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 is not real, we often say it is imaginary. So let us 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 us 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
The answer is 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.
The answer is 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 is 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.
The answer is −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+4i3.
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 will see more of that later.
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
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 as a+0i in complex form. Similarly, any imaginary number can be expressed as a complex number. By making a=0, any imaginary number bi can be written as 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. 83.6+bi
A real number does not contain any imaginary parts, so the value of b is 0.
The answer is 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.
a–3i
This imaginary number has no real parts, so the value of a is 0.
The answer is 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.