| {{ 'ml-lesson-number-slides' | message : article.intro.bblockCount}} |
| {{ 'ml-lesson-number-exercises' | message : article.intro.exerciseCount}} |
| {{ 'ml-lesson-time-estimation' | message }} |
no solution.For example, x2=-4 has no solution because no real number exists such that squaring it results in a negative number. Wait, what about numbers that are not real? Are there numbers other than real ones? This lesson will teach and explore such
non-realnumbers.
Here are a few recommended readings to do before beginning this lesson.
Try these practice exercises to warm up for this lesson.
Equation | Unsolvable in | Solvable in |
---|---|---|
x+2=1 | Natural numbers N | Integer numbers Z |
2x−3=0 | Integer numbers Z | Rational numbers Q |
x2=5 | Rational numbers Q | Irrational numbers |
x2=-2 | Real numbers R | ? |
Equations like x2=-2 do not have real solutions. At this point, the big question is: Does a number system more general than the real number system in which such equations can be solved exist?
Is it possible to expand the real number system so that x2=-2 has solutions?
Mathematicians' minds were occupied with such questions for years. Finally, they figured out that calling i the solution of x2=-1 allowed them to solve any equation — the solutions could be real numbers or combinations of real numbers and i. This led them to create the imaginary unit. The term imaginary was coined by René Descartes in 1637.
The imaginary unit i is the principal square root of -1, that is, i=-1. From this definition, it can also be said that i2=-1.
Imaginary Unit
i=-1 or i2=-1
-a=a⋅-1=a⋅i
Tadeo just learned that imaginary numbers are given that name because they do not exist in the real world — they are imaginary. Therefore, if an equation that models a real-life situation has imaginary solutions, then it cannot be solved in the real world. To illustrate this concept, Tadeo's math teacher drew the following polygons and asked three questions.
Square | Triangle | |
---|---|---|
Area Formula | A□=s2 | A△=21bh |
Substitution | A□=x2 | A△=21⋅3⋅4=6 |
A□=x2, A△=6
LHS−6=RHS−6
LHS=RHS
a2=±a
Square | Triangle | |
---|---|---|
Area | A□=x2 | A△=6 |
A□=x2, A△=6
Multiply
LHS−12=RHS−12
LHS=RHS
a2=±a
A□=x2, A△=6
Multiply
LHS−18=RHS−18
LHS=RHS
a2=±a
-a=ia
Split into factors
a⋅b=a⋅b
Calculate root
With the purpose of mastering the calculus of powers of i, Tadeo asked his teacher to give him a homework problem. As such, the teacher asked him to find i34.
Initially, Tadeo plans to find i8, i9, i10, all the way to i34, but he realizes that would take forever! For that reason, he decides to analyze the values of the first 8 powers of i — values he already knows. Because the cycle repeats every four iterations, he thinks there is a relation between the exponents to the number 4.Finding in | |
---|---|
If the remainder of 4n is | Then, in is equal to |
0 | i0=1 |
1 | i1=i |
2 | i2=-1 |
3 | i3=-i |
Rewrite 34 as 32+2
am+n=am⋅an
Rewrite 32 as 4⋅8
am⋅n=(am)n
i4=1, i2=-1
1a=1
\IdPropMult
Compute the required power of i.
The imaginary unit i, which is equal to -1, not only allows square roots to be used in calculations of negative numbers, it also allows for the construction of the set of imaginary numbers.
Excited to continue learning about complex numbers, Tadeo ran to his brother's room and asked if he knew of any real-life applications. His brother, an electrical engineer, reached for his favorite book with a diagram of a series circuit. In the case of resistors, the number next to each component indicates its resistance. In the case of capacitors and inductors, it indicates its reactance.
Tadeo's brother went on telling him that the impedance, or opposition to the current flow, of the circuit shown is equal to the sum of the impedances of each component.
Component | Impedance |
---|---|
Resistor | 8Ω |
Capacitor | -6iΩ |
Inductor | 10iΩ |
The impedance of a resistor equals its resistance, the impedance of a capacitor equals its reactance multiplied by -i, and the impedance of an inductor equals its reactance multiplied by i. All of these quantities are measured in ohms.
Component | Resistance or Reactance | Impedance |
---|---|---|
Resistor 1 | 6Ω | 6Ω |
Capacitor | 9Ω | -9iΩ |
Inductor | 4Ω | 4iΩ |
Resistor 2 | 3Ω | 3Ω |
a+(-b)=a−b
\CommutativePropAdd
\AssociativePropAdd
Factor out i
Add and subtract terms
Tadeo is feeling great about complex numbers so far but wants to learn even more. He suspects that complex numbers can also be multiplied, which causes him to wonder if there is a method to do that. Thirsty for knowledge, he looked in his e-book and found the answer.
C1=4+3i, I1=8−3i
Multiply
\CommutativePropMult
am⋅an=am+n
i2=-1
Multiply
\CommutativePropAdd
\AssociativePropAdd
Factor out i
Add and subtract terms
C1=5+2i, I1=7+5i
Multiply
\CommutativePropMult
am⋅an=am+n
i2=-1
Multiply
\CommutativePropAdd
\AssociativePropAdd
Factor out i
Add and subtract terms
V2=25+39i
\CommutativePropAdd
Factor out i
Add terms
V2=25+39i, V1=41+12i
Distribute -1
\CommutativePropAdd
Factor out i
Subtract terms
The complex conjugate of a complex number has the same real part, but the imaginary part is the opposite of its original sign. Therefore, changing the sign of the imaginary part of a complex number creates its complex conjugate. It is denoted by a line drawn above the complex number.
z=a+bi
a+bi=a−bi
Multiply
Add terms
i2=-1
-(-a)=a
Now that Tadeo knows about complex conjugates, there is nothing that can stop him from learning how to divide complex numbers.
Multiply
i2=-1
Multiplication Property of -1
Add and subtract terms
Write as a sum of fractions
c+dia+bi=c2+d2ac+bd+(c2+d2bc−ad)i
The weekend is here and Tadeo still wants to continue practicing operations with complex numbers. Unfortunately, his brother is not at home to keep giving him cool examples. However, this does not stop Tadeo from picking up a book and looking for exercises.
From the book, he chose three exercises that he found interesting.
z=-8+4i, zˉ=-8−4i
Multiply parentheses
Multiply
Subtract terms
i2=-1
-a(-b)=a⋅b
Add terms
Multiply
i2=-1
Multiplication Property of -1
Add terms
Write as a sum of fractions
Calculate quotient
ba=b/10a/10
Just as Tadeo thought he knew all about complex numbers, his teacher told him that unlike real numbers, complex numbers cannot be represented on a number line. However, they can be represented on the complex plane — similar to the coordinate plane but the horizontal axis represents the real part and the vertical axis the imaginary part of a complex number.
Note that the number -3+2i is represented by the point (-3,2).