Complex numbers

Imaginary and complex numbers. Abscissa and ordinate

complex number. Conjugate complex numbers.

Operations with complex numbers... Geometric

representation of complex numbers. Complex plane.

The modulus and argument of a complex number. Trigonometric

complex number form. Operations with complex

numbers in trigonometric form. Moivre's formula.

Initial information about imaginary and complex numbers are given in the section "Imaginary and complex numbers". The need for these numbers of a new type arose when solving quadratic equations for the caseD< 0 (здесь D Is the discriminant of the quadratic equation). For a long time, these numbers did not find physical application, so they were called "imaginary" numbers. However, now they are very widely used in various fields of physics.

and technology: electrical engineering, hydro- and aerodynamics, theory of elasticity, etc.

Complex numbers are written as: a + bi... Here aand b – real numbers , a i – imaginary unit, i.e.e. i 2 = –1. Number acalled abscissa, a b - ordinate complex numbera + bi.Two complex numbersa + bi and a - bi are called associated complex numbers.

Basic agreements:

1. Real number acan also be written in the form complex number:a +0 ior a -0 i. For example, records 5 + 0 i and 5 - 0 imean the same number5 .

2. Complex number 0 + bi called purely imaginary number. Recordingbimeans the same as 0 + bi.

3. Two complex numbers a + bi andc + diare considered equal ifa \u003d cand b \u003d d... Otherwise complex numbers are not equal.

Addition. The sum of complex numbersa + bi and c + diis called a complex number (a + c ) + (b + d ) i.In this way, when adding complex numbers, their abscissas and ordinates are added separately.

This definition follows the rules for dealing with ordinary polynomials.

Subtraction. Difference of two complex numbersa + bi (decreasing) and c + di (subtracted) is called a complex number ( a - c ) + (b - d ) i.

In this way, when subtracting two complex numbers, their abscissas and ordinates are subtracted separately.

Multiplication. The product of complex numbersa + bi and c + di called a complex number:

( ac - bd ) + (ad + bc ) i.This definition follows from two requirements:

1) numbers a + bi and c + dimust be multiplied like algebraic binomial,

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

PRI me r. ( a + bi )( a - bi) \u003d a 2 + b 2 . Hence, composition

two conjugate complex numbers is equal to the real

a positive number.

Division. Divide a complex numbera + bi (divisible) by another c + di(divider) - means to find the third numbere + f i (chat), which being multiplied by a divisorc + di, results in the dividenda + bi.

If the divisor is not zero, division is always possible.

PRI me r. Find (8 + i ) : (2 – 3 i) .

Solution. Let's rewrite this ratio as a fraction:

Multiplying its numerator and denominator by 2 + 3i

AND after performing all the transformations, we get:

Geometric representation of complex numbers. Real numbers are represented by dots on the number line:

Here the point Ameans number –3, pointB - number 2, and O - zero. In contrast, complex numbers are represented by dots on coordinate plane... For this we choose rectangular (Cartesian) coordinates with the same scales on both axes. Then the complex number a + bi will be represented by a dot P with abscissa a and ordinate b (see fig.). This coordinate system is called complex plane .

Module complex number is the length of the vector OPrepresenting a complex number on the coordinate ( an integrated) plane. Complex number module a + bi denoted by | a + bi | or letter r

Complex numbers are the minimum extension of the set of real numbers we are used to. Their fundamental difference is that an element appears that gives -1 in the square, i.e. i, or.

Any complex number has two parts: real and imaginary:

Thus, it can be seen that the set of real numbers coincides with the set of complex numbers with zero imaginary part.

The most popular model of a set of complex numbers is the regular plane. The first coordinate of each point will be its real part, and the second one will be imaginary. Then vectors with the origin at point (0,0) will act as the complex numbers themselves.

Operations on complex numbers.

In fact, if we take into account the model of a set of complex numbers, it is intuitively clear that addition (subtraction) and multiplication of two complex numbers are performed in the same way as the corresponding operations on vectors. And we mean the vector product of vectors, because the result of this operation is again a vector.

