Complex Numbers
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Complex Numbers
Origin of Complex Numbers
- The concept of complex numbers was first identified by the Greek mathematician,
Leonhard Euler (1707-1783), while he was trying to find the square root of the Quadratic
Equation x + 1 = 0.
Definition of Complex Numbers
- A complex number is an ordered pair of real numbers. The set of all complex numbers
is denoted by the symbol ‘C‘. We have C = {(a, b) / a, b R} = R X R.
- A Complex Number is a number of the form z = a + ib , where ‘a’ and ‘b’ are
real numbers and ‘i’ is the imaginary unit, with the property i = (- 1).
Z = a + ib can also be represented as z = (a , b)
- The real number ‘a’ is called the real part [Re(z)] of the complex number
and the real number ‘b’is the imaginary part [Im(z)].
- Real numbers may be considered to be complex numbers with an imaginary part of Zero;
that is, the real number a is equivalent to the complex number ‘a + i0′
Example
- Consider the complex number 3 + i5, its real part is 3 and imaginary part is 5 .
- Consider the complex number 7 – i2, its real part is 7 and imaginary part is -2.
- 7 can be considered as a complex number with its imaginary part as zero.
Example
- Consider the complex number 3 + i5, its real part is 3 and imaginary part is 5 .
- Consider the complex number 7 – i2, its real part is 7 and imaginary part is -2.
- 7 can be considered as a complex number with its imaginary part as zero.
Arithmetic Operations on Complex Numbers
- All the four operations, addition, subtraction, multiplication and division can be
performed on complex numbers.
Addition of Complex Numbers
- If z1 = (a , b) and z2 = (c , d) then z1+ z2 = (a + c , b + d).
- For Example: z1 = 8 + i5 and z2 = 6 + i2 then z1 + z2 = 14 + i7 = (14 , 7)
Negative of a Complex Number
- If z = (a, b) then we define negative of a complex number as – z = (- a , – b) = (- a) + i(- b).
- For Example: z = 2 + i4, then – z = (- 2) + i(- 4) = (- 2 , – 4).
Subtraction of Complex Numbers
- If z1 = (a , b) and z2 = (c , d) then z1-z2 = (a – c , b – d).
- For Example: z1 = 4 + i7 and z2 = 2 + i5 then z1 – z2 = 2 + i2 = (2 , 2).
Multiplication of Complex Numbers
- If z1 = (a , b) and z2 = (c , d) then z1 . z2 = (a , b) . (c , d) = (ac – bd , ad + bc)
- For Example: z1 = 2 + i3 and z2 = 4 + i5 then z1 . z2 = – 7 + i22 = (- 7 , 22).
Division of Complex Numbers
- If z1 = (a , b) and z2 = (c , d) then =
- For Example: z1 = 2 + i3 and z2 = 3 + i4 then = =
Conjugate of a Complex number:
- For any complex number z= a + bi, we define the conjugate of z as a + (-b)i and
- denote this by and = (a – bi), that is, = (a – bi)
Geometrical Representation of Complex Numbers
- Carl Friedrich Gauss (1777-1855) was one of the mathematicians who first thought
that complex numbers can be represented on a two.dimensional plane called a Complex
Plane or a z-Plane. The Complex Plane is also known as the Argand Plane or Argand
diagram,named after Jean-Robert Argand. The geometrical representation of complex
number z and its conjugate are shown in the figure given.
The figure shows the representation of z = x + iy.Point z is obtained on the Cartesian
plane by taking the real part ‘x’ along the horizontal line/axis (as the x coordinate)
and then the imaginary part ‘y’ along the vertical line/axis, (as the y coordinate).
Hence the Horizontal line/axis is known as the ‘real axis’ and the vertical line/axis
is known as the ‘imaginary axis’.
As seen in the figure , the conjugate of z, is the
reflection of z along the real axis. oz = r, is called the modulus of the complex
number Z, where r = The angle of inclination of oz, with the positive real axis is
and is called the amplitude of the complex number, where =tan The complex number
z can also be represented in terms of r and as z = r(cos + isin ) = x + iy
where x = r cos and y = r sin . This notation is referred to as the polar form or
the trigonometric form.
Further reading on Complex Numbers
- Square root of a complex number
- Cube root of a complex number
- nth root of a complex number
- DeMovire’s Theorem
- Application of complex numbers
Additional Links for Complex Numbers
Watch the video related to complex
they are really crazy. like two animals dance!!!