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Imaginary Numbers

Imaginary Numbers

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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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Tutorial
This tutorial covers imaginary numbers, through the definition and discussion of:
  1. Squaring and Square Roots: A Review
  2. Imaginary Numbers
  3. Writing Imaginary Numbers


1. Squaring and Square Roots: A Review

The square root of a number x is the number whose product with itself is x.

If you square the number -2, it equals 4. If you square the number 2, it also equals 4.
left parenthesis negative 2 right parenthesis squared equals 4 space space space space space space space space space space 2 squared equals 4

As you can see from the examples above, when you square any real number, the result will never be a negative number.


2. Imaginary Numbers

Since the square of a real number cannot be negative, the square root of a negative number must be a non-real number, otherwise known as an imaginary number. The imaginary unit, i, is defined as the square root of -1.

KEY FORMULA
square root of negative 1 end root equals i
Imaginary Unit
The square root of -1, denoted by i

Imaginary numbers may be the result of solving a quadratic equation using the quadratic formula.

KEY FORMULA
x equals fraction numerator negative b plus-or-minus square root of b squared minus 4 a c end root over denominator 2 a end fraction

3. Writing Imaginary Numbers

Imaginary numbers are written using the imaginary unit i in the form bi (b times i), where b is a real number. Recall that the product property for square roots states 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:

square root of a b end root equals square root of a times square root of b

You can also use the product property for square roots of negative numbers in the form bi.

The square root of -9 can be written as the square root of 9 times the square root of -1. The square root of 9 is 3, and the square root of -1 is defined as i. Therefore, you can write the square root of -9 as 3i.
square root of negative 9 end root equals square root of 9 times square root of negative 1 end root equals 3 i

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. For example, suppose you want to simplify the expression:

square root of 12 minus left parenthesis 7 minus 3 right parenthesis squared end root

You can start by simplifying in your parentheses.

square root of 12 minus 4 squared end root

Next, you square the 4, and subtract your terms, which equals -4.

table attributes columnalign left end attributes row cell square root of 12 minus 16 end root equals end cell row cell square root of negative 4 end root end cell end table

Using the product property for square roots, you can rewrite the square root of -4 as the square root of 4 times the square root of -1. The square root of 4 is 2, and the square root of -1 is i, so your final answer is 2i.

square root of 4 times square root of negative 1 end root equals 2 i
Now, suppose you want to simplify the expression:
square root of 4 times 3 minus 15 end root
Simplify underneath the square root, starting with multiplication, followed by subtraction.
table attributes columnalign left end attributes row cell square root of 12 minus 15 end root equals end cell row cell square root of negative 3 end root end cell end table
Now you can rewrite your expression using the product property of square roots. Note that since the square root of 3 is not an integer, you would leave it as the square root of 3. Since the square root of -1 is i, your final answer is the square root of 3 times i.
square root of 3 times square root of negative 1 end root equals square root of 3 i

Today you reviewed squaring and square roots, recalling that the square root of a number x is the number whose product with itself is x. Remember, the square of any real number will never be a negative number, and the square root of a negative number must be a non-real or imaginary number. You learned that this imaginary unit i is defined as the square root of -1. Lastly, you learned that when writing imaginary numbers, you use the imaginary unit, i, in the form bi, where b is a real number.

Source: This work is adapted from Sophia author Colleen Atakpu.

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.