1.1 Addition.

(As you can see, this operation exactly matches)

1.2 Subtraction, similarly, is performed according to the following rule:

2. Multiplication.

3. Division.

Defined simply as the inverse of multiplication.

Trigonometric form.

The modulus of a complex number z is the following quantity:

,

obviously this is, again, just the modulus (length) of the vector (a, b).

Most often, the modulus of a complex number is denoted as ρ.

It turns out that

z \u003d ρ (cosφ + isinφ).

The following immediately follow from the trigonometric form of notation for a complex number. formulas :

The last formula is called Moivre formula. The formula is derived directly from it nth root of a complex number:

thus, there are n roots of the nth degree of the complex number z.

Lesson plan.

1. Organizational moment.

2. Presentation of the material.

3. Homework.

4. Summing up the lesson.

During the classes

I. Organizational moment.

II. Material presentation.

Motivation.

Expansion of the set of real numbers is that new numbers (imaginary) are added to the real numbers. The introduction of these numbers is associated with the impossibility of extracting a root from a negative number in the set of real numbers.

Introduction of the concept of a complex number.

The imaginary numbers with which we supplement the real numbers are written as biwhere i Is an imaginary unit, and i 2 \u003d - 1.

Based on this, we get following definition complex number.

Definition... A complex number is an expression of the form a + biwhere aand b - real numbers. In this case, the following conditions are met:

a) Two complex numbers a 1 + b 1 iand a 2 + b 2 iare equal if and only if a 1 \u003d a 2, b 1 \u003d b 2.

b) The addition of complex numbers is determined by the rule:

(a 1 + b 1 i) + (a 2 + b 2 i) \u003d (a 1 + a 2) + (b 1 + b 2) i.

c) Multiplication of complex numbers is determined by the rule:

(a 1 + b 1 i) (a 2 + b 2 i) \u003d (a 1 a 2 - b 1 b 2) + (a 1 b 2 - a 2 b 1) i.

Algebraic form of a complex number.

Writing a complex number as a + biis called the algebraic form of a complex number, where a - real part, bi Is the imaginary part, and b Is a real number.

Complex number a + biconsidered equal to zeroif its real and imaginary parts are equal to zero: a \u003d b \u003d 0

Complex number a + biat b \u003d 0considered to coincide with real number a: a + 0i \u003d a.

Complex number a + biat a \u003d 0is called purely imaginary and is denoted bi: 0 + bi \u003d bi.

Two complex numbers z \u003d a + bi and \u003d a - bithat differ only in the sign of the imaginary part are called conjugate.

Actions on complex numbers in algebraic form.

You can perform the following actions on complex numbers in algebraic form.

1) Addition.

Definition... The sum of complex numbers z 1 \u003d a 1 + b 1 i and z 2 \u003d a 2 + b 2 icalled a complex number z, the real part of which is equal to the sum of the real parts z 1 and z 2, and the imaginary part is the sum of the imaginary parts of the numbers z 1 and z 2, i.e z \u003d (a 1 + a 2) + (b 1 + b 2) i.

Numbers z 1 and z 2 are called terms.

The addition of complex numbers has the following properties:

1º. Commutativity: z 1 + z 2 \u003d z 2 + z 1.

2º. Associativity: (z 1 + z 2) + z 3 \u003d z 1 + (z 2 + z 3).

3º. Complex number –A –bi called the opposite of a complex number z \u003d a + bi... Complex number opposite to complex number z, denoted -z... Sum of complex numbers z and -z is equal to zero: z + (-z) \u003d 0

Example 1. Perform addition (3 - i) + (-1 + 2i).

(3 - i) + (-1 + 2i) \u003d (3 + (-1)) + (-1 + 2) i \u003d 2 + 1i.

2) Subtraction.

Definition. Subtract from a complex number z 1 complex number z 2 z, what z + z 2 \u003d z 1.

Theorem... The difference of complex numbers exists and, moreover, is unique.

Example 2. Perform subtraction (4 - 2i) - (-3 + 2i).

