Fractions For Finickity Folk.
["preamble"] ...

Fractions are extremely useful and can be found within many important formulas throughout mathematics.

Learning how to add, subtract, multiply and divide fractions will enable you to better understand and combine formulas as you advance in your study of Maths.

Special thanks to : Misha, Taye , Ani, for your support and inspiration for this project.

["~>d@♥£<~" , "London (UK)"]

Fractions For Finickity Folk.

1 : Anatomy of a Fraction.

A fraction is a composite number that consists of up to five parts :
+12
3
 

... • a plus or a minus sign ( which if omitted implies a plus-sign ) , • a whole-number ( called the "integer-part" ) , ( Please note : zero is usually omitted from the integer-part of a fraction if the total value represented by the fraction amounts to less-than one ) ... • a whole-number stacked above a horizontal-line ( called the "numerator" ) , • a horizontal-line , • a whole-number placed beneath a horizontal line ( called the "denominator" ) ; ...

2 : First Principles.


If we replace all whole-numbers ( 1 , 2 , 3 , above ), with labels : a , b , c , then, in general : ["2.1" , "expansion"]
ab
c
=
+ a + b
c
 
and ...
-ab
c
=
- a - b
c
 
...and please note : ["2.2" , "conversion"]
a
=
+ a
1
 
and ...
-a
=
- a
1
 
...Also ... ["2.3" , "multiplication & inversion"]

b
c
 
×
 

b
c
=
 b × b
c × c
and ...

b
c
÷
b
c
=
b
c
×
c
b
=
b × c
c × b
=
c × b
c × b
=
1
 
...and ... ["2.4" , "construction"]
a
a
=
b
b
=
c
c
= +1
  and ...
-a
a
=
-b
b
=
-c
c
= -1
 
...Also, please note : ["2.5" , "signs"]

+1
×
+1
=
+1
+1
×
-1
=
-1
-1
×
+1
=
-1
-1
×
-1
=
+1

and...

+1
+1
=
+1
÷
+1
=
+1
+1
-1
=
+1
÷
-1
=
-1
-1
+1
=
-1
÷
+1
=
-1
-1
-1
=
-1
÷
-1
=
+1
 
.........

3 : Addition and Subtraction.

If parts labelled : a , b , c and d , e , f , are all positive numbers (or zero) then, in general :
["3.1" , "addition of fractions"]
ab
c
+
de
f
=
ab
c
× f
f
+
de
f
× c
c
...
=
ab × f
c × f
+
de × c
f × c
...
=
a + b × f
c × f
+ d + e × c
f × c
...
=
(
a + d
)
+
(
b × f
c × f
+ e × c
f × c
)
...
=
( a + d )
+ (
b × f + e × c
f × c
)

... and further more ...

...
=
(
a
1
+
d
1
)
+
(
b × f + e × c
f × c
)
...
=
(
a
1
×
f × c
f × c
+
d
1
×
f × c
f × c
)
+
(
b × f + e × c
f × c
)
...
=
(
a × f × c
f × c
+
d × f × c
f × c
)
+
(
b × f + e × c
f × c
)

Note : ( ( 1 × f × c ) = ( f × c ) ) ...

...
=
a × f × c
f × c
+
d × f × c
f × c
+
b × f + e × c
f × c

Therefore :

ab
c
+
de
f
=
a × f × c + d × f × c + b × f + e × c
f × c
 
...and ... ["3.2" , "subtraction of fractions"]
ab
c
-
de
f
=
ab
c
×
f
f
-
de
f
×

 c
c
...
=
ab × f
c × f
-
de × c
f × c
...
=
a + b × f
c × f
- d - e × c
f × c
...
=
(
a - d
)
+
(
b × f
c × f
- e × c
f × c
)
...
=
( a - d )
+ (
b × f - e × c
f × c
)

... OK, deep breath ...

...
=
(
a
1
-
d
1
)
+
(
b × f - e × c
f × c
)
...
=
(
a
1
×
f × c
f × c
-
d
1
×
f × c
f × c
)
+
(
b × f - e × c
f × c
)
...
=
(
a × f × c
f × c
-
d × f × c
f × c
)
+
(
b × f - e × c
f × c
)

Note : ( ( 1 × f × c ) = ( f × c ) ) ...

...
=
a × f × c
f × c
-
d × f × c
f × c
+
b × f - e × c
f × c

Therefore :

ab
c
-
de
f
=
a × f × c - d × f × c + b × f - e × c
f × c
 
......

4 : Multiplication and Division.

Now, if parts labelled : a , b , c and d , e , f are all positive numbers (or zero) then, in general : ["4.1" , "multiplication of fractions"]

ab
c
×
de
f
=
(
a
1
+
b
c
)
×
(
d
1
+
e
f
)
...
=
(
a
1
×
c
c
+
b
c
)
×
(
d
1
×
f
f
+
e
f
)
...
=
(
a × c
1 × c
+
b
c
)
×
(
d × f
1 × f
+
e
f
)
...
=
(
a × c
c
+
b
c
)
×
(
d × f
f
+
e
f
)
...
=
(
a × c + b
c
)
×
(
d × f + e
f
)
...
=
(a × c + b) × (d × f + e)
c × f

... using grid multiplication to expand to top-part...

× (a × c)(b)
(d × f)(a × c) × (d × f)(b) × (d × f)
(e)(a × c) × (e)(b) × (e)


ab
c
×
de
f
=
(a × c × d × f)+(b × d × f)+(a × c × e)+(b × e)
c × f
 
...and ... ["4.2" , "division of fractions"]
ab
c
÷
de
f
=
(
a
1
+
b
c
)
÷
(
d
1
+
e
f
)
...
=
(
a
1
×
c
c
+
b
c
)
÷
(
d
1
×
f
f
+
e
f
)
...
=
(
a × c
1 × c
+
b
c
)
÷
(
d × f
1 × f
+
e
f
)
...
=
(
a × c
c
+
b
c
)
÷
(
d × f
f
+
e
f
)
...
=
(
a × c + b
c
)
÷
(
d × f + e
f
)

... flip the fraction on the right ...

...
=
(
a × c + b
c
)
×
(
f
d × f + e
)
...
=
(
(a × c + b) × f
c × (d × f + e)
)
...
=
(
a × c × f + b × f
c × d × f + c × e
)

... rearranging the terms ...

ab
c
÷
de
f
=
(
a × c × f + b × f
d × c × f + e × c
)
 
............

"Title" : ( "Fractions For Finickity Folk." ) ,
"Created" : ( "Tue Nov 28 2023 00:40:38 GMT+0000 (Greenwich Mean Time)" ) ,
"Published" : ( "Fri Dec 01 2023 12:55:00 GMT+0000 (Greenwich Mean Time)" ) ,
"Updated" : ( "Fri Dec 01 2023 12:55:00 GMT+0000 (Greenwich Mean Time)" ) ,
"Author" : [ "Dave Auguste" ] ,
"Commissions" : [ "@dave_on_fiverr" ] ,
"Support" : [ "buymeacoffee.com/daveauguste" ] ,