All existing numbers were created according to human needs at the time of creation, as is the case with natural numbers, which were created to count and control “stocks”, and irrational numbers, which were established to solve problems in relation to roots. It was precisely the problems involving roots that started the knowledge about the complex numbers.
The quadratic equation x2 + 4x + 5 = 0 has no real roots. This means that, within the set of real numbers, it is impossible to find values for x that equal the first term of this equation to the second. We observe this phenomenon from the beginning of Bhaskara's formula:
Δ = 42 – 4·1·5
Δ = 16 – 20
Δ = – 4
Once a negative value is found for Δ, it becomes impossible to proceed with Bhaskara's formula, as it requires that √Δ (root of delta) be calculated. Now, we know that √– 4 cannot be calculated because there is no real number that, multiplied by itself, results in – 4.
Complex numbers were created to meet these needs. From its creation, the √– 4 can be developed as follows:
√– 4 = √(– 1·4) = √(– 1)·22 = 2√(– 1)
A √(– 1) is understood as a new type of number. The set of all these numbers is known as the set of complex numbers, and each representative of this new set is defined as follows: Let A be a complex number, then,
A = The + Bi, where Theand B are real numbers and i = √(– 1)
In this definition, The It is known as real part of A and B It is known as imaginary part of A.
Properties of complex numbers
Real numbers represent, in their entirety and geometrically, a line. Complex numbers, in turn, represent an entire plane. The Cartesian plane used to represent the complex numbers is known as the Argand-Gauss plane.
Every complex number can be represented on the Argand-Gauss plane as a point of coordinates (a, b). The distance from the point representing a complex number to the point (0,0) is called the modulus of the complex number., which is defined:
Let A = a + bi be a complex number, its modulus is |A| = a2 + b2
Complex numbers also have an inverse element, called a conjugate. It is defined as:
Let A = a + bi be a complex number,
Ā = a – bi is the conjugate of this number.
Property 1: The product of a complex number and its conjugate is equal to the sum of the squares of the real part and the imaginary part of the complex number. Mathematically:
AĀ = a2 + b2
Example: What is the product of A = 2 + 5i by its conjugate?
Just do the calculation: a2 + b2 = 22 + 52 = 4 + 25 = 29. If we chose to write the conjugate of A and, after that, perform the multiplication AĀ, we would have:
AĀ = (2 + 5i) (2 - 5i)
AĀ = 4 – 10i + 10i + 25
AĀ = 4 + 25
AĀ = 29
That is, using the proposed property, it is possible to avoid a long calculation as well as errors during these calculations.
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Property 2: If a complex number A is equal to its conjugate, then A is a real number.
Let A = a + bi. If A = Ā, then:
a + bi = a - bi
bi = - bi
b = - b
Therefore, b = 0
Therefore, it is mandatory that every complex number equal to its conjugate is also a real number.
Property 3: The conjugate of the sum of two complex numbers is equal to the sum of the conjugates of these numbers., that is:
_____ _ _
A + B = A + B
Example: What is the conjugate of the sum of 7 + 9i and 2 + 4i?
____ ____
7 + 9i + 2 + 4i = 7 – 9i + 2 – 4i = 9 – 13i
You can add first and then calculate the conjugate of the result, or do the conjugates first and then add the results later.
Property 4: The conjugate of the product between two complex numbers is equal to the product of their conjugates, i.e:
__ _ _
AB = A·B
Example: What is the product of the conjugates of A = 7i + 10 and B = 4 + 3i?
(10 + 7i)·(4 + 3i) = (10 – 7i)·(4 – 3i) = 40 – 30i – 28i – 21 = 19 – 58i
Depending on the need for the exercise, it is possible to multiply first and calculate the conjugate afterwards, or display the conjugates before performing the multiplication.
Property 5: The product of a complex number A and its conjugate is equal to the square of the modulus of A, i.e:
AĀ = |A|2
Example: A = 2 + 6i, then AĀ = |A|2 = (√a2 + b2)2 = (√22 + 62)2 = 22 + 62 = 4 + 16 = 20. Note that it is not necessary to find the conjugate and carry out a multiplication through the distributive property of multiplication over addition (known as little showerhead).
Property 6: The modulus of a complex number is equal to the modulus of its conjugate. In other words:
|A| = |Ā|
Example: Find the modulus of the conjugate of the complex number A = 3 + 4i.
Note that it is not necessary to find the conjugate, as the modules are the same.
|A| = √(a2 + b2)= √(32 + 42) = √(9 + 16) = √25 = 5
If |Ā| were calculated, the only change would be a B negative squared, which has a positive result. Thus, the result would still be the root of 25.
Property 7: If A and B are complex numbers, then the modulus product of A and B is equal to the modulus of the product of A and B., i.e:
|AB| = |A||B|
Example: Let A = 6 + 8i and B = 4 + 3i, how much is |AB|?
Note that it is not necessary to multiply complex numbers before calculating the modulus. It is possible to calculate the modulus of each complex number separately and then just multiply the results.
|A| = √(62 + 82) = √(36 + 64) = √100 = 10
|B| = √(42 + 32) = √(16 + 9) = √25 = 5
|AB| = |A||B| = 10·5 = 50
By Luiz Paulo Moreira
Graduated in Mathematics
Would you like to reference this text in a school or academic work? Look:
SILVA, Luiz Paulo Moreira. "Properties involving complex numbers"; Brazil School. Available in: https://brasilescola.uol.com.br/matematica/propriedades-envolvendo-numeros-complexos.htm. Accessed on June 28, 2021.
