*Complex numbers* are written in the shape: *a+ bi*. Here * a* and *b* – *real numbers*, and * i* – an *imaginary unit, i.e. ** i * ^{ 2}* = * –1.* *A real number *a * is called an *abscissa *of complex number * a+ bi*, and *b – * an *ordinate *of complex number *a+ bi. * Two

complex numbers * a+ bi * and * a – bi * are called the *conjugate complex* numbers.

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*Main agreements:*

1. A real number * a * can also be written in the shape of a complex number: * a+ * 0* i * or * a – *0* i*. For example, the records 5 + 0 *i* and 5 – 0 *i* mean the same real number 5 .

2. A complex number 0*+ * *bi* is called a *pure imaginary number. *The record * bi *means the same as 0*+ **bi*.

3. Two complex numbers * a+ bi* and *c+ di* are considered as equal ones, if * a=c * and* b=d*. Otherwise, the complex numbers aren’t equal.

*Addition. * A sum of complex numbers * a+ bi* and *c+ di * is called a complex number ( * a+ c * ) + (* b+ d *)* i. *So, *at addition of complex numbers their * *abscissas and ordinates are added separately.* This definition corresponds to the rules of operations at usual polynomials.

*Subtraction.* A difference of two complex numbers * a+ bi * ( a minuend ) and *c+ di* ( a subtrahend ) is called a complex number ( * a – c *) + (* b – d *)* i.* So, *at subtraction of two complex numbers their abscissas and ordinates are subtracted separately.*

*Multiplication.* A product of complex numbers * a+ bi* and *c+ di *is called a complex number: ( * ac – bd * ) + (* ad + bc *)* i. *

This definition follows from two requirements:

1) the numbers * a+ bi* and *c+ di * must be multiplied as algebraic binomials,

2) a number * i* has a main property: *i *² = *–*1.

E x a m p l e . ( * a+ bi * )(* a – bi *)*= a *² *+ b * ²*. *Hence it follows, that

*a product * *of two conjugate complex numbers is a real positive number!*

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*Division.** *To divide a complex number *a+ bi * ( a dividend ) by another * c+ di * ( a divisor )* *means to find the third number *e+ f i* ( a quotient ), which being multiplied by the divisor* c+ di*, results the dividend* a+ bi. *If divisor isn’t equal to zero, then division is always valid.

E x a m p l e . Find ( 8 + *i *) : ( 2 – 3*i* ) .

S o l u t i o n . Rewrite this quotient as a fraction:

Multiplying its numerator* *and denominator by 2 + 3*i *and

executing all operations, we’ll receive:

*Geometric representationof complex numbers.* Real numbers are represented by points in a numerical line:

Here a point *A* means a number –3, a point *B* – a number 2, and *O * – zero. In contrast this complex numbers are represented by points in a “numerical” ( coordinate ) plane. For this we select a system of rectangular ( Cartesian ) coordinates with the same scale in both axes. Then, a complex number * a+ bi * will be represented by point *P * with abscissa *a* and ordinate *b* ( see figure ). This coordinate system is called a *complex plane*.

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*Modulus * of a complex number is a length of vector *OP*, representing this complex number in a coordinate(*complex*) plane. Modulus of complex number *a+ bi * is signed as |* a+ bi* | or by letter *r* and equal to :

Conjugate complex numbers havethe same modulus.* *

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*Argument* of a complex number is the angle between *x*-axis and vector *OP*, representing this complex number.

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*Trigonometric form of a complex number.* Abscissa *a* and ordinate *b *of the complex number * a + bi * can be expressed by its modulus *r* and argument ** ** :

* Operations with complex numbers, represented in the trigonometric form.*

This is the famous * **Moivre’s formula.*

Here * k* is any integer. To receive * n* different values of the *n*-th degree root of *z * it’s necessary to give * n* consecutive values for *k *( e.g., *k* = 0, 1, 2,…, *n* – 1) .