Complex numbers are the numbers that are expressed in the form of a+ib where, a,b are real numbers and ‘i’ is an imaginary number called “iota”. The value of i = (√-1). For example, 2+3i is a complex number, where 2 is a real number (Re) and 3i is an imaginary number (Im).
Combination of both the real number and imaginary number is a complex number.
Examples of complex numbers:
- 1 + j
- -13 – 3i
- 0.89 + 1.2 i
- √5 + √2i
An imaginary number is usually represented by ‘i’ or ‘j’, which is equal to √-1. Therefore, the square of the imaginary number gives a negative value.
| Since, i = √-1, so, i2 = -1 |
The main application of these numbers is to represent periodic motions such as water waves, alternating current, light waves, etc., which rely on sine or cosine waves, etc.
Definition
The complex number is basically the combination of a real number and an imaginary number. The complex number is in the form of a+ib, where a = real number and ib = imaginary number. Also, a,b belongs to real numbers and i = √-1.
Hence, a complex number is a simple representation of addition of two numbers, i.e., real number and an imaginary number. One part of it is purely real and the other part is purely imaginary.
What are Real Numbers?
Any number which is present in a number system such as positive, negative, zero, integer, rational, irrational, fractions, etc. are real numbers. It is represented as Re(). For example: 12, -45, 0, 1/7, 2.8, √5, etc., are all real numbers.
What are Imaginary Numbers?
The numbers which are not real are imaginary numbers. When we square an imaginary number, it gives a negative result. It is represented as Im(). Example: √-2, √-7, √-11 are all imaginary numbers.
The complex numbers were introduced to solve the equation x2+1 = 0. The roots of the equation are of form x = ±√-1 and no real roots exist. Thus, with the introduction of complex numbers, we have Imaginary roots.
We denote √-1 with the symbol ‘i’, which denotes Iota (Imaginary number).
Notation
An equation of the form z= a+ib, where a and b are real numbers, is defined to be a complex number. The real part is denoted by Re z = a and the imaginary part is denoted by Im z = ib.
| Z = a + i b |
Examples
See the table below to differentiate between a real number and an imaginary number.
| Complex Number | Real Number | Imaginary Number |
| -1+2i | -1 | 2i |
| 7-9i | 7 | -9i |
| -6i | 0 | -6i (Purely Imaginary) |
| 6 | 6 | 0i (Purely Real) |
Is 0 a complex Number?
As we know, 0 is a real number. And real numbers are part of complex numbers. Therefore, 0 is also a complex number and can be represented as 0+0i.
Graphical representation
In the graph below, check the representation of complex numbers along the axes. Here we can see, the x-axis represents real part and y represents the imaginary part.
Absolute Value
The absolute
value of a real number is the number
itself. The absolute value of x is represented by modulus, i.e. |x|. Hence, the
modulus of any value always gives a positive value, such that;
|3|
= 3
|-3|
= 3
Now,
in case of complex numbers, finding the modulus has a different method.
Suppose,
z = x+iy is a complex number. Then, mod of z, will be:
|z|
= √(x2+y2)
This
expression is obtained when we apply the Pythagorean theorem in a complex
plane. Hence, mod of complex number, z is extended from 0 to z and mod of real
numbers x and y is extended from 0 to x and 0 to y respectively. Now these
values form a right triangle, where 0 is the vertex of the acute angle. Now,
applying Pythagoras theorem,
|z|2 = |x|2+|y|2
|z|2 = x2 +
y2
|z|
= √(x2+y2)
Algebraic Operations on
Complex Numbers
There can be four types of algebraic operation on complex numbers which are mentioned below. Visit the linked article to know
more about these algebraic operations along with solved examples. The four
operations on the complex numbers include:
·
o
- Addition
- Subtraction
- Multiplication
- Division
Roots
of Complex Numbers
When we solve a quadratic
equation in the form of ax2 +bx+c
= 0, the roots of the equations can be determined in three forms;
- Two Distinct Real Roots
- Similar Root
- No Real roots (Complex Roots)
Complex Number Formulas
While
performing the arithmetic operations of complex numbers such as addition and
subtraction, combine similar terms. It means that combine the real number with
the real number and imaginary number with the imaginary number.
