# 3.1 Complex numbers

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In this section, you will:
• Express square roots of negative numbers as multiples of  i.
• Plot complex numbers on the complex plane.
• Add and subtract complex numbers.
• Multiply and divide complex numbers.

The study of mathematics continuously builds upon itself. Negative integers, for example, fill a void left by the set of positive integers. The set of rational numbers, in turn, fills a void left by the set of integers. The set of real numbers fills a void left by the set of rational numbers. Not surprisingly, the set of real numbers has voids as well. For example, we still have no solution to equations such as

${x}^{2}+4=0$

Our best guesses might be +2 or –2. But if we test +2 in this equation, it does not work. If we test –2, it does not work. If we want to have a solution for this equation, we will have to go farther than we have so far. After all, to this point we have described the square root of a negative number as undefined. Fortunately, there is another system of numbers that provides solutions to problems such as these. In this section, we will explore this number system and how to work within it.

## Expressing square roots of negative numbers as multiples of i

We know how to find the square root of any positive real number. In a similar way, we can find the square root of a negative number. The difference is that the root is not real. If the value in the radicand is negative, the root is said to be an imaginary number     . The imaginary number $\text{\hspace{0.17em}}i\text{\hspace{0.17em}}$ is defined as the square root of negative 1.

$\sqrt{-1}=i$

${i}^{2}={\left(\sqrt{-1}\right)}^{2}=-1$

We can write the square root of any negative number as a multiple of $\text{\hspace{0.17em}}i.\text{\hspace{0.17em}}$ Consider the square root of –25.

We use $\text{\hspace{0.17em}}5i\text{\hspace{0.17em}}$ and not $\text{\hspace{0.17em}}-\text{5}i\text{\hspace{0.17em}}$ because the principal root of $\text{\hspace{0.17em}}25\text{\hspace{0.17em}}$ is the positive root.

A complex number is the sum of a real number and an imaginary number. A complex number is expressed in standard form when written $\text{\hspace{0.17em}}a+bi\text{\hspace{0.17em}}$ where $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ is the real part and $\text{\hspace{0.17em}}bi\text{\hspace{0.17em}}$ is the imaginary part. For example, $\text{\hspace{0.17em}}5+2i\text{\hspace{0.17em}}$ is a complex number. So, too, is $\text{\hspace{0.17em}}3+4\sqrt{3}i.$

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.

## Imaginary and complex numbers

A complex number    is a number of the form $\text{\hspace{0.17em}}a+bi\text{\hspace{0.17em}}$ where

• $a\text{\hspace{0.17em}}$ is the real part of the complex number.
• $bi\text{\hspace{0.17em}}$ is the imaginary part of the complex number.

If $\text{\hspace{0.17em}}b=0,\text{\hspace{0.17em}}$ then $\text{\hspace{0.17em}}a+bi\text{\hspace{0.17em}}$ is a real number. If $\text{\hspace{0.17em}}a=0\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}b\text{\hspace{0.17em}}$ is not equal to 0, the complex number is called an imaginary number . An imaginary number is an even root of a negative number.

Given an imaginary number, express it in standard form.

1. Write $\text{\hspace{0.17em}}\sqrt{-a}\text{\hspace{0.17em}}$ as $\text{\hspace{0.17em}}\sqrt{a}\sqrt{-1}.$
2. Express $\text{\hspace{0.17em}}\sqrt{-1}\text{\hspace{0.17em}}$ as $\text{\hspace{0.17em}}i.$
3. Write $\text{\hspace{0.17em}}\sqrt{a}\cdot i\text{\hspace{0.17em}}$ in simplest form.

## Expressing an imaginary number in standard form

Express $\text{\hspace{0.17em}}\sqrt{-9}\text{\hspace{0.17em}}$ in standard form.

$\sqrt{-9}=\sqrt{9}\sqrt{-1}=3i$

In standard form, this is $\text{\hspace{0.17em}}0+3i.$

Express $\text{\hspace{0.17em}}\sqrt{-24}\text{\hspace{0.17em}}$ in standard form.

