COMPLEX NUMBER

Complex no has both real and imaginary part.

It is represent by A+Bi

where A is real part and B is imaginary part.

i is the iota and its value i=√-1

Complex no. also represent by e^iθ or cosθ+isinθ

1. Addition of complex no.

 z1=x+yi and z2=x1+y1i

then

z1+z2=x+yi+x1+y1i

z1+z2=x+x1+yi+y1i

z1+z2=x+x1+yi+y1i

z1+z2=x+x1+(yi+y1)i

2. Subtraction of two complex no.

 z1=x+yi and z2=x1+y1i

then

z1-z2=x+yi-(x1+y1i)

z1-z2=x-x1+yi-y1i

z1-z2=x-x1+yi-y1i

z1-z2=x-x1+(yi-y1)i

3. Product of two complex no.

 z1=x+yi and z2=x1+y1i

then

Z1.Z2=(x+yi).(x1+y1i)

Z1.Z2=(xx1+x1yi+xy1i+yy1i)

Z1.Z2=(xx1+(x1y+xy1)i+yy1i²)

Z1.Z2=(xx1+(x1y+xy1)i-yy1)

Z1.Z2=(xx1-yy1+(x1y+xy1)i

4. Product of two complex no.

 z1=x+yi and z2=x1+y1i

then

z1/z2=(x+yi)/(x1+y1i)

RATIO AND PROPORTION

Ratio and ProportionA ratio is a comparison of two or more quantities of similar kind.
It is represent by A:B or A/B
Properties:

i) If both the terms of a ratio are multiplied or divided by same non zero quantity then the value of ratio remains same.
If A/B=p/q then A/B=kp/kq

ii) If the ratio is square or multiply by a constant then ratio will be remains same.

iii)