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# Complex Number Formula

A complex number is the sum of a real number and an imaginary number. So, a complex number is of the form x + iy. A complex number is nothing but a combination of a real number and an imaginary number. In this section, we will be discussing the complex number formulas to do addition, subtraction, multiplication, and division on the complex numbers. Let us learn the complex number formula with a few solved examples.

## What Is Complex Number Formula?

Operations on complex numbers are similar to polynomials. To add or subtract two complex numbers, use the following complex number formula.

**(a + ib) + (c + id) = (a + c) + i(b + d)**

**(a + ib) - (c + id) = (a - c) + i(b - d)**

To multiply two complex numbers, use the following complex number formula.

**(a + ib) × (c + id) = (ac - bd) + i(bc + ad)**

To divide two complex numbers, use the following complex number formula.

**(a + ib) ****÷ ****(c + id) = [(ac + bd)/(c ^{2} + d^{2})] + i[(bc - ad)/(c^{2} + d^{2})**

To multiply complex conjugates, use the following complex number formula.

**(a + ib) × (a - ib) = a ^{2} + b^{2}**

### Complex Conjugate Formula

The complex conjugate of a complex number, 'z', is its mirror image with respect to the horizontal axis (or x-axis). The complex conjugate of 'z' is denoted by \(\bar{z}\). Here 'z' and \(\bar{z}\) are the complex conjugates of each other.

## How to Find the Conjugate of a Complex Number?

From the above figure, we can notice that** **the complex conjugate of a complex number is obtained by just changing the sign of the imaginary part.

- The complex conjugate of \(x+iy\) is \(x-iy\).
- The complex conjugate of \(x-iy\) is \(x+iy\).

Let's take a quick look at a few solved examples to understand the complex number formulas better.

## Examples Using Complex Number Formula

**Example 1:** **Find the** sum **of 4 - 6i and -2 + 4i using the complex number formula.**

**Solution:**

(4 - 6i) + (-2 + 4i) = (4 - 2) + i(-6 + 4) = 2 - 2i

**Answer: The required sum is 2 - 2i.**

**Example 2: Find the** product **of 1+2i and its conjugate using the complex number formula.**

**Solution:**

(1+2i) × (1 - 2i) = 1^{2} + 2^{2} = 1 + 4 = 5

**Answer: The required product is 5.**

**Example 3: Find the complex conjugate of \(4 z_{1}-2 i z_{2}\) given that \(z_{1}=2-3 i\) and \(z_{2}=-4-7 i\).**

**Solution: **

We will first find \(4 z_{1}-2 i z_{2}\).

\( 4 z_{1}-2 i z_{2} = 4(2-3i) -2i (-4-7i)\)

\(= 8-12i+8i+14i^2\)

\(= 8-12i+8i-14 \,\,\,[ \because i^2=-1]\)

\(= -6 -4i \)

The complex conjugate of \(4 z_{1}-2 i z_{2\)} is obtained just by changing the sign of its imaginary part.

**Answer:** **That is, \(\overline{4 z_{1}-2 i z_{2}}\) is \(-6 + 4i\)**.

## FAQs on Complex Number Formula

### What Is a Complex Number Formula?

The complex number formula gives the various operations that can be performed on a complex number. A complex number is the sum of a real number and an imaginary number. So, a complex number is of the form x + iy. A complex number is nothing but a combination of a real number and an imaginary number.

### What Is the Multiplication of Complex Numbers' Formula?

To multiply two complex numbers, use the following complex number formula, (a + ib) × (c + id) = (ac - bd) + i(bc + ad).

### Is 0 a Complex Number Using Complex Number Formula?

From the definition of complex number, we can conclude that 0 is a complex as well as a real number.

### How To Multiply Complex Conjugates Using Complex Number Formula?

To multiply complex conjugates, use the following complex number formula, (a + ib) × (a - ib) = a^{2} + b^{2}.

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