what is zero times infinity?

---

Clarification of Question by arisandino-ga on 19 Oct 2005 16:13 PDT

(This question is from my 6 year old son and I cannot answer it for him:)

---

Request for Question Clarification by leapinglizard-ga on 19 Oct 2005 17:56 PDT

This is like asking, "What is zero times blue?" There is no answer
because the question doesn't make sense. Although mathematics does
have the notion of infinity -- indeed, of infinities, for there is the
countable infinity and then there are uncountable infinities -- there
is no infinity in arithmetic. We cannot use infinity in the arithmetic
operations of addition, subtraction, multiplication, and division, nor
is infinity the result of any arithmetic operation.

Anyone who tells you that infinity is the result of one divided by
zero, or that zero times infinity is zero, is not speaking the
language of mathematics. There is simply no such arithmetic operation
as "zero times infinity", just as there is no "two minus fish" or
"three kick two".

Does this answer satisfy your son?

leapinglizard

"Zero Times Infinity" is the name of a group of experimental sound artists:

http://www.zerotimesinfinity.net/

I'll go way out on a limb and say that zero times anything is zero. 
Just tested it with my calculator; it doesn't have infinity, but it
gave zero as a result for the number that I used.
But maybe Pinkfreud's answer is what your son wants.  She usually has
the right answer.  :)

Remembering that multiplication comes from addition. ie 6x5= 6+6+6+6+6
       (5 times) so multiplying 0 by infinity is the same as
0+0+.....+0 (infinity times) which is zero.

I am no mathematician, but I don't think "zero times infinity" has an
answer, since infinity isn't really a number in the usual sense. It's
more of a concept.

I think your question is related to this one:
http://answers.google.com/answers/threadview?id=581852
by this logic, 4/2 = 2 because 2 * 2 = 4.
And 1/0 = undefined, so 0 * infinity = undefined.

Ancient hindu upanishats indicate the same as Infinity. 

Om, purnamidam, purnamadam,
purnatpurna mudachyate,
Purnasya purnamadaya,
Purnameva vasishyatey (Isavsyopanishat)

This is Infinity, That is Infinity, Infinity is born out of Infinity,
When Infinity is subtracted from Infinity, Infinity remains.

I disagree with our friendly Squamata... For mathamatical proof, you
can build a definition off of an assumption:
infinity is a real number (new assumption)
0 * n = 0, if n = a real number, therefore 0 * infinity = 0

The probablem with 0 * infinity is not whether or not it is a real
number. It is how you define infinity when you put it into a real
number system. Some functions will always work, regardless of the
value you give to infinity.
n / infinity = 0 in the case of all real numbers and all possibly
methods of finding infinity (limits). Infinity + infinity = infinity;
infinity + n = infinity; etc all make sense.

The problem is, that there are a number of functions that will change
depending on how you find infinity. These equations are indeterminate.
I'll show you what I mean --

n / infinity = 0 This is our premis based on the fact that anything
divided an infinite number of times will eventually reach 0

n / infinity = 0   PREM
(a/b = c) = (a = bc)  Based on the definition of multiplication/division
n = infinity * 0 (Prem inserted into line 2)

So, based on one definition of infinity, and the rules of
multiplication/division we determine that 0 * infinity is equal to any
real number. That is what is meant by indeterminate.

If you want other definitions of infinity that will give you different
answers you will have to learn calc. and some discreet math. Start
with this one:

lim_(m --> oo) sum_(n = 1)^m (9)/(10^n) = 1

This infinity limit can be used to show that .99999... = 1

Infinity can be used in a lot of functions and limits, and will show
all kinds of things depending on which function it comes out of when
you try and use the traditional rules of algebra.

You can tell your kid one of two things:
1. Any number times zero is equal to zero, therefore infinity times zero is zero.
2. There are an infinte number of answers, therefore infinity times
zero is infinity.

Either one you could back up with some basic math, because your kid is
unlikely to accept the fact that it is an indeterminate, non real
number.

You seem to be complicating things needlessly. Here are the laws.

Multiplication by Zero Law--> (a) X (0) = 0 

Zero Factor Law--> If (a)(b) = 0, then a = 0 or b = 0

Nowhere is there an exception for infinity. 

So (a)(b) = 0 where a = 0 and b = infinity, the law works (as well as
where a = infinity and b = 0).

Son: "Papa, what is zero times infinity?"
Father: "What?"

