Complex numbers are written in their algebraic form as follows: a + bi, we know that a and b are numbers reals and that the value of a is the real part of the complex number and that the value of bi is the imaginary part of the number. complex.
We can then say that a complex number z will be equal to a + bi (z = a + bi).
With these numbers we can carry out the operations of addition, subtraction and multiplication, obeying the order and characteristics of the real part and the imaginary part.
Addition
Given any two complex numbers z1 = a + bi and z2 = c + di, adding together we will have:
z1 + z2
(a + bi) + (c + di)
a + bi + c + di
a + c + bi + di
a + c + (b + d) i
(a + c) + (b + d) i
Therefore, z1 + z2 = (a + c) + (b + d) i.
Example:
Given two complex numbers z1 = 6 + 5i and z2 = 2 - i, calculate their sum:
(6 + 5i) + (2 - i)
6 + 5i + 2 - i
6 + 2 + 5i - i
8 + (5 - 1)i
8 + 4i
Therefore, z1 + z2 = 8 + 4i.
Subtraction
Given any two complex numbers z1 = a + bi and z2 = c + di, by subtracting we will have:
z1 - z2
(a + bi) - (c + di)
a + bi - c - di
a - c + bi - di
(a – c) + (b – d) i
Therefore, z1 - z2 = (a - c) + (b - d) i.
Example:
Given two complex numbers z1 = 4 + 5i and z2 = -1 + 3i, calculate their subtraction:
(4 + 5i) - (-1 + 3i)
4 + 5i + 1 – 3i
4 + 1 + 5i – 3i
5 + (5 - 3)i
5 + 2i
Therefore, z1 - z2 = 5 + 2i.
Multiplication
Given any two complex numbers z1 = a + bi and z2 = c + di, by multiplying we will have:
z1. z2
(a + bi). (c + di)
ac + adi + bci + bdi2
ac + adi + bci + bd (-1)
ac + adi + bci - bd
ac - bd + adi + bci
(ac - bd) + (ad + bc) i
Therefore, z1. z2 = (ac - bd) + (ad + bc) i.
Example:
Given two complex numbers z1 = 5 + i and z2 = 2 - i, calculate their multiplication:
(5 + i). (2 - i)
5. 2 - 5i + 2i - i2
10 – 5i + 2i + 1
10 + 1 – 5i + 2i
11 – 3i
Therefore, z1. z2 = 11 – 3i.
by Danielle de Miranda
Graduated in Mathematics
Source: Brazil School - https://brasilescola.uol.com.br/matematica/adicao-subtracao-multiplicacao-numero-complexo.htm