Complex Numbers : Addition and Subtraction of Complex numbers

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Addition and Subtraction of Complex numbers

what you'll learn...

Overview

Complex Number Addition

» $(a + i b) + (c + i d)$ $(a+ib) + (c+id)$
by associative, commutative, distributive laws of real numbers and by considering $i$ $i$ as a variable
→ $= (a + c) + i (b + d)$ $=(a+c) + i(b+d)$

Complex Number Subtraction

» $(a + i b) - (c + i d)$ $(a+ib) - (c+id)$
by associative, commutative, distributive laws of real numbers and by considering $i$ $i$ as a variable
→ $= (a - c) + i (b - d)$ $=(a-c) + i(b-d)$

addition

Consider two complex numbers $z_{1} = a_{1} + i b_{1}$ $z_1=a_1+ib_1$ and $z_{2} = a_{2} + i b_{2}$ $z_2 = a_2+ib_2$ . Then $z_{1} + z_{2}$ $z_1+z_2$ is ' $(a_{1} + a_{2}) + i (b_{1} + b_{2})$ $(a_1+a_2) + i (b_1+b_2)$ '. This is from the associative and distributive laws of real numbers extended to numbers with $\sqrt{- 1}$ $sqrt(-1)$ .

Complex number Addition:
$z_{1} + z_{2}$ $z_1+z_2$
$= a_{1} + i b_{1} + a_{2} + i b_{2}$ $quad quad =a_1+ib_1 + a_2+ib_2$
$= a_{1} + a_{2} + i b_{1} + i b_{2}$ $quad quad =a_1+a_2+ib_1+ib_2$ (associative law of addition)
$= a_{1} + a_{2} + i (b_{1} + b_{2})$ $quad quad = a_1+a_2+ i(b_1+b_2)$ (distributive law of multiplication over addition)
$= (a_{1} + a_{2}) + i (b_{1} + b_{2})$ $quad quad =(a_1+a_2) + i (b_1+b_2)$ (real and imaginary parts of result)

Given two complex numbers $3 + 2 i$ $3+2i$ and $- 1 + i$ $-1+i$ , what is the sum?

The answer is ' $2 + 3 i$ $2+3i$ '

summary

Addition of Complex numbers : For any complex number $z_{1} = a_{1} + i b_{1} \in ℂ$ $z_1=a_1+ib_1 in CC$ and $z_{2} = a_{2} + i b_{2} \in ℂ$ $z_2 = a_2+ib_2 in CC$
$z_{1} + z_{2} = (a_{1} + a_{2}) + i (b_{1} + b_{2})$ $z_1+z_2 = (a_1+a_2)+i(b_1+b_2)$

Addition of two complex numbers is the addition of real and imaginary parts individually.

Subtraction

Consider two complex numbers $z_{1} = a_{1} + i b_{1}$ $z_1=a_1+ib_1$ and $z_{2} = a_{2} + i b_{2}$ $z_2 = a_2+ib_2$ . Then $z_{1} - z_{2}$ $z_1-z_2$ is ' $(a_{1} - a_{2}) + i (b_{1} - b_{2})$ $(a_1-a_2) + i (b_1-b_2)$ '. This is from the associative and distributive laws of real numbers extended to numbers with $\sqrt{- 1}$ $sqrt(-1)$ .

Complex number Subtraction:
$z_{1} - z_{2}$ $z_1-z_2$
$= a_{1} + i b_{1} - (a_{2} + i b 2)$ $quad quad =a_1+ib_1 - (a_2+ib2)$
$= a_{1} - a_{2} + i b_{1} - i_{b} 2$ $quad quad =a_1-a_2+ib_1-i_b2$ (associative law)
$= a_{1} - a_{2} + i (b_{1} - b 2)$ $quad quad = a_1-a_2+ i(b_1-b2)$ (distributive law)
$=$ $quad quad =$ (a_1-a_2) + i (b_1-b_2) (real and imaginary parts of result)`

Given $z_{1} = 2.1 + i$ $z_1 = 2.1+i$ and $z_{2} = - 2.1 + i$ $z_2 = -2.1+i$ What is $z_{1} - z_{2}$ $z_1-z_2$ ?

The answer is ' $4.2$ $4.2$ '

summary

Subtraction of Complex numbers : For any complex number $z_{1} = a_{1} + i b_{1} \in ℂ$ $z_1=a_1+ib_1 in CC$ and $z_{2} = a_{2} + i b_{2} \in ℂ$ $z_2 = a_2+ib_2 in CC$
$z_{1} - z_{2} = (a_{1} - a_{2}) + i (b_{1} - b_{2})$ $z_1-z_2 = (a_1-a_2)+i(b_1-b_2)$

Subtraction of two complex numbers is the subtraction of real and imaginary parts individually.

Outline

The outline of material to learn "complex numbers" is as follows.

Note : Click here for detailed overview of Complex-Numbers

→ Complex Numbers in Number System

→ Representation of Complex Number (incomplete)

→ Euler's Formula

→ Generic Form of Complex Numbers

→ Argand Plane & Polar form

→ Complex Number Arithmetic Applications

→ Understanding Complex Artithmetics

→ Addition & Subtraction

→ Multiplication, Conjugate, & Division

→ Exponents & Roots

→ Properties of Addition

→ Properties of Multiplication

→ Properties of Conjugate

→ Algebraic Identities