1. Complex Numbers and Their Geometry Complex Numbers and Their Geometry
Another Kind of Number
We introduce complex numbers as an expansion of the real number system. While complex numbers have many applications in the real world of science and engineering, they also provide a path to a deeper understanding of polynomials and their roots.
Every major branch of mathematics has a “fundamental theorem” which provides much of the theoretical underpinnings for that branch. Algebra is no exception and the theorem called The Fundamental Theorem of Algebra was so named because, at the time, finding solutions of polynomial equations had been in the forefront of mathematical research for quite some time. The introduction of complex numbers made possible a simple statement of the Fundamental Theorem.
Had mathematicians’ interest been focused on other mathematical questions what became the Fundamental Theorem might well have been something very different than the theorem we see today.
Complex Numbers
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2. Complex Numbers and Their Geometry
The Complex Plane
Complex Numbers and Their Geometry
Complex Numbers and the Imaginary i
Definition:
The number x such that
x2 = –1
is defined to be i
Applying the square root property,
Complex Numbers and the Imaginary i
The complex numbers are sometimes divided into all-real and all-imaginary numbers. That is, a + bi is real if b = 0, and is all-imaginary if b ≠ 0 and a = 0. Some authors use the definition shown in the illustration allowing a to be zero for an imaginary number but not requiring it. This minor difference does not produce any different mathematics at the application level since operations with complex numbers do not depend on the naming of the numbers.  –1 = x
so that –1  i =
and –i – –1  =
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Complex Numbers
Complex Numbers
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3. Complex Numbers and Their Geometry
The Complex Plane
Complex Numbers and Their Geometry
Complex Numbers and the Imaginary i
Standard Form x iy
a + bi , b = 0 ●
Real Numbers
a + bi
a + bi
Complex Numbers
a + bi , b ≠ 0
The Complex Plane
Imaginary Numbers
Sometimes b ≠ 0 and a = 0
Complex Numbers and the Imaginary i
The complex numbers are sometimes divided into all-real and all-imaginary numbers. That is, a + bi is real if b = 0, and is all-imaginary if b ≠ 0 and a = 0. Some authors use the definition shown in the illustration allowing a to be zero for an imaginary number but not requiring it. This minor difference does not produce any different mathematics at the application level since operations with complex numbers do not depend on the naming of the numbers.
The complex plane allows us to plot complex numbers in graphical fashion. The vertical axis is normally taken as the pure imaginary axis and is sometimes designated as such by the “iy” tag shown in the illustration.
Some authors do not plot complex numbers at all and some use merely the “y” designation with the understanding that all y components are considered to be imaginary. Still other authors define a complex number as an ordered pair of real numbers, the first of which is the “real” component and the second is the “imaginary” component. All of these views of complex numbers lead to essentially the same mathematical results.
Note:
a + bi is also written a + ib for real a and b
… especially if b is a radical or function
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Complex Numbers
Complex Numbers
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4. Complex Numbers and Their Geometry
The Complex Plane
Complex Numbers and Their Geometry
Complex Numbers in the Plane
Numbers as points in a plane …
instead of points on a line
Complex numbers as ordered pairs of real numbers
Complex number z
Point in complex plane x iy
Imaginary Axis
Complex Numbers and the Imaginary i
The complex numbers are sometimes divided into all-real and all-imaginary numbers. That is, a + bi is real if b = 0, and is all-imaginary if b ≠ 0 and a = 0. Some authors use the definition shown in the illustration allowing a to be zero for an imaginary number but not requiring it. This minor difference does not produce any different mathematics at the application level since operations with complex numbers do not depend on the naming of the numbers.
The complex plane allows us to plot complex numbers in graphical fashion. The vertical axis is normally taken as the pure imaginary axis and is sometimes designated as such by the “iy” tag shown in the illustration.
