Denominator = 2 or 1/2
Date: 01/16/97 at 13:17:47 From: Ray Subject: Fractions Dear Dr. Math, I think I understand fractions, and how denominators and numerators work, but there is still one problem. My teacher says that when the denominator is 2, that means I cut the pie (or whatever) in 2. Same with 3 or 4, or any number. My teacher also says that the denominator can be another fraction! She says that if the denominator is 1/2, then the pie is multiplied by 2. I think she's from Mars! Can you help me?
Date: 01/20/97 at 11:20:11 From: Doctor Toby Subject: Re: fractions What you have here are two different concepts (very closely related to each other) which are both being called `fractions'. One concept is that of a *rational number*. This is something whose numerator and denominator are both integers (ordinary numbers: positive, negative, or - in the numerator - zero). This is what people usually mean when they say `fraction'. To get 3/5 of a pie, you cut it into 5 pieces and take 3 of the pieces. The other concept is that of *division*. You probably already know about division; it's one of the four basic operations, along with addition, subtraction, and multiplication. People often write division like `25 @ 5'; except instead of `@' they use a symbol like this: . _____ . But that is not the only way to write division! Another way people write 25 divided by 5 is as a fraction, with 25 in the numerator and 5 in the denominator. So when your teacher is talking about 1/2 in the denominator, she's not talking about cutting a pie into pieces any more; she's talking about dividing something by 1/2. Cutting a pie and dividing numbers are different concepts, but they use the same notation. You might wonder why this confusing state of affairs exists. But when you think about it, it actually makes some sense. Suppose you see `3/5' written down somewhere. Does this mean the rational number 3/5 or does it mean 3 divided by 5? You might spend all day trying to decide, but it doesn't really matter; the answer to 3 divided by 5 IS the rational number 3/5! These two kinds of fractions, rational numbers and division, are really the same thing being thought about in different ways. So it makes sense that they use the same notation. Now you know that your teacher really meant dividing by 1/2 when she talked about 1/2 being in the denominator: `If the denominator is 1/2, then the pie is multiplied by 2' just means that dividing by 1/2 is the same as multiplying by 2. If you don't understand this, think about what division means; 37 (for example) divided by 1/2 is whatever number you have to multiply by 1/2 in order to get 37. If you get 37 when you multiply this mystery number by 1/2, the number must have been twice as big as 37 to begin with. That's why dividing by 1/2 is the same as multiplying by 2. -Doctor Toby, The Math Forum Check out our web site! http://mathforum.org/dr.math/
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