(9.6.1) – Define imaginary and complex numbers

Up to now, you’ve known it was impossible to take a square root of a negative number. This is true, using only the real numbers. But here you will learn about a new kind of number that lets you work with square roots of negative numbers! Complex numbers are made from both real and imaginary numbers.  Imaginary numbers are called imaginary because they are impossible and, therefore, exist only in the world of ideas and pure imagination. Imaginary numbers result from taking the square root of a negative number.

Here we will first define and perform algebraic operations on complex numbers, then we will provide examples of quadratic equations that have solutions that are complex numbers.

Express roots of negative numbers in terms of [latex]i[/latex]

You really need only one new number to start working with the square roots of negative numbers. That number is the square root of [latex]−1,\sqrt{-1}[/latex]. The real numbers are those that can be shown on a number line—they seem pretty real to us! When something’s not real, you often say it is imaginary. So let’s call this new number [latex]i[/latex] and use it to represent the square root of [latex]−1[/latex].

[latex] i=\sqrt{-1}[/latex]

Because [latex] \sqrt{x}\,\cdot \,\sqrt{x}=x[/latex], we can also see that [latex] \sqrt{-1}\,\cdot \,\sqrt{-1}=-1[/latex] or [latex] i\,\cdot \,i=-1[/latex]. We also know that [latex] i\,\cdot \,i={{i}^{2}}[/latex], so we can conclude that [latex] {{i}^{2}}=-1[/latex].

[latex] {{i}^{2}}=-1[/latex]

The number [latex]i[/latex] allows us to work with roots of all negative numbers, not just [latex] \sqrt{-1}[/latex]. There are two important rules to remember: [latex] \sqrt{-1}=i[/latex], and [latex] \sqrt{ab}=\sqrt{a}\sqrt{b}[/latex]. You will use these rules to rewrite the square root of a negative number as the square root of a positive number times [latex] \sqrt{-1}[/latex]. Next you will simplify the square root and rewrite [latex] \sqrt{-1}[/latex] as [latex]i[/latex]. Let’s try an example.

You may have wanted to simplify [latex] -\sqrt{-72}[/latex] using different factors. Some may have thought of rewriting this radical as [latex] -\sqrt{-9}\sqrt{8}[/latex], or [latex] -\sqrt{-4}\sqrt{18}[/latex], or [latex] -\sqrt{-6}\sqrt{12}[/latex] for instance. Each of these radicals would have eventually yielded the same answer of [latex] -6i\sqrt{2}[/latex].

In the following video, we show more examples of how to use imaginary numbers to simplify a square root with a negative radicand.

Express imaginary numbers as [latex]bi[/latex] and complex numbers as [latex]a+bi[/latex]

A complex number is the sum of a real number and an imaginary number. A complex number is expressed in standard form when written [latex]a+bi[/latex] where [latex]a[/latex] is the real part and [latex]bi[/latex] is the imaginary part. For example, [latex]5+2i[/latex] is a complex number. So, too, is [latex]3+4\sqrt{3}i[/latex].

Imaginary numbers are distinguished from real numbers because a squared imaginary number produces a negative real number. Recall, when a positive real number is squared, the result is a positive real number and when a negative real number is squared, again, the result is a positive real number. Complex numbers are a combination of real and imaginary numbers.

You can use the usual operations (addition, subtraction, multiplication, and so on) with imaginary numbers. You’ll see more of that, later. When you add a real number to an imaginary number, however, you get a complex number. A complex number is any number in the form [latex]a+bi[/latex], where [latex]a[/latex] is a real number and [latex]bi[/latex] is an imaginary number. The number [latex]a[/latex] is sometimes called the real part of the complex number, and [latex]bi[/latex] is sometimes called the imaginary part.

In a number with a radical as part of [latex]b[/latex], such as [latex]\displaystyle -\frac{3}{5}+i\sqrt{2}[/latex] above, the imaginary [latex]i[/latex] should be written in front of the radical. Though writing this number as [latex]\displaystyle -\frac{3}{5}+\sqrt{2}i[/latex] is technically correct, it makes it much more difficult to tell whether [latex]i[/latex] is inside or outside of the radical. Putting it before the radical, as in [latex]\displaystyle -\frac{3}{5}+i\sqrt{2}[/latex], clears up any confusion. Look at these last two examples.

By making [latex]b=0[/latex], any real number can be expressed as a complex number. The real number [latex]a[/latex] is written [latex]a+0i[/latex] in complex form. Similarly, any imaginary number can be expressed as a complex number. By making [latex]a=0[/latex], any imaginary number [latex]bi[/latex] is written [latex]0+bi[/latex] in complex form.

In the next video we show more examples of how to write numbers as complex numbers.

(9.6.2) – Algebraic Operations on Complex Numbers

Add and Subtract Complex Numbers

Any time new kinds of numbers are introduced, one of the first questions that needs to be addressed is, “How do you add them?” In this topic, you’ll learn how to add complex numbers and also how to subtract.

First, consider the following expression.

[latex](6x+8)+(4x+2)[/latex]

To simplify this expression, you combine the like terms, [latex]6x[/latex] and [latex]4x[/latex]. These are like terms because they have the same variable with the same exponents. Similarly, [latex]8[/latex] and [latex]2[/latex] are like terms because they are both constants, with no variables.

[latex](6x+8)+(4x+2)=10x+10[/latex]

In the same way, you can simplify expressions with radicals.

[latex] (6\sqrt{3}+8)+(4\sqrt{3}+2)=10\sqrt{3}+10[/latex]

You can add [latex] 6\sqrt{3}[/latex] to [latex] 4\sqrt{3}[/latex] because the two terms have the same radical, [latex] \sqrt{3}[/latex], just as [latex]6x[/latex] and [latex]4x[/latex] have the same variable and exponent.

The number [latex]i[/latex] looks like a variable, but remember that it is equal to [latex]\sqrt{-1}[/latex]. The great thing is you have no new rules to worry about—whether you treat it as a variable or a radical, the exact same rules apply to adding and subtracting complex numbers. You combine the imaginary parts (the terms with [latex]i[/latex]), and you combine the real parts.

In the following video we show more examples of how to add and subtract complex numbers.

Multiply and Divide Complex Numbers

Multiplying complex numbers is much like multiplying binomials. The major difference is that we work with the real and imaginary parts separately.

Multiplying a Complex Number by a Real Number

Let’s begin by multiplying a complex number by a real number. We distribute the real number just as we would with a binomial. So,for [latex]3(6+2i)[/latex], 3 is multiplied to both the real and imaginary parts. So we have [latex](3)(6)+(3)(2i) = 18 + 6i[/latex].

How To: Given a complex number and a real number, multiply to find the product.

1. Use the distributive property.
2. Simplify.

Example

Find the product [latex]4\left(2+5i\right)[/latex].

Show Answer

Distribute the 4.

[latex]\begin{array}{cc}4\left(2+5i\right)&=&\left(4\cdot 2\right)+\left(4\cdot 5i\right)\hfill \\ &=&8+20i\hfill \end{array}[/latex]

 Multiplying Complex Numbers Together