Question:

Find the multiplicative inverse of the complex number 5+3i√5+3i

Updated On: Dec 23, 2024
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Solution and Explanation

Let z=5+3iz=√5+3i

Then,

z¯=53iz¯=√5-3i

z2=(5)2+(3)2=14|z|^{2}=(√5)^2+(3)^2 =14

Therefore the multiplicative inverse of 43i4-3i

z1=z¯z2z^{-1}=\dfrac{z¯}{|z|^{2}}

=53i14=\dfrac{√5-3i}{14}

=514314i=\dfrac{√5}{14}-\dfrac{3}{14}i   (Ans.)

 

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Concepts Used:

Complex Numbers and Quadratic Equations

Complex Number: Any number that is formed as a+ib is called a complex number. For example: 9+3i,7+8i are complex numbers. Here i = -1. With this we can say that i² = 1. So, for every equation which does not have a real solution we can use i = -1.

Quadratic equation: A polynomial that has two roots or is of the degree 2 is called a quadratic equation. The general form of a quadratic equation is y=ax²+bx+c. Here a≠0, b and c are the real numbers.