This problem should be completely unimportant to me, but it’s been in the back of my mind for quite some time.
8/(x-3)=(x+5)/(x-3)
I naturally said the answer was three, but my math teacher said since (8/0) is undefined, the problem has no solution.
(8/0)=(8/0) So it shouldn’t even matter that it’s undefined, right?
It does matter, and there’s no solution.
I suppose I should expand more.
Because 8/0 is undefined, and therefore meaningless, 8/0 = 8/0 is not a true sentence. It’s meaningless, and should more or less be ignored.
From the latest issue of Discover :
Assume a = b
Then ab = a2 = b2
So ab + (a2 - 2ab) = a2 + (a2 - 2ab)
Simplifying, a2 - ab = 2a2 - 2ab
Factoring, 1(a2 - ab) = 2(a2 - ab)
Therefore 1 = 2
I agree with you all that the answer is no solution. However, just to make sure, I got all the stuff on one side and zero on the other side and entered it into my TI-83 like so:
(8/(x-3))-((x+5)/(x-3))
And it graphed it! No error message (like it normally gives when there’s no solution) or anything! So now I’m confused.
~Monica
Note that (8/(x-3))-((x+5)/(x-3)) is equal to -1 when x != 3. At every point other than x = 3, there is a solution, so that’s what your caculator graphed. If you try to move the cursor to x = 3, you’ll find that it won’t give you that point.
You’re being taught the dogmas of mathematics. In that world 8/0 is called “undefined”. All they are saying is that there is no NUMERICAL answer.
But it is intuitively obvious that
8/0 = 8/0
and
undefined = undefined
So you are right. Or, I would say that you are logically right, and semantically right. But these are not answers that mathematics has defined numerically. So, math says the answer is “undefined”. Human understanding is bigger than mathematics sometimes (I know this viewpoint will outrage some SDMB members).
In my mind, 8/0 = infinity (if you think about it, as the denominator gets closer to zero, the result gets bigger).
Infinity is not a number: I can only conceive of it as a process – whatever I number I can imagine, I can always add one. Infinity is the process of being bigger than any number you can name or write or think of, and therefore there is no number than can represent it. It is not a number. Hence 8/0 is not a number. But still, 8/0 = 8/0, OBVIOUSLY; it’s just not within the discipline of mathematics.
But then again, I’m not mathematician.
I’m glad you didn’t dismiss the problem. It’s not such a “Simple Algebra Problem,” by the way. Nothing in math is simple.
You’re thinking that when you take the difference between equal quantities, you always get zero. That’s true. But N/0 is not equal to N/0 (unless maybe N is zero; someone has to check me on that). Once you divide by zero, all bets are off.
Why? Well, basically to preserve the rules of algrebra. The rules we work when solving equations (with addition, subtraction, multiplication and all that) are defined by an ‘algebra’ (in high school, you take algebra class; you’re actually just learning about one specific algebra). One rule of algebra is that certain operations are associative (+ and *). For example, if you have 4+9+2 it doesn’t matter if you add the 4 and 9 first or the 9 and 2 first. Same answer: 15.
But if you let N/0 - N/0 = 0, you’ll quickly have to start answering a lot of other question or you’ll be breaking this rule of associativity (and others). Consider 1 + N/0 - N/0. If you subtract the N/0 from N/0 first, you’ll get 1-0 which is 1. But if you do the 1 + N/0 first? What does that equal?
Zero is often hard to understand. It has a pretty long history as a placeholder when writing numbers (to tell the difference between 30 and 3000 for instance) but zero as a number itself is a much more recent invention. To get a better understanding of it, the place to go is Abstract Algebra (and perhaps specifically Group Theory), which are generally upper level college or lower level graduate classes.
That’s exactly what I was thinking, Frantic Mad. I guess for mathematical purposes, though, I’ll stick with no solution. Thanks to everyone for your replies.
Not at all.
Mathematics is simply a method of human expression, and it’s being invented as we go along and get better at it. Calculus, which every college and university student even brushing against science has to study, didn’t even exist 500 years ago. It had to be invented.
This is an example of something mathematics cannot presently express.
Have I missed something here?
if a = b, then a2 - ab = 0
It appears to me as though all you have proved is that one group of zero is equal to two groups of zero.
-Oli
No, we can express it exactly. It’s undefined.
If 8/0 were anything, it would have to have the property that 8/0 * 0 = 8. But zero times anything is zero. Ergo, there’s no such beast.
Intuition doesn’t count for anything in higher mathematics. Otherwise, there’d be a lot fewer headaches. Since 8/0 doesn’t mean anything, the sentence “8/0 = 8/0” is meaningless.
starman: That proof shows that if you can divide by zero, then you can prove that 1 = 2. That’s why division by zero is such a problem.
Another issue with trying to extend the reals by [symbol]a[/symbol] = 1/0 is that [symbol]a[/symbol]/[symbol]a[/symbol] can’t be defined in a way consistent with all the properties of reals that we’d like to keep.
FranticMad, if 8/0 is so clearly = infinity, what happens to
8/(x-3) if x is just a teensy bit less than 3?
Then you’d get a negative number.
My post is related to the OP, because it also involves dividing by zero.
I don’t think so. In the second line you claim that ab = 2a = 2b. Plug in 1 for a and b. You get:
1 = 2 = 2 ?
For a = b = 3
9 = 6 = 6 ?
The only values that ab = 2a = 2b where a = b are 2 and zero. But both of those force you do divide by zero in your next to last step.
No, it says ab=a^2=b^2.
NB: To make a[sup]2[/sup] and b[sup]2[/sup], use the superscript vB. To do this, surround the “2” with “sup” and “/sup” (without the quotation marks, of course, and both “sup” and “/sup” enclosed in [brackets] ).
In other words, use the vB tag “sup” for superscripts (and “sub” for subscripts) in exactly the same way that you use the “i” tag for italics, the “b” tag for bold, etc.
You were, after all, in a math class.
FranticMad, if you think undefined should equal undefined, then do you also think that the squareroot of an apple is equal to 8/0? Just wondering.