What does zero multiplied by infinity equal?

Question

Is zero not infinitly small and infinity infinitly big. Maybe the answer is 1?

Proof that 0 X Infinity = 1

All you people who thought 0 was the answer you are wrong.

Imagine you are walking in a straight line to a destination. You will eventually reach a half way point. This will have taken half the total time of the walk to reach. You can now assume this to be your new starting point and again the above will apply. Every time you will reach a half way point. So how come you reach your destination and you don’t just keep reaching half way points for ever and ever.

Easy it is because zero multiplied by infinity is one.

1= the distance to your destination.

0 = the final distance of the last half

And Infinity = the number of times a half way point is met.


If it the was zero then you would never get there.

Answer 1

You cannot multiply something by inifinity because infinity is not a number, it's a concept, so technically it is undefined or "indeterminate," however, anything multiplied by zero is zero.

Answer 2

For many years the number system didn't include the number zero. It was introduced as a place holder ie to indicate the values of numbers placed before it. As a number it is used to indicate nothing or the absence of other values. Therefore 0 multiplied by infinity means what is the value of no infinity. The answer is 0 by definition.

The problem you give in your example is called Zeno's First Paradox.

In your example the distances get closer and closer to zero but the amount of time you need to complete that distance also tends to zero. I'm afraid in your additional information you have fallen into the trap of thinking that the sum of an infinite series must be infinite.

The total distance which can be seen as the sum of the infinite series 1/2 + 1/4 + 1/8 +1/16 etc converges to 1. You might then say ahh, it only converges to 1 but never reaches 1. But the time taken to do that additional distance becomes infinitely small and doesn't become a physical barrier to completing the total distance.

Answer 3

John,

PLEASE be careful about some of the answers you have received. Most of them are, quite simply, totally wrong.

You have asked a good question, but one that is simply NOT as straightforward as you, and many others, seem to think. You can't just say zero times anything is zero! Zero times any number is zero but infinity is not a number. So, in one sense, the answer to your question is that the question does not make sense!

However, there is a sort of way in which it does make sense but you need to understand the concept of the limit, which is quite a subtle concept. Imagine 2 numbers, a and b. a is getting closer and closer to 0 while b is getting larger without bound. What happens to the product ab? Well, surely, that depends on how quickly a is approaching 0 compared to how quickly b is getting larger without bound. For example, suppose a = sinx and b = 1/x and we allow x to get closer to 0 (so a gets closer to 0 while b get large without bound). You can show that the limit, as x gets closer to 0, of (sinx)/x is one. So, in a sense, that 0 times that infinity = 1. However, if you let a = (sinx)^2 while b = 1/x then the limit will be zero so that 0 times that infinity = 0. You could let a = sinx and b = 2/x so that this time the limit will be 2. You could let a = (sinx) and b = 1/x^2. This time, ab will not have a limit - b just gets too big too quickly (or you could say that ab tends to infinity as x tends to 0). In other words, depending on the a and the b, ab could have ANY limit.

I hope this goes some way to explaining things!

Answer 4

The simple PROOF.

“God I see in you no end and no beginning.” This is not a problem, for we do not need to see, but only to manipulate it.

It is not practical to think of infinity as a (really, really big) number. It is a concept indeed. Let us not try to understand Infinity, or unboundedness, for the human mind is insuited for this task.

However, mathematics has come a long way, and has developed tools to deal with concepts such as infinity. In fact, the whole calculus is based on the concept of limits and infinity.

The PROOF:

Using limits:
OK
Lets define zero and infinity in terms of limits.

0 = Lim x-->inf. (k/x), where k is a constant and k is not 0.
inf. = Lim x-->inf. (x), pretty straight forward.

Now:
0 * infinity = [ Lim x-->inf. (k/x) ] * [Lim x-->inf. (x) ];

Using the multiplicative property of limits: Lim x-->a (x) * Lim x-->a (y) = Lim x-->a (xy)
[ Lim x-->inf. (k/x) ] * [Lim x-->inf. (x) ] = Lim x-->inf. (k/x * x);
k/x * x = k, or a constant;
Lim x-->inf. (k/x * x) = Lim x-->inf. (k) = k;
~~~~~~~~~~~~~ 0 * infinity = k, a constant ~~~~~~~~~~~~~~~

Misconceptions:

Infinity is not a number, and therefore cannot be multiplied by zero directly. So the answer can not be zero.

Infinity is not a/0, rather it is a/infinitesimal value (that is no 0).

Samagreen stated that:
“Therefore, if we already understand that One(1) over Infinity is Zero(0), then we come to the answer that:-
Zero multiplied by Infinity is equal to ONE !!”
This is not true because, even thought One(1) over Infinity = 0, any other real number over Infinity = 0, so you see that the answer must be a constant.

The answer is not necessarily 1, because “the distance to your destination” is not necessarily 1. The proportion of space covered is 1, but the total distance can be any number. Also, the distance could be negative due to backward movement.

“God I see in you no end and no beginning,” but this I do. It is k.

Answer 5

Zero

Answer 6

I disagree.
0 x 1 = 0
0 x 1000 = 0
0 x 1000000000 = 0
0 x 1000000000000000000 = 0
So as n tends to infinity, the product 0 x n does not converge to 1.
0 x infinity cannot be calculated as infinity is not a number, but the answer is not 1.

Answer 7

Zero

Answer 8

Zero

Answer 9

Is it not zero as infinity is a concept to label something that is not quantifiable in real terms so you cannot put it in an equation - it's like saying what is zero times a finger?

Answer 10

It's true that infinity isn't a number and is only a concept. And "multiplied by" is the same thing as the word "of," as we all had beaten into our noggins when we were learning story problems back in elementary school. So take any other intangible thing... say, the scent of freshly baked bread, and have "zero of" that thing. You still have nothing.

My hero zero! Powerful.

Answer 11

Zero. Even though infinity is a concept, it doesn't matter what you multiply by zero, it could be 0*x or 0*n any other variable, zero times anything is zero.