complex numbers 2-4. imaginary numbers 1.designed so negative numbers can have square roots., i 2 =...
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Complex Numbers 2-4
Imaginary Numbers
1. Designed so negative numbers can have square roots.
, i 2 = -1
2. Imaginary numbers consist of all numbers bi, where b is a real number and i is the imaginary unit, with the property that i 2 = -1.
Example
i 1
551515
27492492198 ii
Practice
yy
x
53 .5
5 .4
18 .3
9-- .2
11 .1
2
15||
5
23
3
11
iy
ix
i
i
i
Powers of i
i 1 = i
i 2 = -1
i 3 = -i
i 4 = 1
Higher Powers of iDivide the exponent by 4, then determine
the remainder.
If the remainder is
1 i 1 = i
2 i 2 = -1
3 i 3 = -i
0 i 0 = i 4 = 1
Practice
1. i 20 =
2. i 27 =
3. i 71 - i 49 =
4. i 4444484044844444441 =
5. i -2 =
6. i -27 =
Negative Exponents
11
11
1
1
11-
1
1
1
44-
3-
2-
1-
ii
ii
i
i
-ii
i
ii
i
i
ii
Multiplying Imaginary Numbers
Practice
1010
182
155
33
3646
i
iii 24i
-18
6
-10
Complex Numbers• Complex numbers are the sum of a real
number and an imaginary number.
• They are written in the form a + bi where a is a real number and bi is an imaginary number.
• Complex numbers include real and imaginary numbers since 3 = 3 + 0i or 4i = 0 + 4i
• Imaginary numbers follow the properties we have learned (commutative, associative, distributive…)
Complex Numbers a + bi
Real Numbersb=0
Imaginary Numbersb≠0
PUREImaginaryNumbersa=0
The Big Picture where does everything fit
Pure
복소수
허수
순허수
Addition and Subtraction
• Combine the real parts and combine the imaginary parts.
• (6 + 3i) + (8-2i) = (6+8) + (3 – 2)i = 14 + i
• (6 + 3i) - (8-2i) = (6-8) + (3 – -2)i = -2 + 5i
• (6 - 3i) + (8-2i) = (6+8) + (-3 – 2)i = 14 - 5i
Practice
• 3i – (5 – 2i)=
• (-2 + 8i) – (7+3i)=
• 4 – 10i + 3i – 2 =
-5 + 5i
-9 +5i
2 – 7i
Graphing Complex Numbers
Finding Absolute Values
The absolute value of a complex number | a + bi | is its distance from the origin. So we use the distance formula or simplified as
we ignore the i for the formula
)b a (from 22 iba
Finding Absolute Values
Practice - Find
1)|6 – 4i |
2)|-2 + 5i |
3)|4i| 3) 4
Multiplying Complex Numbers
• Multiply like real numbers and treat i like a variable but i2 = -1
(3 + 2i)(4-7i) = 12 – 21i +8i -14i2 =
12 – 13i -14(-1) = 12-13i +14 =
26 – 13i
Practice
1) (3+2i )(2-i )
2) (2-i )(2+i )
3) (6-5i )(3-2i )
1) 6 – 3i +4i – 2i2 = 6+i+2= 8+ i
2) 4 + 2i -2i – i2 = 4 – (-1) = 5
3) 18 – 12i -15i -10= 8 - 27i
Division
Complex Conjugates
i
i
23
35
13
199
)4(9
)1(61915
23
23
23
35 ii
i
i
i
i
Practice
Equations with complex numbers
• Two complex numbers are equal if their real part is equal and their imaginary part is equal.
• If a+bi = c+di then a=c and b=d
• 5x+1 + (3+2y)i = 2x-2 + (y-6)i
• real part 5x+1 = 2x-2, 3x = -3, x=-1
• imaginary part 3+2y = y-6, y=-9
Conjugates 켤레복소수
In algebra, a conjugate is a binomial formed by negating the second term of a binomial. The conjugate of x + y is x − y,
where x and y are real numbers.
If y is imaginary, the process is termed complex conjugation: the complex conjugate of a + bi is a − bi,
where a and b are real.
Complex Conjugates
i
i
23
35
13
199
)4(9
)1(61915
23
23
23
35 ii
i
i
i
i
Practice
Equations with complex numbers
• Two complex numbers are equal if their real part is equal and their imaginary part is equal.
• If a+bi = c+di then a=c and b=d
• 5x+1 + (3+2y)i = 2x-2 + (y-6)i
• real part 5x+1 = 2x-2, 3x = -3, x=-1
• imaginary part 3+2y = y-6, y=-9
Graphing Points
• You have the real axis (x-axis) and the imaginary axis (y-axis).
• Plot point (a, b)
• You can plot inequalities by shading areas of the graph.
• How would we graph
{a + bi | a≤3 and b ≤2} ?
Graph {a + bi | a≤3 and b ≤2}
What happens when we multiply a complex number, a + bi by i ?
4 + 2i -2 + 4i
-4 -2i
2 – 4i
More about Complex NumbersTo solve an equation involving complex numbers,
equate the real parts and equate the imaginary parts. 3 1 2 2 2
2
1 2 2
2 1 2 2
3 1 4 2 4
2 4 2 4
x y i x yi
y i yi
x yi i yi
i yi
i i
i i
i i
( )
( ) ( )
set up two equations - real and imaginary
real imaginary
3x +1 = 2x ( + 2)
2
y = 2
check : 3(-1) +1+ (2 + 2)
Product of a Complex Number and its Conjugate
• What happens when you multiply a complex number by it’s conjugate?
(3+4i )(3-4i ) =
So (a + bi )(a – bi ) =
9-(-16) = 25
a2 – (b2)(i2) = a2 – (-1)(b2)
= a2 + (b2)
Dividing and Reciprocals
• When you divide by a complex number (a fraction with a complex number as a denominator) you multiply both the numerator and denominator by 1 by multiplying both by the conjugate of the denominator,
Example3 2
2
3 2
2
2
2
6 3 4 2
4 2 2
6 2
4 1
8
5
8
5
1
5
2
2
i
i
i
i
i
i
i i i
i i i
i ii
( )
( )
Finding the Reciprocal
• To find the reciprocal of a complex number, you divide 1 by that complex number.
• The reciprocal of 3 + 2i is
• But now you need to rationalize the denominator by multiplying the numerator and denominator by the conjugate.
• So find the reciprocal of 3 + 2i
1
3 2 i
1
3 2
1
3 2
3 2
3 2
3 2
13
3
13
2
13
i i
i
i
ii