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Is there such a thing as an "unsolvable" question? There have been times throughout history where entirely new branches of mathematics have been created in service of puzzling out equations most mathematicians thought impossible to solve. Math is hard: sometimes, the same tools we use to ask a question aren't enough to answer it. So can we ever know if something is truly "unsolvable," or is there always just more, new math, out there waiting to be rearranged in ways that allow us to solve what we once thought impossible?
In your math education thus far, you may not have encountered any big, unsolved problems. But you've certainly encountered problems that were unsolvable to you until you learned the techniques necessary to figure them out.
In this section, we introduce some such problems -- and the techniques required to solve them.
Discussion
Consider the equation x2=−1.
What is this equation describing? Specifically, what would need to be true about the values of x that satisfy it?
Can you solve it with what you've learned about math thus far? Why or why not?
We want all numbers x such that
“Squaring” means multiplying a number by itself. For any real number r, we know
because a positive times positive is positive, and a negative times negative is also positive (zero squared is zero). So no real r can give r2=−1.
That tells us two things:
With the real numbers alone, the equation x2=−1 has no solution.
The fact that it “asks” for a square to be negative suggests we need to extend our number system—introduce a new kind of number whose square can be −1.
With what we’ve learned so far (the real number system), there is no solution to x2=−1. To solve it, we must introduce a new type of number so that its square can be negative.
Historically, mathematicians define a number i by exactly this rule:
Once i exists, the original equation has the two solutions
Mathematicians have defined a new kind of number as a method of answering this specific kind of question. An entirely new number system and method of approaching math results from the simple desire to figure out how to take the square root of a negative.
Imaginary number i
The imaginary number i is the square root of −1:
In general, imaginary numbers are of the form bi,b∈R∖{0}. Notice that
Conclusion: the square of any imaginary number is negative.
Checkpoint
It is given that z2=−9.
Find the value of z.
Select the correct option
Even when working with imaginary numbers, however, we need not work with imaginary numbers exclusively. We saw already that we can multiply i by any real coefficient. We can also add or subtract real and imaginary numbers to form what we call complex numbers, an important extension of the real number system.
Complex Numbers a+bi
A complex number
is the sum of a real number a and an imaginary number bi.
We call a the real part of z and b the imaginary part of z:
For example, z=2−3i is a complex number with a real part Re(z)=2 and imaginary part Im(z)=−3.
If two complex numbers are equal, then both their real and imaginary parts are equal:
The expression x+iy is often referred to as the Cartesian form of z.
Discussion
Consider two complex numbers, z and w, with z=−1+3i and w=2−i.
Try calculating the values of:
Re(z)+Re(w)
(Im(z)+Im(w))×i
(Re(z)+Re(w))+(Im(z)+Im(w))×i
z+w
What do you notice?
We have z=−1+3i and w=2−i.
Notice that
We can see that adding complex numbers is just like combining “like terms”: real parts add with real parts, imaginary parts add with imaginary parts.
As you just saw, when adding and subtracting complex numbers, we combine like terms to deal with the real and imaginary parts separately. It can be helpful to think of the relationship between real, imaginary, and complex numbers as adding more layers to the kind of "nested" structure between number systems that we've already worked with.
The sets ℂ, ℝ and ℝ\ℂ
Real numbers are a subset of complex numbers a+bi where b=0. Imaginary numbers are also a subset of complex numbers with a=0.
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Checkpoint
It is given that z=3i−5 and w=−5i+2.
Find z−w.
Select the correct option
Multiplication and division of complex numbers follows a similar process. These also involve the combination of like terms, but due to the properties of the imaginary unit i, it's important to be mindful of which terms are real and which are imaginary.
Discussion
Let z=2+3i,w=1−2i.
Recall that i2=−1. Can you find the product zw?
Recall that by definition
Write the product:
Expand using the distributive (FOIL) method:
Substitute i2=−1:
So
Combine like terms:
The real parts: 2+6=8, and the imaginary parts: −4i+3i=−i.
Thus,
We can follow a similar process for the product of any complex numbers given in Cartesian form.
Product of complex numbers
The product of two complex numbers in Cartesian form is
Checkpoint
It is given that z=−1+2i,w=3+i.
Find the product zw.
Select the correct option
There is a graphical reason that we call x+iy the Cartesian form of z. Much like plotting real numbers on the coordinate plane, we can plot complex numbers on the complex plane. Similar to plotting x and y coordinates on these respective axes, plotting a complex number z=x+iy on the complex plane involves plotting its real and imaginary parts on the real and imaginary axes, labeled x and iy respectively.
Try dragging the point z around the complex plane to see what different coordinates represent.
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The complex plane
Complex numbers can be visualized in the complex plane, also known as the Argand Diagram.
To plot a complex number, the real part determines the x-coordinate and the imaginary part determines the y-coordinate. Therefore the complex number a+bi has coordinates (a,b) on the plane.
It is conventional to use arrows from the origin to the point (a,b) to represent complex numbers.
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One of the most crucial concepts in the complex number system is that of the conjugate.
Complex conjugates
The conjugate of a complex number z is the complex number with the same real component and the opposite imaginary component:
Since the real components of z and z∗ are the same, and the imaginary components are opposite, on the complex plane z∗ is the reflection of z in the x-axis.
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Example
Find the complex conjugate of z=−5−2i.
We simply change the −2 to a +2:
Discussion
To see why the complex conjugate is so useful to us, let's work through some examples to see how it behaves when we do arithmetic with it.
Let z=2+i and w=−1+3i, so z∗=2−i and w∗=−1−3i.
What do you expect the following expressions to evaluate to? Can you find the exact values?
z+z∗? z−z∗?
z+w? (z+w)∗? (z∗+w∗)?
zw? (zw)∗? z∗w∗?
