maths > complex-number

Understanding Complex Arithmetic and equivalence

what you'll learn...

Overview

Complex Arithmetic as Numerical Expression

»  Real Number arithmetic

→  Addition with closure, commutative, associative, identity, inverse properties

→  Multiplication with closure, commutative, associative, distributive, identity, inverse properties

→  subtraction and division as inverse of addition and multiplication respectively

→  PEMDAS / BODMAS Precedence rule

»  Complex number arithmetic
→  extension of real arithmetic with additional number $i$$i$
→  $i$$i$ is handled like a variable
→  ${i}^{2}=-1$${i}^{2} = - 1$ maps $i$$i$ to a real number

When Two Complex numbers are Equal?

»  if two complex numbers $a+ib$$a + i b$ and $c+id$$c + i d$ are equal then

→  $a=c$$a = c$ and

→  $b=d$$b = d$

understanding complex arithmetics

The complex number system can be understood as :

•  Real number system extended to include solutions to polynomials

•  Real number system added with an additional number $i=\sqrt{-1}$$i = \sqrt{- 1}$

•  Mathematical Model of amplitude and phase of sine waves and elements interacting with sine waves

For the complex number the following definitions and properties
•  multiplication and division
•  exponents and roots
•  properties like commutative, distributive, associative, etc.

One need not learn each of these afresh. Just extend the definitions and properties of real numbers. Let us see what this means.

analogy in real numbers

One person decides to write all real numbers in the form $3.1+7×2.4$$3.1 + 7 \times 2.4$ - that is, two real numbers with $7$$7$ appearing as in the expression. How would the representation affect the arithmetic operations, like addition, subtraction, associative, etc?

All the definitions and properties will hold true for the numerical expressions too, as long as the expression is kept in tact. (This is one of the properties -- closure. The expression itself is a realnumber.)

Consider the example of writing real numbers in the form $a+7b$$a + 7 b$:
•  Two identical numerical expressions may not be evident from the numbers in the expressions. eg: $3.1+7×2.4=4.5+7×2.2$$3.1 + 7 \times 2.4 = 4.5 + 7 \times 2.2$, though the expression looks different, they evaluate to the same.
•  There can be additional properties that is a combination of already existing properties. eg: To add $3.1+7×2.4+1.1+7×0.2$$3.1 + 7 \times 2.4 + 1.1 + 7 \times 0.2$, add the coefficients $3.1+1.1$$3.1 + 1.1$ and $2.4+0.2$$2.4 + 0.2$. This looks to be a new definitions for addition. But it is derived from the distributive and associative properties of addition and subtraction.

Complex number is a numerical expression with $\sqrt{-1}$$\sqrt{- 1}$ as an element. When learning rules and properties of complex numbers, visualize the complex number as a numerical expression of real numbers with a new element $i$$i$.

summary

Complex Arithmetic Fundamentals: The definitions and properties of real number arithmetic is extended to include $\sqrt{-1}$$\sqrt{- 1}$ as a number that cannot be added or multiplied to other numbers.

Complex arithmetic is the extension of Real numbers arithmetic.

equivalence

The two complex numbers ${z}_{1}=a+ib$${z}_{1} = a + i b$ and ${z}_{2}=p+iq$${z}_{2} = p + i q$ are equal only if the real and imaginary parts are equal. That is, if ${z}_{1}={z}_{2}$${z}_{1} = {z}_{2}$ then $a=p$$a = p$ and $b=q$$b = q$

If two complex numbers $3-i4$$3 - i 4$ and $1+x$$1 + x$ are equal, What is x?

The answer is '$2-i4$$2 - i 4$'

summary

Complex numbers Equivalence: Two complex numbers ${z}_{1}={a}_{1}+i{b}_{1}$${z}_{1} = {a}_{1} + i {b}_{1}$ and ${z}_{2}={a}_{2}+i{b}_{2}$${z}_{2} = {a}_{2} + i {b}_{2}$ are equal only if ${a}_{1}={a}_{2}$${a}_{1} = {a}_{2}$ and ${b}_{1}={b}_{2}$${b}_{1} = {b}_{2}$

Two complex numbers are equal if real and imaginary parts are equal.

Outline