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