COMPLEX NUMBER. COMPLEX NUMBERS IN TRIGONOMETRIC AND EXPONENTIAL FORM

LECTURE PLAN:

1. Complex numbers

2. Complex numbers in trigonometric and exponential form

Complex numbers

The main theorem of algebra. An algebraic equation of the nth degree

x^{n}+a_{n-1}x^{n-1}+…+a_{1}x+a_{0}=0

has n zeros. (Without a proof.)

Example.

x^{3}-1=0, (x-1)(x^{2}+x+1)=0, x_{1}=1,

x^{2}+x+1=0; D=1-4=-3,

- not real numbers.

Definition. The number is called the imaginary unit and is denoted by

; i^{3}=-i;

i^{4}=i^{2.}i^{2}=(-1)^{.}(-1)=1; i^{5}=i;

i^{6}=-i and so on. i.e. is a multiple of 4.

Example. Find

i^{35}= i^{32+3 }=i^{3}=-i.

Definition. A number of the form a=a+bi is called a complex number, where, i is the imaginary unit and à, b are real numbers. Numbers à and bi are called the real and the imaginary parts of a complex number.

The following is a geometric interpretation of a complex number

ó

Ì(a,b)

a=a+bi

b

0 x

a

Each complex number corresponds to a point of the coordinate plane õOó. Each real number is also a complex number with zero imaginary part, i.e. b=0. For example, 32=32+0i.

Geometrically, addition and subtraction of complex numbers is performed exactly in the same way as addition and subtraction of the vectors with corresponding coordinates.

Multiplication of complex numbers is performed as usual multiplication of algebraic expressions; after the multiplication the real and imaginary parts are grouped separately.

The numbers a =a +bi, a =a –bi are called complex conjugates or simply conjugates.

The product of two conjugates is a nonnegative real number a^{2}+bia-abi-i^{2}b^{2}=a^{2}+b^{2}.

4) Division. Definition. The quotient of two complex numbers is a

complex number b such that .

Suppose, , then by the definition

;

.

Due to equality of real and imaginary parts,

; .

Solve the resulting system of linear equations

,

and find the real numbers c and d

,

, .

The denominator of both expressions is the product of two conjugates,

so let us multiply both the numerator and the denominator of by the conjugate of the denominator

- as expected, the result equals b.

Rule. In order to find the quotient of two complex numbers, it is enough to multiply both the numerator and the denominator by the conjugate of the denominator, and then separate the real and imaginary parts.

Examples. 1. Divide

.

2. Solve the equation õ^{2}-2õ+10=0 and check Viete's theorem for the complex zeros

;

; x_{1}+x_{2}=2; x_{1 }^{.}x_{2}=1^{2}+3^{2}=10.

The magnitude and argument of a complex number. Plot the complex number a=a+bi on the plane.

ó

r b

j

0 a x

It is easy to see that r^{2}=a^{2}+b^{2 }and^{ } ; the numbers ρ and j are called the absolute value (or magnitude) and the argument of a complex number respectively.