(4 - 2i) - (-3 + 2i) \u003d (4 - (-3)) + (-2 - 2) i \u003d 7 - 4i.

3) Multiplication.

Definition... The product of complex numbers z 1 \u003d a 1 + b 1 i and z 2 \u003d a 2 + b 2 i called a complex number zdefined by the equality: z \u003d (a 1 a 2 - b 1 b 2) + (a 1 b 2 + a 2 b 1) i.

Numbers z 1 and z 2 are called factors.

Multiplication of complex numbers has the following properties:

1º. Commutativity: z 1 z 2 \u003d z 2 z 1.

2º. Associativity: (z 1 z 2) z 3 \u003d z 1 (z 2 z 3)

3º. Distributiveness of multiplication relative to addition:

(z 1 + z 2) z 3 \u003d z 1 z 3 + z 2 z 3.

4º. z \u003d (a + bi) (a - bi) \u003d a 2 + b 2is a real number.

In practice, multiplication of complex numbers is performed according to the rule of multiplying the sum by the sum and separating the real and imaginary parts.

In the following example, we will consider multiplication of complex numbers in two ways: by rule and multiplication of sum by sum.

Example 3. Perform multiplication (2 + 3i) (5 - 7i).

1 way. (2 + 3i) (5 - 7i) \u003d (2 × 5 - 3 × (- 7)) + (2 × (- 7) + 3 × 5) i \u003d \u003d (10 + 21) + (- 14 + 15 ) i \u003d 31 + i.

Method 2. (2 + 3i) (5 - 7i) \u003d 2 × 5 + 2 × (- 7i) + 3i × 5 + 3i × (- 7i) \u003d \u003d 10 - 14i + 15i + 21 \u003d 31 + i.

4) Division.

Definition... Divide a complex number z 1 on a complex number z 2, then find such a complex number z, what z z 2 \u003d z 1.

Theorem. The quotient of complex numbers exists and is unique if z 2 ≠ 0 + 0i.

In practice, the quotient of complex numbers is found by multiplying the numerator and denominator by the conjugate of the denominator.

Let z 1 \u003d a 1 + b 1 i, z 2 \u003d a 2 + b 2 ithen

.

In the following example, divide by the formula and the rule of multiplication by the conjugate of the denominator.

Example 4. Find the quotient .

5) Raising to a positive integer.

a) The powers of the imaginary unit.

Using the equality i 2 \u003d -1, it is easy to define any positive integer power of the imaginary unit. We have:

i 3 \u003d i 2 i \u003d -i,

i 4 \u003d i 2 i 2 \u003d 1,

i 5 \u003d i 4 i \u003d i,

i 6 \u003d i 4 i 2 \u003d -1,

i 7 \u003d i 5 i 2 \u003d -i,

i 8 \u003d i 6 i 2 \u003d 1 etc.

This shows that the values \u200b\u200bof the degree i nwhere n - a positive integer, periodically repeated when the indicator increases by 4 .

Therefore, to raise the number i to a whole positive degree, the exponent must be divided by 4 and erect i to a power whose exponent is equal to the remainder of the division.

Example 5. Calculate: (i 36 + i 17) i 23.

i 36 \u003d (i 4) 9 \u003d 1 9 \u003d 1,

i 17 \u003d i 4 × 4 + 1 \u003d (i 4) 4 × i \u003d 1 i \u003d i.

i 23 \u003d i 4 × 5 + 3 \u003d (i 4) 5 × i 3 \u003d 1 · i 3 \u003d - i.

(i 36 + i 17) i 23 \u003d (1 + i) (- i) \u003d - i + 1 \u003d 1 - i.

b) Raising a complex number to a positive integer power is performed according to the rule of raising a binomial to the appropriate power, since it is a special case of multiplying the same complex factors.

Example 6. Calculate: (4 + 2i) 3

(4 + 2i) 3 \u003d 4 3 + 3 × 4 2 × 2i + 3 × 4 × (2i) 2 + (2i) 3 \u003d 64 + 96i - 48 - 8i \u003d 16 + 88i.