Complex Numbers
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ALGEBRA OF COMPLEX NUMBERS
What is a complex number?
A complex number can be described as a number of the form x + iy where a and b are real numbers and i is given by root of -1 (i.e.-11/2). A complex number is denoted by z (i.e. z = x +iy), and the real part of the complex number z, i.e. x, is given by R(z) and the imaginary part, i.e. iy, is given by I(z). We can also describe a complex number as purely real or purely imaginary as I(z) or R(z) is zero, respectively. If a complex number z = 0 + i0 occurs then it is said to be both purely real as well as purely imaginary.
Two complex numbers z and z’ can be equal if and only R(z) = r(z’) and I(z) = i(z’), and the conjugate of a complex number z = x + iy is given by Z = x - iy.
NOTE: Geometrically the conjugate of a complex number z is is the reflection or its image in the real axis.
NOTE: usually the symbol i is denoted by the number (0,1) in the argand plane i.e. the complex number (0,1). i is given by -1, i3 by -i, i4 by 1 and so on.
BASIC MATHEMATICAL OPERATIONS WITH COMPLEX NUMBERS
1. ADDITION
If two complex numbers z1 = x1 + iy1 and z2 = x2- iy2 are to be added, then we add, the real and imaginary parts. i.e.
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z1 + z2 = (x1+x2) + i(y1+y2) |
Similarly for more than two complex numbers.
2. SUBTRACTION
If two complex numbers z1 = x1 + iy1 and z2 = x2- iy2 are to be subtracted, then we subtract the real and imaginary parts. i.e.
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z1 - z2 = (x1-x2) + i(y1-y2) |
3. MULTIPLICATION
If two complex numbers z1 = x1 + iy1 and z2 = x2- iy2 are to be multiplied then
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z1z2 = (x1y1 -x2y2 ) + i(x1y2 + x2y1) |
If two complex numbers are to be divided then we multiply both the numerator and denominator by the conjugate of the denominator and on solving we get,
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z1/z2 = (x1x2)/(x22+y22) + i(x2y1 - x1y2)/(x22+y22) |
If z = x + iy denotes a complex number then its modulus is given by
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|z| = (x2 + y2)1/2 |
- |z1 + z2| <= |z1| + |z2|
- |z1 - z2| = |z1| - |z2|
- |z1/z2| = |z1| / |z2|
- |z1 z2| = |z1| |z2|
- |z|2 = |z2|