Date: 08/20/98 at 20:02:40 From: Axel Ishmael Subject: Half empty or half full? I have been studying inverses. It is my understanding that the same situation can be described using an inverse statement. For example, say that there are 3 + 4 times as many girls as boys. Is this the same as saying there are -3 + 1/4 as many boys as girls? But I still don't quite understand why the inverse of (3 + 4X) must equal -(3 + 4X) which is -3 - 4X. Do you see that the definition of inverse seems to be in conflict here? Would it be correct to define say that there are -3 - 4times as many boys? Does -(3 + 4X) also equal (1/3 + 1/4X) because the negative sign means inverse? I am trying to describe the same situations using inverses. Thank you for all your help.
Date: 08/21/98 at 16:13:00 From: Doctor Peterson Subject: Re: Half empty or half full? Hi, Axel. I think it might help if we take out some of the words and write a complete equation to clarify what you are saying. Talking about "3 + 4times as many girls as boys" is confusing, because it's not really proper grammar in either English or math. Let's say that there are X boys and Y girls. Now I think you are saying that Y = 3 + 4X Now you want to invert this statement, to say how many boys there are compared to girls, rather than girls compared to boys. That means we want to solve for X in terms of Y. You're right that we want to use inverses, and that -3 is the additive inverse of +3 and 1/4 is the multiplicative inverse of 4. Notice that there are two kinds of inverses, so the potential for confusion is obvious if we don't discuss them in the right context. A better way to describe this is that subtracting 3 is the inverse of adding 3, and dividing by 4 is the inverse of multiplying by 4. The other trouble you run into is that if you want to invert a whole expression, you have to invert its parts in the right order. It's as if I wanted to take off my clothes, and I tried to remove my shirt first, then my sweater, then my coat. It wouldn't work. I'd get all tangled up, just the way you did. An algebraic expression comes in layers - we call this the order of operations. In your expression, we first multiply 4 times X, then add 3 to the result. Now we want to peel off those layers one by one and wrap them (inside out - inverting them!) around the other side of the equation. So since the "outside layer" is adding 3, we invert it by subtracting three from both sides: Y - 3 = 4X Now to unwrap the next layer, multiplication by 4, we divide each side by 4: Y - 3 ----- = X 4 So now we have inverted your expression, and the result is not -3 + 1/4Y, or -3 - 4Y, or 1/3 + 1/4Y, but (Y - 3)/4. Order makes a big difference. Now, I don't know whether you are learning about inverse functions, but if I want to say this in terms of functions, I can define a function: f(x) = 3 + 4x and what we have found out is that the inverse of this function is: f^-1(y) = (y - 3)/4 This is the function that undoes what f did. Function f dressed x, and f^-1 undresses it. Function f changed the number of boys into the number of girls, and f^-1 changes the number of girls back into the number of boys. I hope this helps. You're obviously thinking hard and enjoying it, and keeping things in order should make your thinking more satisfying. - Doctor Peterson, The Math Forum Check out our web site! http://mathforum.org/dr.math/
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