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Complex numbers

Kapitoly: Complex numbers, Graphical representation of complex numbers, The goniometric form of a complex number

Complex numbers are an extension of real numbers. In the real number field, we can do most of the classical operations such as addition, subtraction, multiplication and division (except division by zero). In real numbers, we can also subtract, but only non-negative numbers. This presents a crack, for example when we calculate a quadratic equation and come up with a negative discriminant. This crack is patched by complex numbers.

What is a complex number

Complex numbers differ from ordinary numbers mainly in that they contain two parts - a real part and an imaginary part. A complex number is a pair of ordered numbers [x, y], where the number x represents the real part and the number y represents the imaginary part. If the real part is zero, it is a purely imaginary complex number. The numbers x and y are themselves real.

The set of complex numbers is denoted by the capital letter cé: $\mathbb{C}$.

Equality of complex numbers: unlike ordinary numbers, complex numbers contain two components. Thus, if two complex numbers are to be equal, they must be equal in both components. These complex numbers are thus different from each other: [2, 3]≠[0, 3]≠[2, 0]. The complex numbers z1 = [1,2] and z2 = [1,2] are equal: z1 = z2.

Algebraic form and imaginary unit

Complex numbers are more often written in algebraic form, which looks like this. The complex number [x, y] has the algebraic form x + yi, where i is called the imaginary unit. A very important relationship holds for the square of the imaginary unit:

$$i^2=-1$$

This equation will be used frequently hereafter, along with other powers. Higher powers can already be calculated in the usual way. For example, if we want to simplify i3, we can decompose the expression according to the rules of calculating with powers into i2 · i. We already know that i2 is equal to minus one. We don't decompose the second E any more, so we get: i3 = −i.

$$\Large i^3=\underbrace{i^2}_{-1}\cdot \underbrace{i}_{i}=-i$$

Similarly for i4. We can decompose this into i2 · i2, taking i2 = −1, so we get: −1 · − 1 = 1.

$$\Large i^4=\underbrace{i^2}_{-1}\cdot\underbrace{i^2}_{-1}=-1\cdot-1=1$$

So i4 = 1. We can take advantage of this if we are calculating even higher powers of the imaginary unit. For example, when we compute i7, we can decompose this into i4 · i3. We know that i4 = 1, so we get: 1 · i3. And we know that i3 = −i. The result is: 1 · (−i) = −i.

Addition and multiplication

We can add and multiply complex numbers. When adding two complex numbers z1 and z2, we just add the real parts and the imaginary parts separately:

$$z_1+z_2=(x_1+y_1i) + (x_2+y_2i)=(x_1+x_2)+(y_1+y_2)i$$

Example: z1 = 3 + 7i and z2 = 5 + 8i. The sum would look like this: z1 + z2 = (3 + 5)+(7 + 8)i = 8 + 15i.

Theproduct of two complex numbers is a bit more complicated, but it can be broken down into the classic multiplication of parentheses. The basic formula looks like this:

$$z_1\cdot z_2=(x_1+y_1i) \cdot (x_2+y_2i)=(x_1x_2-y_1y_2)+(x_1y_2+x_2y_1)i$$

And how did we arrive at it? Let's try multiplying z1 · z2 the same way we would multiply a regular parenthesis. We get:

$$z_1\cdot z_2=x_1x_2+x_1y_2i+x_2y_1i+y_1y_2i^2$$

Because i2 = −1, we get after adjusting the last term:

$$z_1\cdot z_2=x_1x_2+x_1y_2i+x_2y_1i-y_1y_2$$

Now we just put the terms without and with the imaginary unit together:

$$z_1\cdot z_2=(x_1x_2-y_1y_2)+(x_1y_2+x_2y_1)i$$

Example. Let's try multiplying the numbers 5 + 6i and 4 + 7i. We get:

$$(5+6i)\cdot(4+7i)=20+35i+24i+42i^2=20+59i-42=-22+59i$$

Inverse, inverted and complex-associated numbers

  • The reciprocal to the complex number x + yi has the form −x − yi. Similar to the real numbers, we get the reciprocal by multiplying the given complex number by minus one. Example: the reciprocal of 2 + 7i is −2 − 7i. The reciprocal of −5 + 8i is 5 − 8i, etc.

  • Theinverse of the complex number x + yi is $\frac{1}{x+yi}$. The inverse of 4 − 2i is $\frac{1}{4-2i}$.

  • Thecomplex number associated to the number x + yi has the form x − yi. It is usually denoted by a bar as follows: $\overline{z}$, or by an asterisk z*.Such a number is the number complex associated to the complex number z. The complex number associated to the number 2 + 9i is the number 2 − 9i.

Subtraction and division

Once we have defined the reciprocal and inverse of a number, we can also define the operations of subtraction and division. If we want to subtract two complex numbers, z1 − z2, we add the reciprocal of z2 to the number z1. In practice, we get a simple formula:

$$z_1-z_2=(x_1+y_1i) - (x_2+y_2i) = (x_1-x_2)+(y_1-y_2)i$$

Similarly, we convert division z1 / z2 to multiplication by multiplying $z_1\cdot z_2^\prime$, where $z_2^\prime$ is the inverse of z2.

The absolute value

We calculate the absolute value of the complex number z using the formula:

$$|z|=\sqrt{x^2+y^2} = \sqrt{z\cdot \overline{z}}$$

(In the square root, the square z is the complex number associated to z.) The meaning of this formula is well seen in the geometric representation of complex numbers.

Any complex number whose absolute value is equal to one is called a complex unit. Examples of complex units: 1, i, $\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}i$.

Absolute value properties:

  • The absolute value of a complex number is a real number.
  • |z|≥0
  • |z| = |−z| = |z*|, where z* is a complex number.

Video

If you'd rather watch than read, watch the following video about complex numbers by Mirek Olšák.

http://www.youtube.com/watch?v=Ip69mJyF-8s

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