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COMPLEX NUMBERS

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by

Eva Li

on 16 December 2013

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Transcript of COMPLEX NUMBERS

What Are Complex Numbers?
z = a+b
i
Adding and Subtracting Complex numbers
Rafael Bombelli
(1572)
Mathematician
Rene Descartes
(1637)
Philosopher, Mathematician, Writer
Named a+"square root" of -b "imaginary numbers"
COMPLEX NUMBERS

Albert Girard
(1629)
Mathematician
Other Mathematicians Who Contributed to "Complex Numbers"
Bernoulli, Moivre, Euler, Argand, Gauss
The term "complex numbers" originated from C.F.Gauss (1831)
use the FOIL method or the distributive property
replace i^2 with -1
Girolamo Cardano
(1501-1576)
Mathematician and Medicine
First to introduce complex numbers
He referred complex numbers as a "mental torture" (Ars Magna, Chapter 37)
Doubted such expressions such as 5+"square root" of -15
Introduced the symbol "i"
Established rules for calculating in "C"
a+"square root" of -b are solutions impossibles.
Multiplying Complex
numbers
Cyclic Pattern of Powers of i
Questions Involving Complex Numbers
made up of a combination of real and imaginary numbers
real
imaginary
The Imaginary Unit
first add or subtract the real parts of the numbers
then add or subtract the imaginary parts of the numbers
1. (14 + 7
i
)

+ (3 + 12
i
)
= (14 + 3) + (7
i
+ 12
i
)
= 17 + 19
i
2. (-4 +5
i
) - (6 - 52
i
)
= (-4 - 6) + (5
i
- (-52
i
))
= -10 + 57
i
1. (9 + 3i)(-4 + 7i)
= (9 x -4) + (9 x 7i) + (3i x -4) + (3i x 7i)
= (-36) + (27i) + (-12i) + (21i^2)
= -18 +15i +21(-1)
= -39 + 15i
Eva Li, Emily Zhu, Stanley Chen,Amy Ou
Dividing Complex
Numbers
Conjugate: the change of the sign in a binomial
Multiply the numerator and the denominator by the complex conjugate of the denominator
The powers of i repeat in a cyclic pattern.
i to the power of any integer is i, -1, -i, or 1
To simplify powers of 5 or greater for i, divide exponent by 4:
i^56 = (i^4)^14 = 1^14= 1
i^105= (i^4)26 x i^1 = 1 x i = i
i^79= i^76 x i^3= (i^4)^19 x (-1) = 1^19 x (-i) = -i

Simplifying Square Roots with Negative Numbers
not true if both a and b are negative numbers
1. sqrt(-121)
= sqrt(-1) x sqrt 121
= 11
i
2. 7 x sqrt(-24)
= 7 x sqrt(-1) x sqrt24
= 7 x
i
x sqrt4 x sqrt6
= 14
i
sqrt6
Absolute Value



6. (6+3i)/(7-5i) = ?

the absolute value of a complex number is the distance from the origin to a point on the plane
can be represented graphically on complex plane
the absolute value of z when
z = 2 - 3i
(a + bi)(a - bi)= a^2 + b^2
put in standard complex form
2. (10 + 4i) - (2 - (-6)) = ?
Question 2
3. (3 + 6i) x (5 + 2i) = ?
Question 3
Question 1
1. (8 + 3i) + ( -9 - 7i) = ?

4. (4 + 2i) / (3 - i) = ?
Question 4
5. i^1050 = ?
Question 5
Question 6
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