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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


has n zeros. (Without a proof.)


x3-1=0, (x-1)(x2+x+1)=0, x1=1,

x2+x+1=0; D=1-4=-3,

- not real numbers.

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

; i3=-i;

i4=i2.i2=(-1).(-1)=1; i5=i;

i6=-i and so on. i.e. is a multiple of 4.

Example. Find

i35= i32+3 =i3=-i.

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

The following is a geometric interpretation of a complex number





0 x


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


Operations on complex numbers


1) Addition of complex numbers.

Consider complex numbers a1 =a1 +b1i, a2 =a2 +b2i.

Add them and combine the like terms


a1 +a2 =a1 +b1i +a2 +b2i= (a1 +a2) +i(b1+b2),

We add separately the real and the imaginary parts.

2) Subtraction a1a2 = (a1a2) +i(b1b2).

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



b2 a2




0 a1 a2 x

3) Multiplication.

a1 . a2 =(a1 +b1i)(a2 +b2i)= a1a2+b2ia1+b1a2i+b1b2i2=

=(a1a2-b1b2)+i( a1b2 +b1a2), .. i2=-1.


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 a2+bia-abi-i2b2=a2+b2.

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 2-2+10=0 and check Viete's theorem for the complex zeros


; x1+x2=2; x1 .x2=12+32=10.

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


r b


0 a x

It is easy to see that r2=a2+b2 and ; the numbers ρ and j are called the absolute value (or magnitude) and the argument of a complex number respectively.



Date: 2015-01-02; view: 1058

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