$\sqrt{-24}=0+2i\sqrt{6}$

can you not take the square root of a negative number
Suppose P= {-3,1,3} Q={-3,-2-1} and R= {-2,2,3}.what is the intersection
can I get some pretty basic questions
In what way does set notation relate to function notation
Ama
is precalculus needed to take caculus
It depends on what you already know. Just test yourself with some precalculus questions. If you find them easy, you're good to go.
Spiro
the solution doesn't seem right for this problem
what is the domain of f(x)=x-4/x^2-2x-15 then
x is different from -5&3
Seid
All real x except 5 and - 3
Spiro
how to prroved cos⁴x-sin⁴x= cos²x-sin²x are equal
Don't think that you can.
Elliott
how do you provided cos⁴x-sin⁴x = cos²x-sin²x are equal
What are the question marks for?
Elliott
Someone should please solve it for me Add 2over ×+3 +y-4 over 5 simplify (×+a)with square root of two -×root 2 all over a multiply 1over ×-y{(×-y)(×+y)} over ×y
For the first question, I got (3y-2)/15 Second one, I got Root 2 Third one, I got 1/(y to the fourth power) I dont if it's right cause I can barely understand the question.
Is under distribute property, inverse function, algebra and addition and multiplication function; so is a combined question
Abena
find the equation of the line if m=3, and b=-2
graph the following linear equation using intercepts method. 2x+y=4
Ashley
how
Wargod
what?
John
ok, one moment
UriEl
how do I post your graph for you?
UriEl
it won't let me send an image?
UriEl
also for the first one... y=mx+b so.... y=3x-2
UriEl
y=mx+b you were already given the 'm' and 'b'. so.. y=3x-2
Tommy
Please were did you get y=mx+b from
Abena
y=mx+b is the formula of a straight line. where m = the slope & b = where the line crosses the y-axis. In this case, being that the "m" and "b", are given, all you have to do is plug them into the formula to complete the equation.
Tommy
thanks Tommy
Nimo
0=3x-2 2=3x x=3/2 then . y=3/2X-2 I think
Given
co ordinates for x x=0,(-2,0) x=1,(1,1) x=2,(2,4)
neil
"7"has an open circle and "10"has a filled in circle who can I have a set builder notation
Where do the rays point?
Spiro
x=-b+_Гb2-(4ac) ______________ 2a
I've run into this: x = r*cos(angle1 + angle2) Which expands to: x = r(cos(angle1)*cos(angle2) - sin(angle1)*sin(angle2)) The r value confuses me here, because distributing it makes: (r*cos(angle2))(cos(angle1) - (r*sin(angle2))(sin(angle1)) How does this make sense? Why does the r distribute once
so good
abdikarin
this is an identity when 2 adding two angles within a cosine. it's called the cosine sum formula. there is also a different formula when cosine has an angle minus another angle it's called the sum and difference formulas and they are under any list of trig identities
strategies to form the general term
carlmark
consider r(a+b) = ra + rb. The a and b are the trig identity.
Mike
How can you tell what type of parent function a graph is ?
generally by how the graph looks and understanding what the base parent functions look like and perform on a graph
William
if you have a graphed line, you can have an idea by how the directions of the line turns, i.e. negative, positive, zero
William
y=x will obviously be a straight line with a zero slope
William
y=x^2 will have a parabolic line opening to positive infinity on both sides of the y axis vice versa with y=-x^2 you'll have both ends of the parabolic line pointing downward heading to negative infinity on both sides of the y axis
William
y=x will be a straight line, but it will have a slope of one. Remember, if y=1 then x=1, so for every unit you rise you move over positively one unit. To get a straight line with a slope of 0, set y=1 or any integer.
Aaron
yes, correction on my end, I meant slope of 1 instead of slope of 0
William
what is f(x)=
I don't understand
Joe
Typically a function 'f' will take 'x' as input, and produce 'y' as output. As 'f(x)=y'. According to Google, "The range of a function is the complete set of all possible resulting values of the dependent variable (y, usually), after we have substituted the domain."
Thomas
Sorry, I don't know where the "Â"s came from. They shouldn't be there. Just ignore them. :-)
Thomas
Darius
Thanks.
Thomas
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Thomas
It is the Â that should not be there. It doesn't seem to show if encloses in quotation marks. "Â" or 'Â' ... Â
Thomas
Now it shows, go figure?
Thomas