# Identities of Complex Numbers

A complex number is a number that is made up of both real and imaginary numbers. Here we talk about some properties of complex numbers. To familiarize more with them read this post. A complex number is written as $$a+ib$$ and usually represented by $$z$$. Where $$a$$ signifies a real number and $$ib$$ represents an imaginary number. In addition, $$a,b$$ are real values, and $$i^2 = -1$$.

As a result, a complex number is a straightforward representation of the addition of two integers, namely a real and an imaginary number. One side is entirely genuine, while the other is entirely imagined.

## A step-by-step guide to identities of complex numbers

The following are some of the properties of complex numbers:

• The sum of two conjugate complex numbers will result in a real number.
• The multiplying of two conjugate complex numbers will produce a real number as well as a complex number.
• If $$x$$ and $$y$$ are real numbers and $$x+yi =0$$, then $$x =0$$ and $$y =0$$ are the same value.
• When two conjugate complex numbers are added together, the result is a real number.
• A real number can be obtained by multiplying two conjugate complex numbers.
• If $$p$$, $$q$$, $$r$$, and $$s$$ are real numbers, then $$p+qi = r+si$$, $$p = r$$, and $$q=s$$.
• The “basic law” of addition and multiplication applies to complex numbers: $$\color{blue}{z_{1 }+ z_{2 }= z_{2 }+ z_{1 } }$$ , $$\color{blue}{z_{1 }. z_{2 }= z_{2 }. z_{1 } }$$
• The complex number follows the “associative law” of addition and multiplication, which is a mathematical rule: $$\color{blue}{(z_{1 }+ z_{2 })+ z_{3 }= z_{1 }+ (z_{2 }+z_{3 })}$$ ,$$\color{blue}{(z_{1 }. z_{2 }).z_{3 }=z_{1 } (z_{2 }. z_{3 } )}$$
• The “distributive law” applies to complex numbers: $$\color{blue}{z_{1 }( z_{2 }+z_{3 })= z_{1 }. z_{2 }+z_{1 }z_{3 } }$$
• In other words, if the sum of two complex numbers is real, and the product of two complex numbers is also genuine, then these complex numbers are conjugated with one another.
• For any two complex numbers, say $$z_{1 }$$ and $$z_{2 }$$, then $$|z_{1 }+z_{2 }|≤|z_{1 }|+|z_{2 }|$$
• When two complex numbers are multiplied by their conjugate value, the output should be a complex number with a positive value.

### Algebraic identities of complex numbers

All algebraic identities apply equally to complex numbers. The addition and subtraction of complex numbers with the exponents of $$2$$ or $$3$$ can be easily solved using algebraic identities of complex numbers.

• $$\color{blue}{(z_{1 }+ z_{2 })^2= (z_{1 })^2+ (z_{2 })^2+2 z_{1 }× z_{2 } }$$
• $$\color{blue}{(z_{1 }- z_{2 })^2= (z_{1 })^2+ (z_{2 })^2-2 z_{1 }× z_{2 } }$$
• $$\color{blue}{(z_{1 })^2- (z_{2 })^2= (z_{1 }+ z_{2 })( z_{1 }- z_{2 }) }$$
• $$\color{blue}{(z_{1 }+ z_{2 })^3= (z_{1 })^3+ 3(z_{1 })^2 z_{2 } +3 (z_{2 })^2 z_{1 }+ (z_{2 })^3 }$$
• $$\color{blue}{(z_{1 }- z_{2 })^3= (z_{1 })^3- 3(z_{1 })^2 z_{2 } +3 (z_{2 })^2 z_{1 }- (z_{2 })^3 }$$

### Identities of Complex Numbers – Example 1:

Find the sum of the complex numbers. $$z_{1 }=-3+i$$ and $$z_{2 }=4-3i$$

$$z_{1 }$$ $$+$$ $$z_{2 }$$ $$=(-3+i)+(4-3i)=(-3+4)+(i-3i)=1-2i$$

### Identities of Complex Numbers – Example 2:

Solve the complex numbers $$(2+i)^2$$.

To solve complex numbers use this formula: $$\color{blue}{(z_{1 }+ z_{2 })^2= (z_{1 })^2+ (z_{2 })^2+2 z_{1 }× z_{2 } }$$

$$(2+i)^2$$ $$=$$ $$(2)^2$$$$+$$$$(i)^2$$$$+(2×2×i)$$$$=4+i^2+4i$$

Then: $$i^2=-1$$ → $$4+i^2=4-1=3$$

Now: $$4+i^2+4i =3+4i$$

### Identities of Complex Numbers – Example 3:

Solve the complex numbers $$(3-i)^3$$.

First, use this formula: $$\color{blue}{(z_{1 }- z_{2 })^3= (z_{1 })^3- 3(z_{1 })^2 z_{2 } +3 (z_{2 })^2 z_{1 }- (z_{2 })^3 }$$

$$(3-i)^3$$ $$=(3)^3-3(3)^2(i)+3(i)^2(3)-(i)^3$$ $$=27-27i+9i^2-i^3$$

Then: $$i^2=-1$$ → $$9i^2=-9$$

$$=27-27i-9-i^3$$ $$=18-27i-i^3$$

$$i^3=-i$$ → $$18-27i-i^3= 18-27i-(-i)=$$ $$18-27i+i$$

Now: $$18-27i+i =18-26i$$

## Exercises for Identities of Complex Numbers

### Simplify.

1. $$\color{blue}{(4+5i)^2}$$
2. $$\color{blue}{(12+5i)+(3+i^2+6i)}$$
3. $$\color{blue}{(20+7i)-(45i+12)}$$
4. $$\color{blue}{(5-4i)^2(3+3i)}$$
5. $$\color{blue}{(i^2-5i)^3}$$
6. $$\color{blue}{(6-i)^2-(10+i)^2}$$
1. $$\color{blue}{-9+40i}$$
2. $$\color{blue}{14+11i}$$
3. $$\color{blue}{8-38i}$$
4. $$\color{blue}{147-93i}$$
5. $$\color{blue}{74+110i}$$
6. $$\color{blue}{-64-32i}$$

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