Some authors do not plot complex numbers at all and some use merely the “y” designation with the understanding that all y components are considered to be imaginary. Still other authors define a complex number as an ordered pair of real numbers, the first of which is the “real” component and the second is the “imaginary” component. All of these views of complex numbers lead to essentially the same mathematical results. ●
(a, b) =
z = a + ib ib z
a + ib = a
The Complex Plane
Real Axis
z = (a, b)
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Complex Numbers
Complex Numbers
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5. Complex Numbers and Their Geometry
The Complex Plane
Complex Numbers and Their Geometry
Radical Expressions and Arithmetic
The expression
Sum and difference of complex numbers
(a + bi) ± (c + di) = (a ± c) ± (b ± d)i
Examples:
(3 + 4i) – (2 – 5i)
(7 – 3i) + (2 – 5i)  – a
can be written  –1 a = – i =  a
Radical Expressions and Arithmetic
Historically, imaginary numbers were discovered, or rather invented, as a result of studies involving radicals. They were not considered “less real” than the ordinary real numbers, but were just numbers that had the interesting property that their squares were negative real numbers.
That is, if such a number, call it z for the moment, is such that z2 = -a for some positive real number a, then the square root property, along with the properties of radicals, tells us that
We note that we showed only the positive (or principal) root here and note that there will also be a negative root whose square is -a :
We give the value a name, i , to represent the imaginary unit (which corresponds to 1 in the real numbers). We call z a complex number to distinguish it from real numbers whose squares are all positive (except for 0).
The representation of a complex number z as x + iy is convenient and allows for computations that have the form used for binomial conjugates in the real numbers. The examples clearly demonstrate this methodology.
= (3 – 2) + (4 + 5)i
= 1 + 9i
= (7 + 2) – (3 + 5)i
= 9 – 8i
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Complex Numbers
Complex Numbers
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6. Complex Numbers and Their Geometry
The Complex Plane
Complex Numbers and Their Geometry
Radical Expressions and Arithmetic
The expression
Product of Complex Numbers  – a
can be written  –1 a = – i =  a
(a + bi)(c + di)
ac + bdi2 + (ad + bc)i =
(ac – bd) + (ad + bc)i =
Radical Expressions and Arithmetic
Historically, imaginary numbers were discovered, or rather invented, as a result of studies involving radicals. They were not considered “less real” than the ordinary real numbers, but were just numbers that had the interesting property that their squares were negative real numbers.
That is, if such a number, call it z for the moment, is such that z2 = -a for some positive real number a, then the square root property, along with the properties of radicals, tells us that
We note that we showed only the positive (or principal) root here and note that there will also be a negative root whose square is -a :
We give the value a name, i , to represent the imaginary unit (which corresponds to 1 in the real numbers). We call z a complex number to distinguish it from real numbers whose squares are all positive (except for 0).
The representation of a complex number z as x + iy is convenient and allows for computations that have the form used for binomial conjugates in the real numbers. The examples clearly demonstrate this methodology.
Example:
(3 + 4i)(2 – 5i)
= 6 – 20i2 + (8i – 15i)
= 26 – 7i
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Complex Numbers
Complex Numbers
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7. Complex Numbers and Their Geometry
The Complex Plane
Complex Numbers and Their Geometry
Complex Conjugates
Definition: a + bi and a – bi are a
complex conjugate pair
Example:
7 + 3i and 7 – 3i are complex conjugates
Complex Conjugates
Analogous to real binomial conjugates are complex conjugates in the complex number system. These are pairs of complex numbers having the same real component but with opposite-signed imaginary components.
Complex conjugates are important in solving polynomial equations. If a solution is a complex number, then there is a matching complex conjugate that is also a solution. That is, complex solutions always occur in complex conjugate pairs.
A useful fact about complex conjugate pairs is that their product is always a real number. This is especially convenient when finding quotients of complex numbers and expressing them in the standard form a + ib. This is shown in the illustration to convert the denominator to a real number.
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Complex Numbers
Complex Numbers
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8. Complex Numbers and Their Geometry
The Complex Plane
Complex Numbers and Their Geometry
Complex Conjugates
Definition: a + bi and a – bi are a
complex conjugate pair
Fact:
The product of complex conjugates is always real
(a + bi ) • (a – bi) = a2 + b2
Example:
(7 + 3i) • (7 – 3i) =
Complex Conjugates
Analogous to real binomial conjugates are complex conjugates in the complex number system. These are pairs of complex numbers having the same real component but with opposite-signed imaginary components.
