Lance wrote:Who ever said it is a number? I think infinity exists outside the realm of numbers. It is something you can go toward but never reach. Kind of like that one 18 year old blonde girl.
It's a number if we decide it is a number. But more critically than whether we decide to call infinity a number or not, what are the properties of this infinity? Does it behave like a number, or is it somehow quite different than the other numbers? Therein lies the rub.
Lance wrote:Dragon Star wrote:I can use an equation with infinity to prove that 1=0 which is false, so I'm skeptical that it's possible.
Can you show us that equation?
I think it is very important in such discussions to follow one rule, which is, do not believe anything, no matter how obvious it seems, unless you can prove it rigorously, step-by-step. In the world of internet mathematical proofs, if you want to know where the mistake is, look for the statement, "it is obvious that . . ." or "every child knows that . . ." - that's usually where you'll find it.
Dragon has presented an argument, but Llance questions the very first step, which does make an (unproven) assumption about how infinity behaves. Let's see whether we can take a different approach, but end up at the same destination.
Many of the familiar number systems are what are known as fields. A field is set of objects (often called "numbers"), together with two operations called "addition" (denoted by ) and "multiplication" (denoted by ), which follow eleven rules. These rules are not all that complicated, and should be mostly familiar from primary school arithmetic.
A1: If and are numbers, then is a number. (Additive completeness - you can add any two numbers, and the result is a number.)
A2: If and are numbers, then . (Commutativity of addition - it doesn't matter what order you add numbers in, you get the same answer either way.)
A3: If , , and are numbers, then . (Associativity of addition - it doesn't matter which order you do the addition operations in, you get the same answer either way.)
A4: There exists a number called such that, for any number , . (Existence of an additive identity - zero plus any number just gives you the same number right back.)
A5: For every number , there exists a number called , such that . (Existence of an additive inverse - every number, including zero, has a negative.)
M1: If and are numbers, then is a number. (Multiplicative completeness - you can multiply any two numbers, and the result is a number.)
M2: If and are numbers, then . (Commutativity of multiplication - it doesn't matter what order you multiply numbers in, you get the same answer either way.)
M3: If , , and are numbers, then . (Associativity of multiplication - it doesn't matter which order you do the multiplication operations in, you get the same answer either way.)
M4: There exists a number called such that, for any number , . (Existence of an multiplicative identity - one times any number just gives you the same number right back.)
M5: For every number , there exists a number called , such that . (Existence of a multiplicative inverse - every number, except zero, has a reciprocal.)
D1: For all numbers , , and , . (Distributivity of multiplication over addition.)
Note that the equality relation is reflexive (, every number is equal to itself), symmetric (if , then ), and transitive (if and , then ).
The rational numbers, the real numbers, and the complex numbers are all fields (with addition and multiplication defined in the standard way). The integers are not - the only integers with multiplicative inverses are and . So M5 does not hold for the integers.
Do these seem like good rules for numbers to follow? If so, we can have a look at what happens if we relax the annoying exception in M5 - every number except zero has a reciprocal. We can introduce a new number called "infinity" which is the reciprocal of zero, and see what happens if we insist that the new "infinity" number must satisfy the eleven rules above, along with all the other numbers. The conclusion that Monsieur Dragon reached is not far away.