How do we know the formula for fraction multiplication is right ?
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We all learnt in college this formula:
Where and are integers and and are different from zero, or in other words: and
When we apply it, it just seems to work. But how are we sure the formula is always true ?
We will take for granted the properties of the operations on integers, especially for multiplication:
- The associativity of multiplication:
- The commutativity of multiplication
- The equal priorities of multiplication and division, which means the order in which we do multiplication and division does not change the result:
- The neutrality of for multiplication:
- A number divided by itself equals
We can try to use simple (?) algebra to transform into and thus be sure of our formula.
As in many algebra transformations, we will introduce a neutral expression, which means an expression that evaluates to the neutral element (here ) for the operator (here ) and thus will let the value of the formula unchanged. We use property (5) from the prerequisites to introduce a handy neutral expression:
Now we use properties (1), (2) and (3) from the prerequisites to rewrite the equation above:
Dividing by a number then multiplying by that same number does not change the initial number, so and . So we now have:
Using associativity, we write:
Which is the same as:
Yay ! Here we have it ! Q.E.D - Quod Erat Demonstrandum !
Notice that is not exactly the same mathematical object as . The first is a binary operator applied to 2 integers and the second is a number (a rational number). They are equal, meaning they can be interchanged in an expression, because they eventually represent the same value.
See you soon !
Keep learning !
Written on Sun Oct 21st 2018, 10:31 GMT+02:00.
Last updated on Sun Oct 21st 2018, 10:31 GMT+02:00.