Complex conjugates are important in solving polynomial equations. If a solution is a complex number, then there is a matching complex conjugate that is also a solution. That is, complex solutions always occur in complex conjugate pairs.
A useful fact about complex conjugate pairs is that their product is always a real number. This is especially convenient when finding quotients of complex numbers and expressing them in the standard form a + ib. This is shown in the illustration to convert the denominator to a real number. =
= 58
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Complex Numbers
Complex Numbers
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9. Complex Numbers and Their Geometry
The Complex Plane
Complex Numbers and Their Geometry
Complex-Number Quotients
Complex conjugate of c + di
a + bi
c + di
c – di =
a + bi
c + di =
(bc – ad)i +
(ac + bd)
c2 + d 2
Real denominator =
ac + bd
c2 + d 2 +
(bc – ad)i
Complex-Number Quotients
While addition and multiplication of complex numbers are carried out in a fashion analogous to handling of real-valued polynomials, division is accomplished somewhat differently.
We note that if k is any real number, then so is its reciprocal 1/k. Thus
With this in mind, we seek a way to convert complex denominators into real-number denominators. As the illustration shows we can multiply the complex denominator by its complex conjugate, which produces a real-number denominator. Of course, if we multiply the denominator by a value, we must also multiply the numerator by the same value to avoid changing the value of the fraction. That is, we can change the form of fraction without changing the value of the fraction by multiplying by 1 in clever disguise. = + i
ac + bd
c2 + d2 ( )
bc – ad
c2 + d 2
Note:
We can always multiply by 1 in clever
“disguise” to change form NOT value
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Complex Numbers
Complex Numbers
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10. Complex Numbers and Their Geometry
The Complex Plane
Complex Numbers and Their Geometry
Complex-Number Quotients
Quotient Examples 1. i
a + bi
c + di = +
ac + bd
c2 + d2 ( )
bc – ad
c2 + d 2 i
1 + 2i
( i)
(1 + 2i)
(1 – 2i) =
(65 – 30)i =
Quotient Examples
The example above illustrates a basic technique in finding quotients of complex numbers: first convert the denominator to a real number and handle as separate fractions with the same real denominator.
The simplest way to convert the denominator to a real number is to multiply it by its complex conjugate. This of course requires that we also multiply the numerator by the same conjugate.
Note that in the second example we use the complex conjugate of i, which is of course just -i. If this is not obvious, think of i as 0 + i and its conjugate as 0 – i. =
29 + 7i
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Complex Numbers
Complex Numbers
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11. Complex Numbers and Their Geometry
The Complex Plane
Complex Numbers and Their Geometry
Quotient Examples 2. 3 i =
( i )( –i ) 3
( –i ) =
–i 2 3
( –i ) =
–(–1) 3
( –i )
Quotient Examples
The example above illustrates a basic technique in finding quotients of complex numbers: first convert the denominator to a real number and handle as separate fractions with the same real denominator.
The simplest way to convert the denominator to a real number is to multiply it by its complex conjugate. This of course requires that we also multiply the numerator by the same conjugate.
Note that in the second example we use the complex conjugate of i, which is of course just -i. If this is not obvious, think of i as 0 + i and its conjugate as 0 – i. =
–3i
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Complex Numbers
Complex Numbers
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12. Complex Numbers and Their Geometry
The Complex Plane
Complex Numbers and Their Geometry
Quotient Examples 3. =
(1 – i )2
(1 + i )2
(–2 + i)
(1 + i )2
–2 + i =
(1 – i ))2
((1 + i )
(–2i)
(–2 + i) =
( 2 )2
2 + 4i
Quotient Examples
The example above illustrates a basic technique in finding quotients of complex numbers: first convert the denominator to a real number and handle as separate fractions with the same real denominator.
The simplest way to convert the denominator to a real number is to multiply it by its complex conjugate. This of course requires that we also multiply the numerator by the same conjugate.
Note that in the second example we use the complex conjugate of i, which is of course just -i. If this is not obvious, think of i as 0 + i and its conjugate as 0 – i. = 1 2 i +
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Complex Numbers
Complex Numbers
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13. Complex Numbers and Their Geometry
Think about it !
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Complex Numbers
Complex Numbers
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