When studying the properties of the quadratic equation, a restriction was placed - for the discriminant less than zero there is no solution. It was immediately stipulated that this was a set of real numbers. A mathematician will be interested in an inquiring mind - what is the secret contained in the clause on real values?
Over time, mathematicians introduced the concept of complex numbers, where the unit is taken to be the conditional value of the root of the second degree of minus one.
History reference
Mathematical theory develops sequentially, from simple to complex. Let’s see how the concept came to be called the "complex number", and why it is needed.
Since time immemorial, the basis of mathematics has been the usual account. Researchers knew only a natural set of meanings. Addition and subtraction were simple. As economic relations became more complex, instead of adding up the same values, multiplication began to be applied. There was an inverse operation to multiply - division.
The concept of a natural number limited the use of arithmetic operations. On the set of integer values, it is impossible to solve all the division problems. Work with fractions led first to the concept of rational meanings, and then to irrational meanings. If for the rational it is possible to indicate the exact location of the point on the line, then for the irrational it is impossible to indicate such a point. You can only approximately indicate the interval of finding. The combination of rational and irrational numbers formed a real set, which can be represented as some line with a given scale. Each step along the line is a natural number, and between them are rational and irrational meanings.
The era of theoretical mathematics has begun. The development of astronomy, mechanics, and physics required the solution of increasingly complex equations. In general, the roots of the quadratic equation were found. When solving a more complex cubic polynomial, scientists were faced with a contradiction. The concept of the cubic root of the negative makes sense, but for the square one, the uncertainty is obtained. Moreover, the quadratic equation is only a special case of the cubic.
In 1545, the Italian J. Cardano proposed introducing the concept of an imaginary number.
That number was the root of the second degree of minus one. Finally, the term complex number was formed only after three hundred years, in the works of the famous mathematician Gauss. He proposed to formally extend to the imaginary number all the laws of algebra. The real line expanded to a plane. The world has become larger.
Basic concepts
Recall a number of functions that have restrictions on the real set:
- y = arcsin (x), defined in the range of values between negative and positive units.
- y = ln (x), the decimal logarithm makes sense with positive arguments.
- square root y = √x, calculated only for x ≥ 0.
By the notation i = √ (-1), we introduce such a concept as an imaginary number, this will remove all restrictions from the domain of definition of the above functions. Expressions of the type y = arcsin (2), y = ln (-4), y = √ (-5) take on meaning in some space of complex numbers.
The algebraic form can be written in the form of the expression z = x + i × y on the set of real values x and y, and i 2 = -1.
The new concept removes all restrictions on the use of any algebraic function and resembles a graph of a straight line in the coordinates of real and imaginary values.
Complex plane
The geometric shape of complex numbers clearly allows you to imagine many of their properties. On the axis Re (z) we mark real values of x, along Im (z) - imaginary values of y, then the point z on the plane will display the desired complex value.
Definitions:
- Re (z) is the real axis.
- Im (z) - means the imaginary axis.
- z is a conditional point of a complex number.
- The numerical value of the length of the vector from the zero point to z is called the module.
- The real and imaginary axes split the plane into quarters. With a positive value of coordinates - I quarter. When the argument of the real axis is less than 0, and the imaginary is greater than 0 - II quarter. When the coordinates are negative - III quarter. The last, fourth quarter contains many positive real values and negative imaginary values.
Thus, on a plane with x and y coordinates, you can always visualize a point of a complex number. The symbol i is introduced to separate the real part from the imaginary.
The properties
- With the zero value of the imaginary argument, we get just a number (z = x), which is located on the real axis and belongs to the real set.
- A special case when the value of the real argument becomes zero, the expression z = i × y corresponds to the location of the point on the imaginary axis.
- The general form of z = x + i × y will be with nonzero values of the arguments. Indicates the location of the point characterizing the complex number in one of the quarters.
Trigonometric notation
Recall the polar coordinate system and the definition of the trigonometric functions sin and cos. Obviously, using these functions, you can describe the location of any point on the plane. To do this, it is enough to know the length of the polar beam and the angle of inclination to the material axis.
Definition A record of the form ∣z ∣ multiplied by the sum of the trigonometric functions cos (ϴ) and the imaginary part i × sin (ϴ) is called a trigonometric complex number. Here, the designation is the angle of inclination to the real axis
ϴ = arg (z), and r = ∣z∣, the length of the beam.
From the definition and properties of trigonometric functions, a very important Moiré formula follows:
z n = r n × (cos (n × ϴ) + i × sin (n × ϴ)).
Using this formula, it is convenient to solve many systems of equations containing trigonometric functions. Especially when the problem of exponentiation arises.
Module and phase
To complete the description of a complex set, we offer two important definitions.
