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For example, dividing by 2, means we cut something in two.

But dividing by 0.5, can only be explained with multiplying something by 2.

So, is there a "physical" explanation of dividing by 0.5? Is it "I divide by an entity that internally multiplies' or something as so bizarre?

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I once heard it explained as asking the question how many times does m/n fit into 1. For example, how many times does 1/2 fit into 1, that is 1/(1/2) = 2, or 2 times. –  Amzoti Dec 1 '12 at 1:41
    
That's a very good answer, similar to what Peter Tamaroff gave below. –  Lela Dax Dec 1 '12 at 1:43
    
Also, read this, maybe as good an answer as you'll get (has some nice things): mathforum.org/library/drmath/view/58058.html –  Amzoti Dec 1 '12 at 1:43
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4 Answers

up vote 5 down vote accepted

I once heard: "If you feed a kid $Y$ grams of chocolate he would have a density of $X$ $\rm gr/cm^3$ of chocolate in his blood. But if you feed half a boy chocolate he will have $2X$ $\rm gr/cm^3$ of chocolate in his blood".

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It sounds good and simple enough. –  Lela Dax Dec 1 '12 at 1:33
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Chocolate gets into the stomach first, and the concentration of it will vary with time. I guess dissolving the same amount of table salt in water of different volumes is a better example. –  FrenzY DT. Dec 1 '12 at 2:37
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@FrenzYDT. It is odd you took my example seriously. –  Pedro Tamaroff Dec 1 '12 at 2:39
    
Well, as folks say $1+1=2$ doesn't always necessarily hold. AAMOF, I like your explanation. –  FrenzY DT. Dec 1 '12 at 2:41
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$$\frac{1}{0.5} = \frac{2(\not{.5})}{\not{.5}} = 2.$$

$$\frac{1}{\frac12} = \frac{2\cdot\not{\frac{1}{2}}}{\not{\frac12}} = 2.$$


In words, "how many times does one-half fit into the whole?"

And more generally, "how many times does $\;\dfrac1n\;$ fit into $\;1\;$?"

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It sounds good mathematically but I don't know if it's complete in a physical sense since it might have to explain the meaning of 0,5/0,5 = 1. –  Lela Dax Dec 1 '12 at 1:39
    
I guess it can be "I multiply 2 by a number that was divided by that same number hence I multiply 2 by 1". –  Lela Dax Dec 1 '12 at 1:49
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Along the same lines as Pedro Tamaroff's answer:

If you have ten ounces of vodka and each cocktail requires one ounce, then you can make ten cocktails.

If you have ten ounces of vodka and each cocktail requires only half an ounce, then you can make twenty cocktails. In other words, halving the vodka doubles the number of drinks. (Unfortunately, nobody will be happy with these drinks.)

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Modeling 12 / 4,

If 12 marbles fit perfectly into 4 boxes, how many marbles fit into 1 box?

versus modeling 12 / (1/2).

If 12 marbles fit perfectly into half a box, how many marbles fit into 1 box?
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