How infinity?

Question

See if infinity is in the form a/0 then a = infinity*0 isnt it but dats not possible any answers??

Answer 1

The explanation is in the meaning of the expression "a/0".

A small comment: the great french matematician Pascal from XVIIth century (famous for various inventions, as for example the first calculator that ever existed, capable to add, substract, multiply and divide any pair of numbers, when he was a child) used to say that to work in mathematics you had to follow this path: "go to definition and then come back from the definition".

And your question is a perfect example related to it. The central point is the meaning of the expression "a/0". Briefly, this is the situation: when you write "a/b", you want usually say "a divided by b". That's easy. However, when you write "a/0", the meaning is NOT "a divided by zero", because the division by zero is impossible in mathematics.

The expresion "a/0" is an abbreviation to say "a divided by a number extremely close to 0"; or, with even more precision, "a/0" is an abbreviation for: "the limit of a/b, when b --> 0". You must have this meaning in mind to reason about "a/0".

Now, a second step: the expression "a/b = c" means "a divided by b is equal to c", but the expression "a/0 = infinity" means NOT "a divided by 0 is equal to infinite", because again there's no definition for "divided by 0", and for being a number equal to infinite (only limits, NOT NUMBERS, can be equal to infinite; and the meaning of "equal" is not exactly the same when you talk about numbers or limits, e.g. 2 numbers are "equal" in a different way as 2 limits are "equal", for the same reason two triangles are "equal" in a different way than 2 bottles of coca-cola are "equal").

Now, what is the meaning of "a / 0 = infinity"? It means that: "the limit of a / b, when b --> 0, is infinity". Not more (or less) than this.

Now, third step: it is common in arithmetics to do this: if you have that "a / b = c", then you deduce that "a = b*c". This is valid for the operations "divide" and "equal" for NUMBERS, but it is not applied to other meanings of the signs "/" and "=". In particular, you cannot just say, based on the expression "a / 0 = infinity" (that, as I explained, is an abbreviation of a very complicate statement about limits and infinity - not a simple operation with division between numbers), that "a = 0 * infinity".

And anyway, the expression has a certain sense, as we can see in this fourth step:

If you consider the original meaning of "a / 0 = infinity", you have that it meant: "the limit of a/b, when b--> 0, is infinity". Now, let's consider this question:

what is the limit of b * (a/b), when b --> 0?
Obviously, b * (a/b) = a, and so the limit of b*(a/b) when b --> 0 is the limit of a when b --> 0, and this is a.

So we have: the limit of b * (a/b), when b-->0, is a

now, observe that b --> 0 and (a/b) --> infinity, and then we can say that we found that, in a certain sense "0 * infinity = a". Again, this affirmation only has sense if you think about its meaning on limits, but then yes... "a = infnity * 0" is an affirmation that, at the end, had sense... and WAS TRUE! (if correctly interpreted as limits), but was false if treated as simple arithmetics.

As Pascal said, 350 years ago... the secret is always "go to definitions, and come back from definitions"

Hope it helped!

Answer 2

a = infinity * 0 is perfectly possible because infinity*0 is undefined which means it can be any number.

You could try to understand this by approximating infinity and zero.
Take inf = 10^10
and 0 = 10^-10

then inf * 0 = 1

But if you take inf = 10^11 then inf * 0 = 10