Complex numbers are numbers composed of a real and an imaginary part.
They represent the set of all ordered pairs (x, y), whose elements belong to the set of real numbers (R).
The set of complex numbers is indicated by Ç and defined by the operations:
- Equality: (a, b) = (c, d) ↔ a = c and b = d
- Addition: (a, b) + (c, d) = (a + b + c + d)
- Multiplication: (a, b). (c, d) = (ac – bd, ad + bc)
Imaginary Unit (i)
Indicated by the letter i, the imaginary unit is the ordered pair (0, 1). Soon:
i. i = -1 ↔ i2 = –1
Thus, i is the square root of –1.
Algebraic Form of Z
The algebraic form of Z is used to represent a complex number using the formula:
Z = x + yi
Where:
- x is a real number indicated by x = Re(Z), being called real part of z.
- y is a real number indicated by y = Im(Z), being called imaginary part of Z.
Complex Number Conjugate
The conjugate of a complex number is indicated by z, defined by z = a - bi. Thus, the sign of its imaginary part is exchanged.
So if z = a + bi, then z = a – bi
When we multiply a complex number by its conjugate, the result will be a real number.
Equality between Complex Numbers
Being two complex numbers Z1 = (a, b) and Z2 = (c, d), they are equal when a = c and b = d. That's because they have identical real and imaginary parts. Thus:
a + bi = c + di When a = c and b = d
Operations with Complex Numbers
With complex numbers it is possible to perform addition, subtraction, multiplication and division operations. Check out the definitions and examples below:
Addition
Z1 + Z2 = (a + c, b + d)
In algebraic form, we have:
(a + bi) + (c + di) = (a + c) + i (b + d)
Example:
(2 +3i) + (–4 + 5i)
(2 - 4) + i (3 + 5)
–2 + 8i
Subtraction
Z1 – Z2 = (a - c, b - d)
In algebraic form, we have:
(a + bi) - (c + di) = (a - c) + i (b - d)
Example:
(4 - 5i) - (2 + i)
(4 – 2) + i (–5 –1)
2 - 6i
Multiplication
(a, b). (c, d) = (ac – bd, ad + bc)
In algebraic form, we use the distributive property:
(a + bi). (c + di) = ac + adi + bci + bdi2 (i2 = –1)
(a + bi). (c + di) = ac + adi + bci – bd
(a + bi). (c + di) = (ac - bd) + i (ad + bc)
Example:
(4 + 3i). (2 - 5i)
8 – 20i + 6i – 15i2
8 - 14i + 15
23 – 14i
Division
Z1/Z2 = Z3
Z1 = Z2. Z3
In the above equality, if Z3 = x + yi, we have:
Z1 = Z2. Z3
a + bi = (c + di). (x + yi)
a + bi = (cx - dy) + i (cy + dx)
By the system of unknowns x and y we have:
cx - dy = a
dx + cy = b
Soon,
x = ac + bd/c2 + d2
y = bc - ad/c2 + d2
Example:
2 - 5i/i
2 – 5i/. (– i)/ (– i)
-2i +5i2/–i2
5 – 2i
Entrance Exam Exercises with Feedback
1. (UF-TO) Consider i the imaginary unit of complex numbers. The value the expression (i + 1)8 é:
a) 32i
b) 32
c) 16
d) 16i
Alternative c: 16
2. (UEL-PR) The complex number z that checks the equation iz – 2w (1 + i) = 0 (w indicates the conjugate of z) is:
a) z = 1 + i
b) z = (1/3) - i
c) z = (1 - i)/3
d) z = 1 + (i/3)
e) z = 1 - i
Alternative e: z = 1 - i
3. (Vunesp-SP) Consider the complex number z = cos π/6 + i sin π/6. The value of Z3 + Z6 + Z12 é:
there
b) ½ +√3/2i
c) i – 2
d) i
e) 2i
Alternative d: i
Check out more questions, with commented resolution, in Exercises on Complex Numbers.
Video lessons
To expand your knowledge of complex numbers, watch the video "Introduction to Complex Numbers"
History of complex numbers
The discovery of complex numbers was made in the 16th century thanks to the contributions of the mathematician Girolamo Cardano (1501-1576).
However, it was not until the 18th century that these studies were formalized by the mathematician Carl Friedrich Gauss (1777-1855).
This was a major step forward in mathematics, as a negative number has a square root, which until the discovery of complex numbers was considered impossible.
To learn more, see also
- Numerical sets
- Polynomials
- irrational numbers
- 1st Degree Equation
- Potentiation and Radiation
