The conjugate number is extremely important to assist the student in performing calculations involving complex number divisions. Therefore, in this article “What is Conjugate in Math”, we will know a little more about the conjugate number and its main properties.
What is Conjugate in Math?
Conjugate numbers are true allies for making complex number divisions. With their participation, such calculations become simpler, easier, and allow better understanding by the individual.
To make this possible, the first step is to understand the magnitude and structure of the essential number in the midst of this calculation: the conjugate number.
One of the main purposes of the conjugate number is to make it possible to transform the complex denominator number (present in a fraction) into a real type number (an example we will see later).
To get the conjugate of a number may be simpler than you might think. Consider the following example:
Imagine that the number z is a complex number.
In calculating z ∈ C, d must be read as a participant of the set of complex numbers. Thus, it is written in another way:
Z = x + yi, or in coordinate form z = (x, y).
Note: the symbol ∈ means that z is a member of the following set, that is, in the case of the example, an element of group C.
Note 2: The letter C, in turn, is used to represent complex numbers.
In this case, z belongs to complex numbers. Continuing:
The z-number set is now represented by a small band above the letter. To obtain it, the only step is to exchange the imaginary signal at its top. In this sense, we will have:
z (with upper range) = x – yi. When in coordinate form, z would look like: (x, -y).
The use of conjugate numbers becomes fundamental for the realization of a series of divisions involving group C numbers, that is, complex numbers.
If we perform the multiplication between zz (this being a conjugate number, represented by the band at the top), a real number will always be obtained, or a number of the complex type, but with its imaginary part equal to 0.
And if you are still unsure about understanding conjugate numbers, check out a little more about the properties of this type of number.
The properties of the conjugate number will be fundamental to help you understand this type of calculation mainly in the realization of divisions involving not only complex numbers, but also their respective conjugates.
The properties of the conjugate number
To understand the properties of conjugate numbers the first step is to understand that the C (set of complex numbers) is composed of ordered pairs of real numbers.
The most common example (also previously used) is:
Z = x + yi.
In this sense, it should be noted that only x ∈ R, y ∈ rez ∈ R belong to the group C of the complex numbers.
From anybody, any complex number Z = (x, y) can be written as follows:
Real part of number: x
Imaginary part of number: yi
Some examples are: 1 + i; 2 + 3i, 5i, 31 and 2 = 3i.
Considering the presence of a complex number, we have the following account: Z = x – ui.
Its conjugate, in turn, can be represented by z = x – yi (all with the upper range).
To get the conjugate number, you have to swap the upper sign of the imaginary part of the number, making the real part stay the same and the imaginary parts become asymmetric. Some examples in this regard are:
Example 1: Z = 1 + 3i-Z (conjugate) = 1-3i;
Example 2: Z = 2 + 3i- Z (conjugate) = 2 – 3i;
Example 3: Z = -4i- Z (conjugate) = 4i.
When | Z | = | Z | (which may be extremely common), the conjugate module of one of the complex numbers should be the same module as the division number.
On the other hand, when the bill reaches Z. Z = | Z | ², this means that the result of a complex number is the square of the modulus of that number, reaching a real number.
Finally, it should be noted that the operations performed with conjugate numbers occur from the observation that, whenever a complex number is multiplied by its conjugate, the result will be a real number.
Let’s consider a small calculation for better understanding:
(1+ 2i). (1-2i)
1-2 i + 2 i. (2i) ²
1² – (2i) ²
1 + 4 = 5 which belongs to the set of real numbers.
Thus, it should be noted that the conjugate is always used for the realization of mathematical divisions between complex numbers. In denominated accounts, the numerator and denominator must be multiplied by the denominator number conjugate to arrive at the final result of the account.
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