Addition
(a + ib) + (c + id) = (a + c)
+ i(b + d)
Subtraction
(a + ib) – (c + id) = (a – c) +
i(b – d)
Multiplication
When
two complex numbers are multiplied by each other, the multiplication process
should be similar to the multiplication of two binomials. It means that the
FOIL method (Distributive multiplication process) is used.
(a + ib). (c + id) = (ac – bd) +
i(ad + bc)
Division
The
division of two complex numbers can be performed by multiplying the numerator
and denominator by its conjugate value of the denominator, and then applying
the FOIL Method.
(a + ib) / (c + id) = (ac+bd)/ (c2 + d2)
+ i(bc – ad) / (c2 + d2)
Power of Iota (i)
Depending upon the power of “i”, it can take the following values;
i4k+1 = i.i4k+2 = -1 i4k+3 =
-i.i4k = 1
Where k can have an integral value (positive or negative).
Similarly, we can find for the negative power of i,
which are as follows;
i-1 = 1 / i
Multiplying and dividing the above term with i, we have;
i-1 = 1 / i × i/i × i-1
= i / i2 = i / -1 = -i / -1 = -i
Note: √-1 × √-1 = √(-1 ×
-1) = √1 = 1 contradicts to the fact that i2 = -1.
Therefore, for an imaginary number, √a × √b is not equal
to √ab.
Identities
Let us see some of the identities.
·
-
- (z1 +
z2)2 = (z1)2 + (z2)2 +
2 z1 × z2
- (z1 – z2)2 =
(z1)2 + (z2)2 –
2 z1 × z2
- (z1)2 – (z2)2 =
(z1 + z2)(z1 – z2)
- (z1 +
z2)3 = (z1)3 + 3(z1)2 z2 +3(z2)2 z1 +
(z2)3
- (z1 –
z2)3 = (z1)3 – 3(z1)2 z2 +3(z2)2 z1 –
(z2)3
The properties of complex numbers are listed below:
- The addition
of two conjugate complex numbers will result in a real number
- The
multiplication of two conjugate complex number will also result in a real
number
- If x and y
are the real numbers and x+yi =0, then x =0 and y =0
- If p, q, r,
and s are the real numbers and p+qi = r+si, then p = r, and q=s
- The complex
number obeys the commutative law of addition and multiplication.
z1+z2 =
z2+z1
z1. z2 =
z2. z1
- The complex
number obeys the associative law of addition and multiplication.
(z1+z2) +z3 =
z1 + (z2+z3)
(z1.z2).z3 =
z1.(z2.z3)
- The complex
number obeys the distributive law
z1.(z2+z3)
= z1.z2 + z1.z3
- If the sum of
two complex number is real, and also the product of two complex number is
also real, then these complex numbers are conjugate to each other.
- For any two
complex numbers, say z1 and z2, then |z1+z2| ≤
|z1|+|z2|
- The result of
the multiplication of two complex numbers and its conjugate value should
result in a complex number and it should be a positive value.
Let z = a+ib be a complex number.
The Modulus of z is represented by |z|.
Mathematically,
|z|=a2+b2
The conjugate of z is denoted by z¯.
Mathematically,
z¯=a–ib
Similar to the XY plane, the Argand(or complex) plane is a system
of rectangular coordinates in which the complex number a+ib is represented by
the point whose coordinates are a and b.
We find the real and complex components in terms of r and θ, where r is the length of the vector and θ is the angle made with the real axis. Check out the detailed argand plane and polar representation of complex numbers in this article and understand this concept in a detailed way along with solved examples.
Solved Problems
Example
1: Simplify
a)
16i + 10i(3-i)
b)
(7i)(5i)
c)
11i + 13i – 2i
Solution:
a)
16i + 10i(3-i)
=
16i + 10i(3) + 10i (-i)
=
16i +30i – 10 i2
=
46 i – 10 (-1)
=
46i + 10
b)
(7i)(5i) = 35 i2 = 35 (-1) = -35
c)
11i + 13i – 2i = 22i
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