Proof of Multiplying by Fraction = Dividing by Inverse

In summary, the conversation discusses a proof involving division and the use of the definition of division to explain how the proof answers the initial question. It also mentions the concept of a multiplicative inverse and its relationship to division.
  • #1
jimgavagan
24
0
Supposedly this proof answers my question.

8 / 16 = .5
8 / 8 = 1
8 / 4 = 2
8 / 2 = 4
8 / 1 = 8
8 / .5 = 16
8 * 2/1 = 8 / .5

I'm just wondering how this proof answers my question?
 
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  • #2


It doesn't ... and that's not a proof o__O

It follows kind of from "definition" of division: [itex]x \over y[/itex] actually means [itex]x \cdot y^{-1}[/itex] where [itex]y^{-1}[/itex] is the number such that [itex]y \cdot y^{-1} = 1[/itex], called the "multiplicative inverse" or "reciprocal" of y. This number is unique. Obviously [itex]({a \over b})^{-1} = {b \over a}[/itex] (since [itex]{a \over b} \cdot {b \over a} = 1[/itex]) as long as we have [itex]a, b \ne 0[/itex]. (Also of note: [itex]0^{-1}[/itex] does not exist!

So it turns out that since [itex]{x \over y} = x \cdot y^{-1}[/itex], setting [itex]y = {a \over b}[/itex] we get

"[itex]{{x} \over {a \over b}}[/itex]" [itex]= {x} \cdot ({a \over b})^{-1} = {x} \cdot {b \over a}[/itex] (again provided [itex]a, b \ne 0[/itex]). I hope this explanation helps.
 
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  • #3


ooooooooooooooooooooooooooh awesome! :D

Very much appreciated!
 

What is "Proof of Multiplying by Fraction = Dividing by Inverse"?

"Proof of Multiplying by Fraction = Dividing by Inverse" is a mathematical concept that states that multiplying a number by a fraction is equivalent to dividing the same number by the inverse of the fraction.

How does this concept work?

This concept works by understanding the relationship between multiplication and division. When we multiply a number by a fraction, we are essentially dividing the number into smaller parts. In other words, we are finding the fraction of the number. Similarly, when we divide a number by the inverse of a fraction, we are also finding the fraction of the number. Therefore, the two operations are equivalent.

Why is this concept important?

This concept is important because it helps us simplify complex mathematical expressions and solve equations more easily. It also allows us to understand the relationship between multiplication and division better, which is crucial in higher-level mathematics.

Can you provide an example of this concept in action?

Sure! Let's take the expression 2 x 1/2. We can rewrite this as 2 x (1/2)^-1, which is equivalent to 2 ÷ (1/2). We know that dividing by a fraction is the same as multiplying by its inverse, so this expression simplifies to 2 x 2 = 4.

Is there a specific rule or formula for this concept?

Yes, the rule for "Proof of Multiplying by Fraction = Dividing by Inverse" is: a x (b/c) = (a x b) / c. In other words, when multiplying a number by a fraction, we can also divide the number by the fraction's denominator to get the same result.

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