The conjugate of the complex number x + iy is defined as the complex number x - iy.
If z represents a complex number, then the complex conjugate of z is denoted by z̄. To get the conjugate of the complex number z, simply change the sign of i in z. For instance, 2 - 3i is the conjugate of 2 + 3i.
The product of a complex number with its conjugate is a real number. For instance,
(i) (a + ib)(a - ib) :
= a^{2} - (ib)^{2}
= a^{2} - i^{2}b^{2}
= a^{2} - (-1)b^{2}
= a^{2} + b^{2}
(i) (3 + 4i)(3 - 4i) :
= 3^{2} - (4i)^{2}
= 9 - 4^{2}i^{2}
= 9 - 4(-1)
= 9 + 4
= 13
Geometrically, the conjugate of of the complex number z is obtained by reflecting z on the real axis.
Note :
Two complex numbers x + iy and x - iy are conjugates to each other. The conjugate is useful in division of complex numbers. The complex number can be replaced with a real number in the denominator by multiplying the numerator and denominator by the conjugate of the denominator. This process is similar to rationalising the denominator to remove surds.
Problem 1 :
Write the complex number given below in x + iy form, hence find its real and imaginary parts.
Solution :
To express the given complex number in the rectangular form x + iy, multiply the numerator and denominator by the conjugate of the denominator to eliminate i in the denominator.
The complex number above is in the form x + iy.
Problem 2 :
Simplify the following expression into rectangular form.
Solution :
We consider
and
Therefore,
Problem 3 :
In the expression given below, find the complex number z in the rectangular form.
Solution :
2(z + 3) = (z - 5i)(1 + 4i)
2z + 6 = z + 4zi - 5i - 20i^{2}
2z + 6 = z + 4zi - 5i - 20(-1)
2z + 6 = z + 4zi - 5i + 20
2z - z - 4zi = -5i + 20 - 6
z - 4zi = -5i + 14
z(1 - 4i) = 14 - 5i
Problem 4 :
Solution :
Problem 5 :
Find z^{-1}, if z = (2 + 3i)(1 - i).
Solution :
z = (2 + 3i)(1 - i)
= 2 - 2i + 3i - 3i^{2}
= 2 + i - 3(-1)
= 2 + i + 3
= 5 + i
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