Express imaginary numbers as bi and complex numbers as a+bi.
You really need only one new number to start working with the square roots of negative numbers. That number is the square root of −1,−1. 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 i and use itto represent the square root of −1.
i=−1
Because x⋅x=x, we can also see that −1⋅−1=−1 or i⋅i=−1. We also know that i⋅i=i2, so we can conclude that i2=−1.
i2=−1
The number i allows us to work with roots of all negative numbers, not just −1. There are two important rules to remember: −1=i, and ab=ab. You will use these rules to rewrite the square root of a negative number as the square root of a positive number times −1. Next you will simplify the square root and rewrite −1 as i. Let’s try an example.
Example
Simplify. −4
Answer: Use the rule ab=ab to rewrite this as a product using −1.
−4=4⋅−1=4−1
Since 4 is a perfect square (4=22), you can simplify the square root of 4.
4−1=2−1
Use the definition of i to rewrite −1 as i.2−1=2i
Answer
−4=2i
Example
Simplify. −18
Answer: Use the rule ab=ab to rewrite this as a product using −1.
−18=18⋅−1=18−1
Since 18 is not a perfect square, use the same rule to rewrite it using factors that are perfect squares. In this case, 9 is the only perfect square factor, and the square root of 9 is 3.
18−1=92−1=32−1
Use the definition of i to rewrite −1 as i.32−1=32i=3i2
Remember to write i in front of the radical.
Answer
−18=3i2
Example
Simplify. −−72
Answer: Use the rule ab=ab to rewrite this as a product using −1.
−−72=−72⋅−1=−72−1
Since 72 is not a perfect square, use the same rule to rewrite it using factors that are perfect squares. Notice that 72 has three perfect squares as factors: 4,9, and 36. It’s easiest to use the largest factor that is a perfect square.
−72−1=−362−1=−62−1
Use the definition of i to rewrite −1 as i.−62−1=−62i=−6i2
Remember to write i in front of the radical.
Answer
−−72=−6i2
You may have wanted to simplify −−72 using different factors. Some may have thought of rewriting this radical as −−98, or −−418, or −−612 for instance. Each of these radicals would have eventually yielded the same answer of −6i2.
In the following video, we show more examples of how to use imaginary numbers to simplify a square root with a negative radicand.
https://youtu.be/LSp7yNP6Xxc
Rewriting the Square Root of a Negative Number
Find perfect squares within the radical.
Rewrite the radical using the rule ab=a⋅b.
Rewrite −1 as i.
Example: −18=9−2=92−1=3i2
Complex Numbers
A complex number is the sum of a real number and an imaginary number. A complex number is expressed in standard form when written a + bi where a is the real part and bi is the imaginary part. For example, 5+2i is a complex number. So, too, is 3+43i.
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 a+bi, where a is a real number and bi is an imaginary number. The number a is sometimes called the real part of the complex number, and bi is sometimes called the imaginary part.
Complex Number
Real part
Imaginary part
3+7i
3
7i
18–32i
18
−32i
−53+i2
−53
i2
22−21i
22
−21i
In a number with a radical as part of b, such as −53+i2 above, the imaginary i should be written in front of the radical. Though writing this number as −53+2i is technically correct, it makes it much more difficult to tell whether i is inside or outside of the radical. Putting it before the radical, as in −53+i2, clears up any confusion. Look at these last two examples.
Number
Number in complex form:
a+bi
Real part
Imaginary part
17
17+0i
17
0i
−3i
0–3i
0
−3i
By making b=0, any real number can be expressed as a complex number. The real number a is written a+0i in complex form. Similarly, any imaginary number can be expressed as a complex number. By making a=0, any imaginary number bi is written 0+bi in complex form.
Example
Write 83.6 as a complex number.
Answer: Remember that a complex number has the form a+bi. You need to figure out what a and b need to be.
a+bi
Since 83.6 is a real number, it is the real part (a) of the complex number a+bi. A real number does not contain any imaginary parts, so the value of b is 0.83.6+bi
Answer
83.6+0i
Example
Write −3i as a complex number.
Answer: Remember that a complex number has the form a+bi. You need to figure out what a and b need to be.
a+bi
Since −3i is an imaginary number, it is the imaginary part bi of the complex number a+bi. This imaginary number has no real parts, so the value of a is 0.
a–3i
Answer
0–3i
In the next video we show more examples of how to write numbers as complex numbers.
https://youtu.be/mfoOYdDkuyY
Summary
Complex numbers have the form a+bi, where a and b are real numbers and i is the square root of −1. All real numbers can be written as complex numbers by setting b=0. Imaginary numbers have the form bi and can also be written as complex numbers by setting a=0. Square roots of negative numbers can be simplified using −1=i and ab=ab.
Licenses & Attributions
CC licensed content, Original
Write Number in the Form of Complex Numbers.Authored by: James Sousa (Mathispower4u.com) for Lumen Learning.License: CC BY: Attribution.
Simplify Square Roots to Imaginary Numbers.Authored by: James Sousa (Mathispower4u.com) for Lumen Learning.License: CC BY: Attribution.