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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
Identity Property of Multiplication
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.
The set of imaginary numbers, represented by the symbol I, is formed by all the numbers that can be written as a+bi, where a is any real number, b is a non-zero real number, and i is the imaginary unit.
The set of complex numbers, represented by the symbol C, is formed by all numbers that can be written in the form z=a+bi, where a and b are real numbers, and i is the imaginary unit. Here, a is called the real part and b is called the imaginary part of the complex number.
If b=0, the number is an imaginary number. Conversely, if b=0, the number is real. Additionally, if a=0 and b=0, the number is a pure imaginary number. Both real and imaginary numbers are subsets of the complex number set.a+bi=c+di⇔a=c and b=d
Now that Tadeo figured out the pattern for the powers of i, he feels confident in learning the other mathematical operations for complex numbers. He heads to the library, asks for a math textbook, explores the text and charts for a few minutes, and focuses on the following.
Two complex numbers a+bi and c+di can be added or subtracted by using the commutative and associative properties of real numbers. To add or subtract two complex numbers, combine their real parts and their imaginary parts separately.
(a+bi)+(c+di)=(a+c)+(b+d)i
(a+bi)−(c+di)=(a−c)+(b−d)i
z1=5+2i, z2=3−3i
Commutative Property of Addition
Associative Property of Addition