Question

what algorithm (rule) will produce the correct result for the sum "a+b" where a and b each represent any rational number.

Answer 1

To find the sum of a + b where a and b are rational number.

1. when a and b are natural numbers

just add them . for example a =3, b=8

then ,a + b = 11

2. When a and b are whole numbers,

simply add them . for example a= 0, b=8

a+ b = 0 + 8= 8

3. When a and b are integers

for example, a =-1 b=8,

a+ b= -1+ 8 =7,

a=-2, b= -8

a+ b= -2-8=-10

a= -6 , b=2

a+ b= -6 + 2= -4

a= 8, b= -2

a+ b= 8 +(-2) =6

I have written this because Rational number = [Integers{Whole number(Natural number)}]

now when a= Any fraction=
`(p)/(q)` and b = Any fraction=
`(u)/(v)`

now ,

`(p)/(q) + (u)/(v)`

Find L.C.M of q and v

= if q and v are Co-prime , just multiply them to find their L.C.M.

For example 14,9. LCM=14×9=126

Otherwise, Find factors of q and v . Then take out common factors first and then multiply the remaining with with common factors.For example

q=12 and v=18

12 =2×2×3

18=2×3×3

common factor =2,3

non common=2,3

L.C.M= 2×2×3×3=36

Suppose LCM of q and v = r

then ,
`(p)/(q) + (u)/(v)`

=
`\frac{{(r)/(q)}* p+{(r)/(v)* u}}{r}\\`

=
`{{(r)/(q)}* p+{(r)/(v)* u}} = k`

then ,

a + b=
`(k)/(r)`