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Transcript of COMPLEX NUMBERS
z = a+b
Adding and Subtracting Complex numbers
Philosopher, Mathematician, Writer
Named a+"square root" of -b "imaginary numbers"
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
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.
Cyclic Pattern of Powers of i
Questions Involving Complex Numbers
made up of a combination of real and imaginary numbers
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
+ (3 + 12
= (14 + 3) + (7
= 17 + 19
2. (-4 +5
) - (6 - 52
= (-4 - 6) + (5
= -10 + 57
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
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
= sqrt(-1) x sqrt 121
2. 7 x sqrt(-24)
= 7 x sqrt(-1) x sqrt24
= 7 x
x sqrt4 x sqrt6
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)) = ?
3. (3 + 6i) x (5 + 2i) = ?
1. (8 + 3i) + ( -9 - 7i) = ?
4. (4 + 2i) / (3 - i) = ?
5. i^1050 = ?