Is this what you expected? What do you notice about the relationship between expressions and their conjugates?
Given
Compute z+z∗ and z−z∗:
Compute z+w, its conjugate, and z∗+w∗:
Compute zw, its conjugate, and z∗w∗:
Observation:
z+z∗ and zz∗ are real, while z−z∗ is purely imaginary.
(z+w)∗=z∗+w∗.
(zw)∗=z∗w∗.
In general, conjugation distributes over addition and multiplication.
Properties of the complex conjugate
The following properties hold for complex conjugates:
Checkpoint
Complex numbers z and w are given by z=3−2i and w=21+ki.
Find the value of k such that (z−w)∗=25+3i.
Select the correct option
Now that we understand the basics of complex numbers, let's review some conventions and properties that are relevant when working with them.
First, consider the point z=1+i4−2i.
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From the diagram, we can see that z is plotted at the same coordinates as z=1−3i, making them equivalent.
Discussion
Why is it the case that 1+i4−2i=1−3i? Try multiplying 1+i4−2i by (1+i)∗(1+i)∗.
Why might we want to write z in the form 1−3i instead of 1+i4−2i as it's given?
We want to show
and explain why we prefer the form 1−3i.
First, eliminate the i in the denominator by multiplying top and bottom by the conjugate 1−i. Since 1−i1−i=1, this does not change the value:
Thus 1+i4−2i=1−3i.
We write z in the “standard form” a+bi, here 1−3i, because it makes the real part a and imaginary part b explicit. This is far more convenient when you want to add or subtract complex numbers, find the modulus or argument, or plot z in the complex plane.
The process of taking the i term out of a complex fraction's denominator is similar to rationalizing the denominator of a real number, and it is one we often perform to make complex numbers much easier to work with.
Fractions of complex numbers
Fractions with complex denominator can be made real using a process analogous to rationalizing the denominator. For a fraction with a complex denominator c+di, we multiply both the numerator and the denominator by the conjugate c−di to get the fraction in a more workable form:
This allows us to split z into its real and imaginary components.
Example
Exercise
Simplify z=1−3i6+5i.
Select the correct option
Even though i is a unit, not a variable, we've been "combining like terms" to simplify complex expressions and generally make them easier to work with. We can use this idea to solve systems of equations involving complex numbers.
Discussion
Consider the complex number z=1−2i and its conjugate z∗=1+2i.
What is the value of the expression 2z∗+z?
If you set up an equation with 2z∗+z equivalent to the value you calculated above, i.e.
for whatever you found x and y to be, how would you start with such an equation and "work backwards" to find the value of z without it being given explicitly?
Now let's say you were given an equation such as
Can you work in the opposite direction of part (a) above to find the value of z?
Part (a)
We know z=1−2i and z∗=1+2i, so
This confirms that 2z∗+z=3+2i.
Now imagine we only know
and want to “invert” this to find z. Why can’t we just algebraically isolate z as we do with a single real variable? Because z and its conjugate z∗ each mix together the real and imaginary parts of z. To separate those parts, we introduce two real variables:
This makes explicit that z depends on two unknown real numbers, a (the real part) and b (the imaginary part).
Substitute into 2z∗+z=3+2i. This gives one complex equation in the two unknowns a,b:
so our equation becomes
By the definition of equality of complex numbers, a complex number x+iy equals another u+iv if and only if x=u and y=v. In other words, real parts must match real parts, and imaginary parts must match imaginary parts. Applying that here:
Solve this familiar system of two real equations:
Finally, reconstruct z:
which agrees with our original value.
Remark. In general, any time you have an equation involving a complex unknown z and its conjugate, rewriting z=a+bi “unpacks” it into two real unknowns. You then match real with real and imaginary with imaginary, yielding a familiar system of real equations.
Part (b)
We know
and want to “invert” this to find z. Why can’t we just algebraically isolate z as we do with a single real variable? Because z and its conjugate z∗ each mix together the real and imaginary parts of z. To separate those parts, we introduce two real variables:
This is the standard form for a complex number and its conjugate.
Substitute into z∗−2z=−2+23i:
Expand and simplify:
By the definition of equality of complex numbers, a complex number x+iy equals another u+iv if and only if x=u and y=v. In other words, real parts must match real parts, and imaginary parts must match imaginary parts. Applying that here:
Solve this system:
Finally, reconstruct z:
Remark: In general, any time you have an equation involving a complex unknown z and its conjugate, rewriting z=a+bi “unpacks” it into two real unknowns. You then match real with real and imaginary with imaginary, yielding a familiar system of real equations.
Solving complex equations
We can solve complex equations involving z and z∗ by using the fact that a complex number z takes the form a+bi. Recall that for complex numbers z1 and z2,
We use this fact to equate the real and imaginary parts of both sides, which creates a solvable a system of two equations in two unknowns from one given equation.
Example
Solve z+2i=3iz∗−2 for z.
Equating the real and imaginary parts of both sides,
Substituting a=3b−2 into b+2=3a:
and therefore a=1.
So z=1+i.
Exercise
It is given that 2z+5i=3z∗+1.
Find the value of z.
Select the correct option
One of the most common applications of complex numbers is in finding complex roots of quadratic equations. Given that the imaginary unit i is defined as the square root of a negative number, the connection is straightforward.
Finding complex roots of quadratics
When a quadratic
has
it has no real roots since the square root in
is not a real number. Instead, the square root will give an imaginary number, making the roots complex.
Example
Find the roots of z2−2z+5=0.
Checkpoint
Find the roots of 5z2−2z+1=0.
Select the correct option
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