Knowing the Pythagorean theorem, it is easy to calculate the length of the beam in the polar coordinate system.
r = ∣z∣ = √ (x 2 + y 2 ), such a record on a complex space is called a “module” and characterizes the distance from 0 to a point on the plane.
The angle of inclination of the complex beam to the real line ϴ is usually called the phase.
It can be seen from the definition that the real and imaginary parts are described using cyclic functions. Namely:
- x = r × cos (ϴ);
- y = r × sin (ϴ);
Conversely, the phase has a connection with algebraic values through the formula:
ϴ = arctan (x / y) + µ, the µ correction is introduced to take into account the periodicity of geometric functions.
Euler's formula
Mathematicians often use exponential form. The numbers of the complex plane are written as expressions
z = r × e i × ϴ , which follows from the Euler formula.
Such a record is widespread for the practical calculation of physical quantities. The presentation form in the form of exponential complex numbers is especially convenient for engineering calculations, where it becomes necessary to calculate circuits with sinusoidal currents and it is necessary to know the value of the function integrals with a given period. The calculations themselves serve as a tool in the design of various machines and mechanisms.
Definition of operations
As already noted, all algebraic laws of work with basic mathematical functions apply to complex numbers.
Transaction amount
When complex values are added, their real and imaginary parts also add up.
z = z 1 + z 2 , where z 1 and z 2 are complex numbers of a general form. Transforming the expression, after opening the brackets and simplifying the notation, we get the real argument x = (x 1 + x 2 ), the imaginary argument y = (y 1 + y 2 ).
On the graph, it looks like the addition of two vectors, according to the well-known parallelogram rule.
Subtraction operation
It is considered as a special case of addition, when one number is positive, the other is negative, that is, located in the mirror quarter. Algebraic notation looks like the difference between real and imaginary parts.
z = z 1 - z 2 , or, taking into account the values of the arguments, similarly to the addition operation, we obtain for real values x = (x 1 - x 2 ) and imaginary y = (y 1 - y 2 ).
Complex plane multiplication
Using the rules for working with polynomials, we derive a formula for solving complex numbers.
Following the general algebraic rules z = z 1 × z 2 , we paint each argument and give similar ones. The real and imaginary parts can be written as follows:
- x = x 1 × x 2 - y 1 × y 2 ,
- y = x 1 × y 2 + x 2 × y 1.
It looks prettier if we use exponential complex numbers.
The expression looks like this: z = z 1 × z 2 = r 1 × e i ϴ 1 × r 2 × e i ϴ 2 = r 1 × r 2 × e i ( ϴ 1+ ϴ 2) .
Then it’s simple, the modules multiply, and the phases add up.
Division
When considering the operation of division, as the inverse of the operation of multiplication, in exponential form of writing we get a simple expression. The division of the value of z 1 by z 2 is the result of the division of their modules and the phase difference. Formally, when using the exponential form of complex numbers, it looks like this:
z = z 1 / z 2 = r 1 × e i ϴ 1 / r 2 × e i ϴ 2 = r 1 / r 2 × e i ( ϴ 1- ϴ 2) .
In the form of an algebraic record, the operation of dividing the numbers of the complex plane is written a little more complicated:
z = z 1 / z 2.
By writing down the arguments and carrying out transformations of the polynomials, it is easy to obtain the values x = x 1 × x 2 + y 1 × y 2 , respectively y = x 2 × y 1 - x 1 × y 2 , however, this expression makes sense within the space described, if z 2 ≠ 0.
Extract the root
All of the above can be applied in the definition of more complex algebraic functions - raising to any degree and the inverse to it - extracting the root.
Using the general concept of raising to a power of n, we obtain the definition:
z n = (r × e i ϴ ) n .
Using common properties, we rewrite it in the form:
z n = r n × e i ϴ n .
Got a simple formula for raising to the power of a complex number.
From the definition of degree we get a very important consequence. An even degree of an imaginary unit is always 1. Any odd degree of an imaginary unit is always -1.
Now we will study the inverse function - extracting the root.
For simplicity, we take n = 2. The square root w of the complex value z on the complex plane C is considered to be the expression z = ±, which is valid for any real argument greater than or equal to zero. For w ≤ 0, there is no solution.
Let's look at the simplest quadratic equation z 2 = 1. Using the complex number formulas, we rewrite r 2 × e i 2ϴ = r 2 × e i 2ϴ = e i 0 . It can be seen from the record that r 2 = 1 and ϴ = 0, therefore, we have the only solution equal to 1. But this contradicts the concept that z = -1 also corresponds to the definition of the square root.
