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Imaginary Numbers

Imaginary Numbers

Author: Colleen Atakpu
Description:

In this lesson, students will learn how to write imaginary numbers, and how to apply the definition of the imaginary unit i.

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[MUSIC PLAYING] Let's look at our objectives for today. We'll start by reviewing squaring and taking the square root. We'll then look at how to write imaginary numbers. And finally, we'll do some examples writing imaginary numbers.

Let's start by reviewing squaring and taking the square root. The square root of a number x is the number whose product with itself is x. If we square the number negative 2, negative 2 squared gives us a positive 4. And if we square the number positive 2, 2 squared also gives us a positive 4.

Therefore, when we square any real number, the result will never be a negative number. Since the square of a real number cannot be negative, the square root of a negative number must be a nonreal, or what we call an imaginary number. We define the imaginary unit i as the square root of negative 1. Imaginary numbers may be the result of solving a quadratic equation using the quadratic formula.

Now let's look at how to write imaginary numbers. Imaginary numbers are written using the imaginary unit i in the form b times i, where b is a real number. Recall that the product property for square roots says that for positive numbers a and b, the square root of a times b is equal to the square root of a times the square root of b.

We can also use the product property for square roots of negative numbers in the form b times i. For example, the square root of negative 9 can be written as the square root of 9 times the square root of negative 1. The square root of 9 is 3, and we define the square root of negative 1 as i. Therefore, we can write the square root of negative 9 as 3i. Let's do some more examples writing imaginary numbers.

Being able to identify perfect squares and appropriately using the product property for square roots is important when you are simplifying square roots and writing imaginary numbers. Here's our first example. We want to simplify the expression the square root of 12 minus 7 minus 3 squared. We start by simplifying in our parentheses. 7 minus 3 gives us 4. We then square the 4, which gives us 16.

We then subtract 12 minus 16, which gives us a negative 4. Using the product property for square roots, we can rewrite the square root of negative 4 as the square root of 4 times the square root of negative 1. The square root of 4 is 2, and the square root of negative 1 is i. So our final answer is 2i. Here's our second example. We want to simplify the expression the square root of 4 times 3 minus 15.

We start by simplifying with multiplying 4 times 3, which gives us 12. We then subtract 12 minus 15, which gives us a negative 3. We can rewrite the square root of negative 3 using the product property of square roots as the square root of 3 times the square root of negative 1. The square root of 3 is not an integer. So we leave it as the square root of 3. And the square root of negative 1 is i. So our final answer is the square root of 3 times i.

Let's go over our important points from today. Make sure you get these in your notes so you can refer to them later. The square root of a number x is the number whose product with itself is x. The square of any number will never be a negative number. Since the square of a real number cannot be negative, the square root of a negative number must be a non-real or imaginary number.

We define the imaginary unit i as the square root of negative 1. Imaginary numbers are written using the imaginary unit i in the form b times i, where b is a real number. So I hope that these important points and examples helped you understand a little bit more about imaginary numbers. Keep using your notes and keep on practicing, and soon you'll be a pro. Thanks for watching.

Notes on "Imaginary Numbers"

00:00 – 00:31 Introduction

00:32 – 01:31 Review of Squaring and Taking Square Roots

01:32 – 02:26 Writing Imaginary Numbers

02:27 – 04:19 Examples Writing Imaginary Numbers

04:20 – 05:19 Important to Remember (Recap)

TERMS TO KNOW
  • KEY FORMULA

    sqrt(-1) = i

  • KEY FORMULA

    x = [-b ± sqrt(b^2-4ac)]/2a

  • Imaginary Unit

    The square root of -1, denoted by i.