Son: "Zero times infinity. I read from Math/Google Answers that one
divided by zero is cannot be or is one or is zero or is infinity.
Those guys there baffle me. Then I wondered what is zero times
infinity. Cannot trust those guys so I am asking you."
Father: "what answers Google? Nobody answers Google."

"Papa, it is Google Answers. It is a website where Math is discussed."
"(Heck.) You found that website and now you are bugging me? What do I
know about Math?"

"Come on, Papa."
"Okay, okay, zero times infinity is zero. Anything multiplied by zero equals zero."

"But, Papa, guys from Google Answers say infinity is not a number."
"Did I say infinity is a number? I said "anything" and infinity is "anything"."

"Wow, I see. Thanks, Papa.  ....anything multiplied by zero equals zero ,,,"

"Wait, Papa, you still sure there is a Santa Claus? Will he give me a
video iPod this Christmas if I wish for that?"
"(Heck.) Of course, son, there is a Santa Claus, and if you behave, He
will buy, er, give you that thing."

" 'Night, Papa."
" 'Night, son. "

Papa: "Who cares what is zero times infinity. If it ain't zero, it
ain't zero. It could be blue, or fish. Who cares. Sonny Boy is still
too young for that. He will investigate more when he grows old enough.
Let him enjoy his youthful innocence for now. Now for this Answers
Google website, lemme see...."

Very good, Ticbol!  These precocious kids are unnerving.

I think maybe the best answer is that it depends how the particular
infinity and zero you are dealing with came to be.  Without any
context, it's meaningless, like asking for the answer to fish*blue, as
others have already pointed out.  Where it arises in some computation,
0*infinity can be 0, or it can be +/- infinity, or it can be any
number.    Here's an example:

What is sin(x) * (1/x) if x is zero?  sin(0) = 0, and as x gets closer
to zero, (1/x) gets bigger and bigger (i.e. gets closer to infinity),
so it looks like the answer is 0*infinity.  If you plug in small
numbers for x, like .1, .01, .0000000001, you find that sin(x) * (1/x)
gets closer and closer to 1.  So in this case you could say that
0*infinity = 1.

(Really, in order to not make mathematicians mad, you have to talk
about "limits".  So instead of asking 'what does sin(x) * (1/x) EQUAL
when x=0?', you have to ask, 'what is the LIMIT of sin(x) * (1/x) as x
APPROACHES 0?'.)

sin(x) * (1/x^2) also looks like 0*infinity, but as x gets closer to
zero, that function gets bigger and bigger.  So in this case, it looks
like 0*infinity = infinity.

sin(x) * (1/sqrt(x)) also looks like 0*infinity, but this one gets
closer and closer to 0 as x approaches 0.  So now, it appears that
0*infinity = 0.

So it all depends on where your numbers came from.  The same is true
for infinity/infinity.  You might think that should be 1, and
sometimes it is, but it can also be zero, or any other number, or
infinity.  Some infinities are bigger than others.

Mark, you are wrong on that, there are exceptions for infinity. It
doesn't operate like a real number because by definition it is the sum
of all numbers.

Thus Infinity * Infinity oo*oo does not equal 2oo.

oo * oo = oo
Just as oo / oo = oo
oo + 1 = oo
oo - 1 = oo

That is why our reptilian friend said you can't use it in standard
operations. It doesn't follow normal laws. You can have zero groups of
5, but you can't have zero groups of infinity, or two groups of
infinity -- because by definition infinity is infinity and does not
change.

While I listed zero as a possible answer to tell the kid, it isn't
accurate. It could be zero, just as much as it could be any other
number from -oo to oo

Yes, the correct answer depends on the context, but the most natural
answer, in terms a 6 year old can appreciate, is zero.  Multiplication
can be considered counting the number of pairs that can be created
from two sets, each with some definite count of elements.  [Kids are
often taught the concept of multiplication by making an array of items
with some number of rows and some number of columns. You have one item
for each pairing of a row with a column.]

If the first set has zero elements, and the second set has an infinite
number of elements, then the number of pairs is still zero.  (More
technically we'd say this is the cardinality of the Cartesian
product.)

Most other contexts that I can think of are more difficult to share
with a 6 year old and also less definite about the result.

regards, mathtalk-ga

Answer: You'll figure it out when you're older.  And don't forget to
explain it to me when you do.

No solution, The only considerable soltuion is 0. Prove Incorrect:


1/Infinity=0   (THis is True)
Infinifty*1/Infinity=0*infinity(True as well)

1=0?

This a contradiction.