanding of the basics of complex numbers and be able to manipulate them in equations.