'What is 6 divided by 2?' You can rephrase this question as: 'what number, when multiplied by 2, will give me 6?' Here, the answer is obviously 3. There's only one answer, it couldn't be anything else- can we say the same for 6/0 ?
Lets do the same sort of thing: 'what is 6 divided by 0?' Well, we can rephrase this as: 'what number, when multiplied by 0, will give me 6?' That's.......... hmm........ D:
Clearly a problem here. '0 x ? = 6'. Apparently, there are no solutions for '?', so you can't divide a number by 0. How about infinity? If we say '?' = infinity, does that solve the problem? If we look at the hyperbolic graph of 'y=1/x', we find that, as we get to x=0, one side of the graph sort of 'shoots up' towards positive infinity, whilst the other side 'shoots down' towards negative infinity.
However, even if we say it's positive or negative infinity. If 0 x infinity = 6, we could say the same for ANY number divided by 0, e.g. 4/0 = infinity, so 0 x infinity = 4. In fact, using this idea, you can say that 0 x infinity = any number, which doesn't appear to be particularly logical. Or does it? Infinity is a very abstract concept, far beyond the scope of human comprehension; infinity doesn't behave like a number, and it's properties are mostly unknown, so we can't say whether this is a logical idea or not. Basically, we're not sure, so we say it's undefined. Anyway, if you look at this graph, you never actually get to x=0, so how can we define what it is we get when x=0, when this value doesn't even exist on the graph? Or is that what infinity is? Is +infinity the sum of all the positive numbers, and is -infinity the sum of all the negative numbers? Even then, there are complex versions of infinity to deal with (to do with complex numbers, involving the square root of -1). Confusing indeed.
So, how about 0/0? This causes even more headaches, because there are so many more things to consider here. You might say that any number divided by itself equals 1, so 0/0=1. However, there's a VERY good reason why mathematicians don't agree with this. Let's assume that 0/0=1. Then:
2 = 2x1 = 2x (0/0) = (2x0) /0 = 0/0 = 1 Therefore, 2=1
Ah.
As you can see, '0/0=1' clearly doesn't work, as we could then make any number equal to any number. So, why don't we say 0/0=0? Surely, 0 divided by any number is 0? Well, lets use same idea we used for 6/0:
'what is 0 divided by 0?', can be rephrased as: 'what number, when multiplied by 0, will give me 0?' Hmmm..... that's........ EVERY NUMBER, not just 0. Therefore, we can say, for example, that 0/0=3, and 0/0=5. Hence, 3=5. In fact, every number=every number.
Doh.
Another problem. Many mathematicians like to say 0/0= infinity, but, even then, they have to be careful, as infinity is definitely not a number, and must be treated with caution. Indeed, you can't even define infinity as a certain entity, as there are different kinds of infinity, as explained above.
So, basically, dividing by 0 makes no sense at all and it would be illogical to give it any sort of value. We just have to live with the fact that it's 'Undefined' and always will be, or else the entire mathematical system would fall to pieces.
If you want to read about dividing by 0 in a bit more detail, you can visit the Wikipedia page: http://en.wikipedia.org/wiki/Division_by_zero
see ya! :)
Admin. H
sometimes i poop on my dads chest
ReplyDelete