Infinities

I sometimes see or hear conversations where people are speculating upon the effects of an infinity on a system. I find many such conversations misunderstand the concept of infinity.

Definition of Infinity

I would define infinity as something like this:

Infinity is not a very big number; it is the complete absence of number.

Infinity is used when something cannot take a value; it is used to describe something which has no finite value. As such, it is something to which no process can be applied to retrieve an answer since there is no value to be processed.

Possible Misuse

A conversation I had in an A-level Physics class at college was that 0.9 (recurring) is equal to 1.0 due to a simple sum. In words, the sum was that 0.3 (recurring) multiplied three times is equal to 1.0 as this is equivalent to three thirds. However, 0.3 (recurring) multiplied three must also equal 0.9 (recurring) if the number is multiplied digit by digit. Mathematically, the argument could be shown look like this: 

1 ÷ 3 = 0.333...3
    1 = 0.333...3 × 3
    1 = 0.999...9

Alternatively, it can be written as below: 

1/3 = 0.333333...
2/3 = .666666...

1/3 + 2/3 = 0.999999... = 1

The problem with this is that it requires an endless series of operations to be performed in order to multiply the decimals digit by digit. (A recurring decimal has an endless string of digits in its decimal portion.) This process could never be completed, so no value could be returned.

To try and apply any system involving infinity is impossible and will aways return erroneous results as infinity describes any value. One cannot multiply infinities or apply any mathematical process onto them since their quantity is undefined.

Controversial Solutions

There are some proofs of 0.9 (recurring) equalling 1 but they are somewhat disputed. The proofs either employ digit by digit multiplication (which would never be completed) or use a concept of "limits" to artificially terminate the process.

I don't find these solutions convincing but I'm hardly an authoritative resource on the subject!