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Subtraction of Complex numbers

A mathematical operation of subtracting a complex number from another complex number is called the subtraction of complex numbers.

Introduction

In some cases, a negative sign appears between two complex numbers. It means that one complex number should be subtracted from another complex number to find their difference mathematically. Hence, it is very important to learn how to subtract a complex number from another complex number mathematically.

Steps

In this fundamental operation, there are three steps involved to find their difference.

  1. Write the subtraction of complex numbers as an expression by displaying a negative sign between them.
  2. Write the real part of all complex numbers closely. Similarly, write the imaginary part of the all complex numbers closely in the same expression. Now, take out the imaginary unit common from imaginary part of the expression.
  3. Finally, find the difference of the real and imaginary parts in the expression.

Algebraic form

$a+ib$ and $c+id$ are the complex numbers in algebraic form. Let us learn how to subtract them mathematically

Step – 1

Assume, the complex number $c+di$ has to subtract from the complex number $a+bi$. So, write the complex number $a+bi$ first and then $c+di$ in a row but display a minus sign between them for expressing the subtraction.

$(a+bi)-(c+di)$

$\implies$ $(a+bi)-(c+di)$ $\,=\,$ $a+bi-c-di$

Step – 2

Now, bring the real part of the expression closer to separate the imaginary part in the expression.

$\implies$ $(a+bi)-(c+di)$ $\,=\,$ $a-c+bi-di$

Step – 3

In the expression, two terms contain an imaginary unit as a factor. So, take out the imaginary unit common from them.

$\implies$ $(a+bi)-(c+di)$ $\,=\,$ $a-c+i(b-d)$

Now, subtract the real numbers in the real and imaginary parts of the expression to obtain the difference of them.

$\implies$ $(a+bi)-(c+di)$ $\,=\,$ $(a-c)+i(b-d)$

$\implies$ $(a+bi)-(c+di)$ $\,=\,$ $(a-c)+(b-d)i$

In this way, we evaluate the difference of any two complex numbers in mathematics.

Let’s consider $a-c = x$ and $b-d = y$, then write the expression in terms of $x$ and $y$.

$\implies$ $(a+bi)-(c+di)$ $\,=\,$ $x-yi$

It proves that the difference of any two complex numbers is also a complex number.

Example

Subtract $7+2i$ from $3+4i$

$(3+4i)-(7+2i)$

$= \,\,\,$ $3+4i-7-2i$

$= \,\,\,$ $3-7+4i-2i$

$= \,\,\,$ $3-7+i(4-2)$

$= \,\,\,$ $-4+i(2)$

$= \,\,\,$ $-4+i2$

$= \,\,\,$ $-4+2i$

Thus, the subtraction of complex numbers is performed in mathematics and it is proved that the difference of them also a complex number $-4+2i$.

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