We will understand that we do not take into account. If we recall the trigonometric notation, we will restore the statement - with a periodic change in phase ϴ the complex number does not change. Let p denote the period value, then the notation r 2 × e i 2ϴ = e i (0+ p ) is valid, whence 2ϴ = 0 + p, or ϴ = p / 2. Consequently, e i 0 = 1 and e i p / 2 = -1. We got the second solution, which corresponds to a common understanding of the square root.
So, to find an arbitrary root of a complex number, we will follow the procedure.
- We write the exponential form w = ∣w∣ × e i ( arg ( w ) + pk ) , k is an arbitrary integer.
- The desired number can also be represented in the Euler form z = r × e i ϴ .
- We use the general definition of the root extraction function r n * e i n ϴ = ∣w∣ × e i ( arg ( w ) + pk ) .
- From the general properties of the equality of modules and arguments, we write r n = ∣w∣ and nϴ = arg (w) + p × k.
- The final record of the root of the complex number is described by the formula z = √∣w∣ × e i ( arg ( w ) + pk ) / n .
- Comment. The value ∣w∣, by definition, is a positive real number, which means that a root of any degree makes sense.
Field and Pairing
In conclusion, we give two important definitions that are of little value for solving applied problems with complex numbers, but are essential for the further development of mathematical theory.
They say that the expressions of addition and multiplication form a field if they satisfy the axioms for any elements of the complex plane z:
- From the change of place of complex terms, the complex amount does not change.
- The statement is true - in a complex expression, any sum of two numbers can be replaced with their value.
- There is a neutral value of 0 for which z + 0 = 0 + z = z.
- For any z, there is the opposite - z, the addition of which gives zero.
- When changing the places of complex factors, the complex product does not change.
- Multiplication of any two numbers can be replaced by their value.
- There is a neutral value of 1, multiplication by which does not change the complex number.
- For each z ≠ 0, there is an inverse value of z -1 , multiplication by which gives 1 as a result.
- Multiplying the sum of two numbers by the third is equivalent to the operation of multiplying each of them by this number and adding the results.
- 0 ≠ 1.
The numbers z 1 = x + i × y and z 2 = x - i × y are called conjugates.
Theorem. For pairing, the statement is true:
- The conjugation of the sum is equal to the sum of the conjugated elements.
- The conjugation of a product is equal to the product of conjugations.
- Pairing is equal to the number itself.
In general algebra, such properties are called field automorphism.
Examples
Following the above rules and formulas of complex numbers, you can easily operate on them.
Let's consider the simplest examples.
Problem 1. Using the equality 3y +5 xi = 15 - 7i, determine x and y.
Decision. Recall the definition of complex equalities, then 3y = 15, 5x = -7. Therefore, x = -7 / 5, y = 5.
Task 2. Calculate the values 2 + i 28 and 1 + i 135 .
Decision. Obviously, 28 is an even number, from the consequence of the definition of a complex number in degree we have i 28 = 1, which means that the expression 2 + i 28 = 3. The second value, i 135 = -1, then 1 + i 135 = 0.
Problem 3. Calculate the product of the values 2 + 5i and 4 + 3i.
Decision. From the general properties of multiplication of complex numbers, we obtain (2 + 5i) X (4 + 3i) = 8 - 15 + i (6 + 20). The new value will be -7 + 26i.
Problem 4. Calculate the roots of the equation z 3 = -i.
Decision. There are several options for finding a complex number. Consider one of the possible. By definition, ∣ - i∣ = 1, the phase for -i is -p / 4. The original equation can be rewritten as r 3 * e i 3ϴ = e - p / 4 + pk , whence z = e - p / 12 + pk / 3 , for any integer k.
Many solutions have the form (e - ip / 12 , e ip / 4 , e i 2 p / 3 ).
Why complex numbers are needed
History knows many examples when scientists, working on a theory, do not even think about the practical application of their results. Mathematics is first and foremost a game of the mind, strict adherence to causal relationships. Almost all mathematical constructions come down to solving integral and differential equations, and those, in turn, with some approximation, are solved by finding the roots of the polynomials. Here we meet for the first time with the paradox of imaginary numbers.
Scientists, natural scientists, solving perfectly practical problems, resorting to solutions of various equations, discover mathematical paradoxes. The interpretation of these paradoxes leads to completely amazing discoveries. The dual nature of electromagnetic waves is one such example. Complex numbers in the understanding of their properties play a decisive role.
This, in turn, has found practical application in optics, radio electronics, energy, and many other technological fields. Another example, much more difficult to understand physical phenomena. Antimatter was predicted at the tip of the pen. And only after many years, attempts to physically synthesize it begin.
It is not necessary to think that only in physics such situations exist. No less interesting discoveries are made in wildlife, during the synthesis of macromolecules, during the study of artificial intelligence. And all this is due to the expansion of our consciousness, avoiding simple addition and subtraction of natural quantities.