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Question: Write the additive inverse of the complex number of \(4-3i\)....

Write the additive inverse of the complex number of 4−3i4-3i.

Explanation

Solution

We here have been given a complex number as 4−3i4-3i and we need to find its additive inverse. Since it is a complex number, its additive inverse will also be a complex number so we will assume its additive inverse to be a complex number given as x+iyx+iy and then we will keep the sum of this assumed complex number and the given complex number equal to 0 and then we will solve that equation. Then we will compare the real and imaginary parts on both the sides of the equal sign and hence obtain the values of x and y. Once we have those, we can obtain the value of the required inverse.

Complete step by step solution:
Now, we know that the additive inverse of a number, whether it be complex or real, is the number which when added to the given number gives 0 as the result.
Here, we have been given a complex number as 4−3i4-3i whose additive inverse we need to find. For this, we first need to know that the inverse of a complex number is always a complex number and that of a real number is always a real number.
Here, since we have been given a complex number, we will assume the inverse of the given complex number to be another complex number given as x+iyx+iy. Thus, we can say that:
(4−3i)+(x+iy)=0\left( 4-3i \right)+\left( x+iy \right)=0
Now, solving it we get:
(4−3i)+(x+iy)=0 ⇒x+iy=−4+3i \begin{aligned} & \left( 4-3i \right)+\left( x+iy \right)=0 \\\ & \Rightarrow x+iy=-4+3i \\\ \end{aligned}
Now, comparing the real and the imaginary parts of this equation, we get:
x=−4 y=3 \begin{aligned} & x=-4 \\\ & y=3 \\\ \end{aligned}
Thus, putting the values of x and y we get:
x+iy ∴−4+3i \begin{aligned} & x+iy \\\ & \therefore -4+3i \\\ \end{aligned}
Hence, the additive inverse of 4-3i is -4+3i.

Note: We need not assume the complex number in the form of x+iyx+iy to solve this question. We can also straightaway assume the inverse to be a complex number ‘z’ and form the equation and directly get the value of z. This is shown as follows:
(4−3i)+z=0 ∴z=−4+3i \begin{aligned} & \left( 4-3i \right)+z=0 \\\ & \therefore z=-4+3i \\\ \end{aligned}
Also, just like real numbers, the additive inverse of a complex number is just the negation of the given complex number.