3 ms·

0.999... + 0.000...1 = 1 0.000...1 = 1/∞ 0.999... = 1 - 1/∞ 1/∞ is zero or not?

by brutt 6y ago

brabel 6y 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 6y ago
  No, 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 6y ago
    https://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 6y ago
      I'm not quite sure I understand what you're trying to tell me? The example I mentioned is not about surreal numbers.
    - drran 6y ago
      I 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 6y ago
    There are some weird rules like that for floating point numbers too.
empath75 6y ago
What on earth is .000...1 If it terminates it’s not an infinite series. In any case, if you add any finite number to .99999... it’ll be equal to 1 + that number.

0.999... + 0.000...1 = 1 0.000...1 = 1/∞ 0.999... = 1 - 1/∞ 1/∞ is zero or not?

by brutt 6y ago

0.999... + 0.000...1 = 1 0.000...1 = 1/∞ 0.999... = 1 - 1/∞ 1/∞ is zero or not?

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 6y ago
No, 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 6y ago
  https://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 6y ago
    I'm not quite sure I understand what you're trying to tell me? The example I mentioned is not about surreal numbers.
  - drran 6y ago
    I 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 6y ago
  There are some weird rules like that for floating point numbers too.

empath75 6y ago

What on earth is .000...1 If it terminates it’s not an infinite series. In any case, if you add any finite number to .99999... it’ll be equal to 1 + that number.