Multiplication.

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

=(a₁a₂-b₁b₂)+i(a₁b₂ +b₁a₂), т.к. 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 х²- 2 х +10=0 and check Viete's theorem for the complex zeros

;

; x ₁ +x ₂=2; x ₁ ^.x ₂=1²+3²=10.

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

 

у

r b

j

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.

 

 

Trigonometric form of a complex number, de Moivre's formula

Use the above figure to express the real and imaginary parts a,b

of the complex number in terms of the magnitude and argument ρ, j.

a=r^. cos j, b=r ^.sin j.

Hence

a=r( cos j+i sin j)

This is the trigonometric form of a complex number.

Each complex number has a unique trigonometric form because it has a unique magnitude and argument.

Example. α= .

; .

Therefore,

.

1. Multiplication. Consider complex numbers

a₁ =r₁ ( cos j₁ +i sin j₁) and a₂ =r₂ ( cos j₂ +i sin j₂).

The product equals

a₁·a₂ = r₁ ( cos j₁ +i sin j₁)^. r₂ ( cos j₂ +i sin j₂)=

=r₁·r₂ [( cos j₁ cos j₂ -sin j₁ sin j₂)+i( sin j₁ cos j₂ + cos j₁ sin j₂);

a₁·a₂ = r₁·r₂ [ cos (j₁+j₂)+i sin (j₁+j₂)].

Therefore, in order to multiply two complex numbers, it is necessary to multiply their magnitudes and add the arguments.

2. Division. Multiply both the numerator and denominator by the conjugate of the denominator

Therefore, in order to divide two complex numbers, it is necessary to divide their magnitudes and subtract the arguments.

A power of a complex number. If a₁=a₂=a₃=…=a_n=a, then

aⁿ = r ·r ·r·…·r[cos(j+j +j+…+j)+i(sin(j+j +j+…+j)]= =rⁿ[cosnj+isinnj]

or

a^k =r^k[coskj+isinkj] -de Moivre's formula.

Roots of a complex number. Write down de Moivre's formula for using the fact the sine and cosine are periodic functions with the period T=2p:

,

where k may be any of n integers 0,1,2,…, n- 1.

Example. Solve the equation x ³+1=0.

1 -st method: (x +1)(x ²- x +1)=0, x ₁=-1, ;

2 -nd method: x ³=-1, , -1=cosp+ i sinp, which is a trigonometric form of a number.

 

. Using the formula, k=0, ,

k= 1, ,

k=2,

= .