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If you accept that 1/∞ = 0 Then you accept that ∞ * 0 = 1 But the definition of 0 is exactly that anything multiplied by it must be 0. So this cannot be tru
by brabel 7y ago
If you accept that
1/∞ = 0
Then you accept that
∞ * 0 = 1
But the definition of 0 is exactly that anything multiplied by it must be 0. So this cannot be true.
To take a more verbal route: you cannot take nothingness and repeat it. Repeating (or multiplying) nothingness (or 0) is fundamentally nonsense.
Programmer explanation: one cannot loop through `null` even once, let alone a large number.
- aliceryhl 7y agoNo, you don't have to accept that. In my Analysis 2 course we worked a whole bunch with [0, ∞], i.e. the positive real numbers together with infinity, and we defined 1/∞ = 0 and ∞ * 0 = 0. You lose some of your usual rules of arithmetic, but it gets a lot easier to talk about integrals.
- brutt 7y agohttps://en.wikipedia.org/wiki/Surreal_number https://en.wikipedia.org/wiki/Surreal_number Quote: There are also representations like { 0, 1, 2, 3, … | } = ω { 0 | 1, 1/2, 1/4, 1/8, … } = ε where ω is a transfinite number greater than all integers and ε is an infinitesimal greater than 0 but less than any positive real number. Moreover, the standard arithmetic operations (addition, subtraction, multiplication, and division) can be extended to these non-real numbers in a manner that turns the collection of surreal numbers into an ordered field, so that one can talk about 2ω or ω − 1 and so forth.
- aliceryhl 7y agoI'm not quite sure I understand what you're trying to tell me? The example I mentioned is not about surreal numbers.
- drran 7y agoI recommend watching this lecture about surreal numbers by John Conway: https://www.youtube.com/watch?v=1eAmxgINXrE https://www.youtube.com/watch?v=1eAmxgINXrE .
- ramshorns 7y agoThere are some weird rules like that for floating point numbers too.