what is zero multipled by infinity?

Question

simple arthmetric stipulates that any integer from the set of numbers( including real, unreal, imaginary numbers) multplied by zero results in zero. we all know that likewise for infinity. but i think in this special circumstance it results in something else.

my hypothesis
let ý represent infinity,
hence 0 x ý = -1

proof:
we all recall from geometry the product of the gradient of perpendicular lines intersecting at right angles is -1.
now the gradient of lines is is similar to tanø.
the Y-axis gradient is infinity herein shown as ý. and the X-axis gradient is 0.
the angle of the y axis 90degrees and the angle of x axis is 0degrees. thus
since they are perpindicular the product should be negative.
mathematically as so:
tan(90) x tan(0) = -1
but
tan(90) = ý (infinity) and tan(0) = 0
hence
ý(infinity) x 0 = -1
can anyone proof me wrong mathematically????

Answer 1

Nice work! remember now, that your finding might work by the principles of basic analitic geometry, but not when you start adding the further concepts.

I found on the Calculos by Larson, 5th edition, that:
"Two non-vertical lines are perpendiculars only if their gradient follow this relation: (m1) = 1/(m2)" Now, that's what you meant above, that the product of both slopes is -1.

So why doesn't it work for a vertical function? I dont know if you're familiar with some basic calculus but ill try to go easy on this one. If you don't fear calculus, keep reading :p

The problem with vertical lines is that if m = dy/dx , if x is zero, then you have m = α (this means infinite).

So lets use the tan(Θ) function, try to plot it: i already did it for you and you can watch it here:

http://img303.imageshack.us/img303/3083/imagetandy8.jpg

Now, think about you moving along the x-axis. If you try to check

[ lim tan (Θ)] = α
Θ->90º

And that means, "what value the function will have (y, i mean) if you get closer and closer to 90 degrees following this function?" (using radians, you'll find yourself on x = 1.5707...) You will find that it going as vertical as it can, so it tends to infinite. That's easy to see... but now:

If a function tends to infinite, you cant predict the slope, because it tends to infinite, right? If you cant predict the slope, or the limits, or even the value of a point in the function, you cant prove it's Continuous, if you dont know that, you cant derive it.

What does that mean? It proves that the infinite is NOT a number to work with, but a mathematical concept that says: "you can't mess with me". And you can't work with the y-axis because you cant prove its continuous, hence it has no valid slope (gradient).

And your demonstration has been kicked out. Why? Because:

"The Theorem about perpendicular lines will only work IF THOSE FUNCTIONS ARE CONTINUOUS, PLUS DERIVATIVE, PLUS DIFFERENCITABLE, and we just can't prove that, so you have to prove these tree things before you prove your point. If you do, it'll be correct."

Answer 2

Multiplication is just repetative addition.

eg. 3 * 4 means add 4 three times ( or 3, four times).

0 * x will mean add

0 + 0 +0 + 0 +0 + 0 +0 + 0 + ...... x times

if x is infinity it would be
0 + 0 +0 + 0 +0 + 0 +0 + 0 +0 + 0 +0 + 0 +... infinitely

So the answer is 0 (it doesn't matter what x equals in this case)


Note there are some other problems which are undefined
infinity minus infinty
infinity divided by infinity
etc

Answer 3

to disprove your hypothesis - we do not need to bring more math into it. We need to take some away. While geometry is all well and good it quite simply does not apply.

What we have to look at is 0(none) x infinity. 0 or none makes the math go away - like it did not take place at all it will not, because you are multiplying NO times. Therefore nothing is getting multiplied an infinite number of times. This does not mean you take a zero and multiply it forever it means you start with infinity and do nothing to it. So you are left with the idea of infinity with no numeric value attached so it is just an idea with no measurable quantity. Which of course equals zero.

Answer 4

go through your higher mathematics book somewhere (in very small letters) you will find that when the product of the gradient of two straight lines is -1 then it is sure that the lines are perpendicular
i.e. the angle between them is 90 but the reverse is not always true
axes being the only exception. there is no other possible explanation as per your "mathematically" conditions.. just for records 0/0 is an inderminate form you will study it in differentiation( if u haven't already)

Answer 5

good job working all of this out - I'm impressed. Thing is, using tangents will not give you the right answer because it is a different function. if you looked at tan 180, it will be undefined even though it lies on the x-axis, in essence, similar to 0. You can't beat the underlying principle of the zero-product property.

Simply said: if you give 0 apples each to 10 people (0x10) each person essentially gets 0 apples.

Answer 6

0 x any number = 0

Answer 7

Multiplying zero ( 0 ) by infinity the product is zero (0)

Answer 8

It is 0.

Answer 9

Its 1 Not -1. 1 raised to 0 is 1.

Answer 10

0.

Any number, however large, when multiplied by zero results in a zero.

Trial and error:
0 x 9 = 0
0 x 999,999,999 = 0
0 x 999,999,999,999...,999 = 0

So 0 x infinity = 0