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What is infinity?

Infinity is a term with which most people are familiar, but few truly understand. Infinity is not an actual value, like the number 3 -- it is an abstract concept. In math terms, it is used as a "limit", where a value can approach infinity by getting continuously larger, but it will never actually get there. Consider the act of cutting a pizza into slices. You can cut it in half, then cut those halves in half, then cut those halves in half, etc. As the slices get smaller, the number of slices gets larger; therefore, as the size of each slice approaches zero, the number of slices approaches infinity. Again, in math terms, this means that as x approaches zero, the value of 1/x approaches infinity. Some go so far as to say that 1/0 equals infinity, but this would not be entirely correct; nothing can actually "equal" infinity, since it isn't a value, but an abstract limit that can only be "approached".

Here's another example. You are standing a certain distance from a wall. You step forward half that distance. Then half the remaining distance. Then half again, ad infinitum (if you'll pardon the phrase). Theoretically, it would take you an infinite number of steps to reach the wall if you only traversed half the distance with each step. Practically, though, the distance between you and the wall would eventually become immeasurably small, and moving half of that distance would not be feasible. Again, as the steps become smaller, the number of steps can "approach" infinity, but never actually get there.
What about different kinds of infinity? There are twice as many integers as there are odd numbers; therefore, one would think that the number of integers divided by the number of odd numbers would be 2. But these sets are both "infinite", so does that mean that in this case that infinity divided by infinity is 2? What about the number of rational numbers? There are an infinite number of rational numbers greater than 1 and less than 2, and also between any other pair of consecutive integers. Therefore, since there is an infinite number of integers, does this mean that the number of rational numbers is infinity * infinity? And that the number of rational numbers (infinity) divided by the number of integers (infinity) is also infinity? This demonstrates the overall fallacy of trying to use infinity as a distinct value, and the reason why so many such calculations are "undefined" or "indeterminate" (as are things like zero divided by zero, or zero raised to the zero power).
In closing, here is an interesting math trick that "proves" that 2 equals 1. See if you can spot where the logic falls down, and how it is related to what I've talked about above...
Given: A=B
Multiply both sides by A: A*A = A*B
Subtract B*B from both sides: A*A - B*B = A*B - B*B
Factor both sides: (A + B)(A - B) = B(A - B)
Divide both sides by (A - B), which will cancel out that term: A + B = B
Since A=B (as given), substitute B for A: B + B = B
Add the left side: 2B = B
Divide by B: 2 = 1

Comments

A nice explanation...and it has been a year since this was posted and the fallacious proof has yet to be answered; so I shall answer (hopefully not ruining this):
 
Step 4 is incorrect; we are dividing by A-B=0, which leads to an undefined state of being.
 
Also, I would say 1/0= + or - infinity if something must be symbolically assigned to it.
Very good, Suneil!  Indeed, the step where both sides are divided by A-B constitutes a division by zero, causing the "undefined" values in the equality to yield invalid results.
 
As a reward, here's another challenge for you to try to spot the error...
 
Choose two arbitrary and different numbers x and y, where z is the average of x and y, such that x+y=2z.
Given: x+y = 2z
Multiply both sides by (x-y): (x+y)(x-y) = 2z(x-y)
Distribute: x^2 - y^2 = 2xz - 2yz
Add y^2 to both sides: x^2 = y^2 + 2xz - 2yz
Subtract 2xz from both sides: x^2 - 2xz = y^2 - 2yz
Add z^2 to both sides: x^2 - 2xz + z^2 = y^2 - 2yz + z^2
Factor: (x-z)^2 = (y-z)^2
Take the square root of both sides: x-z = y-z
Add z to both sides: x=y

... which "proves" that the two arbitrary numbers chosen are equal!  Or does it?
 

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