 maths > complex-number

Algebraic Identities for Complex numbers

what you'll learn...

Overview

Complex Algebraic Identities

»  All identities of real-numbers holds

→  ${\left(a+b\right)}^{2}={a}^{2}+2ab+{b}^{2}$${\left(a + b\right)}^{2} = {a}^{2} + 2 a b + {b}^{2}$

→  ${\left(a-b\right)}^{2}$${\left(a - b\right)}^{2}$, etc.

algebraic identities

The following are some examples of algebraic identities

${\left(a+b\right)}^{2}={a}^{2}+2ab+{b}^{2}$${\left(a + b\right)}^{2} = {a}^{2} + 2 a b + {b}^{2}$

${\left(a-b\right)}^{2}={a}^{2}-2ab+{b}^{2}$${\left(a - b\right)}^{2} = {a}^{2} - 2 a b + {b}^{2}$

${a}^{2}-{b}^{2}=\left(a+b\right)\left(a-b\right)$${a}^{2} - {b}^{2} = \left(a + b\right) \left(a - b\right)$

If an algebraic identity is true for real numbers, will that be true for complex numbers? The answer is 'Yes, for all identities with four fundamental operations'. Because, the algebraic identities of real numbers are true for numerical expressions.

And complex number can be equivalently considered as a special numerical expression of real numbers.

The four fundamental operations like
•  subtraction,
•  multiplication, and
•  division

have the properties
•  associative,
•  commutative
•  distributive.

The algebraic identities employ the properties of the operations given above. Between real numbers and complex numbers the properties are the same. So the algebraic identities will hold true.

summary

Algebraic Identities: ${\left({z}_{1}+{z}_{2}\right)}^{2}={z}_{1}^{2}+2{z}_{1}{z}_{2}+{z}_{2}^{2}$${\left({z}_{1} + {z}_{2}\right)}^{2} = {z}_{1}^{2} + 2 {z}_{1} {z}_{2} + {z}_{2}^{2}$
${\left({z}_{1}-{z}_{2}\right)}^{2}={z}_{1}^{2}-2{z}_{1}{z}_{2}+{z}_{2}^{2}$${\left({z}_{1} - {z}_{2}\right)}^{2} = {z}_{1}^{2} - 2 {z}_{1} {z}_{2} + {z}_{2}^{2}$
$\left({z}_{1}+{z}_{2}\right)\left({z}_{1}-{z}_{2}\right)={z}_{1}^{2}-{z}_{2}^{2}$$\left({z}_{1} + {z}_{2}\right) \left({z}_{1} - {z}_{2}\right) = {z}_{1}^{2} - {z}_{2}^{2}$
${\left({z}_{1}+{z}_{2}\right)}^{3}={z}_{1}^{3}+{z}_{2}^{3}+3{z}_{1}^{2}{z}_{2}+3{z}_{1}{z}_{2}^{2}$${\left({z}_{1} + {z}_{2}\right)}^{3} = {z}_{1}^{3} + {z}_{2}^{3} + 3 {z}_{1}^{2} {z}_{2} + 3 {z}_{1} {z}_{2}^{2}$
and so on for any complex number.

Outline