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You don't need to understand anything about infinitesimals to understand 0.999... = 1. Perhaps you meant limits? The "standard" approach would be to point out t

by rntz 10y ago

You don't need to understand anything about infinitesimals to understand 0.999... = 1. Perhaps you meant limits? The "standard" approach would be to point out that Σ_{i=1}^∞ 9/(10^i) = 1 (that is, the sum from i = 1 to infinity of 9/(10^i) is 1), and understanding an infinite summation requires the concept of a limit. (Of course, there are simpler proofs that use only basic algebra and intuition about decimals; a limit is just the most direct approach.) You're spot-on about needing to understand there's a distinction between a number and its decimal representation, though.

soVeryTired 10y ago
Limits use a construction that's pretty similar to an infinitesimal. The epsilon-delta definition of a limit is no joke for students.
- alejohausner 10y ago
  Oddly enough, I never understood the epsilon-delta description of limits until I read David Foster Wallace's book on infinity. All through my degree in math I was taught about things without learning the historical context that created those things.
  - geoka9 10y ago
    Getting a degree in math without understanding epsilon-delta is quite an achievement! I am serious.
  - SomeStupidPoint 10y ago
    I'm always super happy that I stumbled in to taking topology before real analysis. It meant that I understood the topological idea of limits before I had to do proofs using just the epsilon (for sequence) or epsilon-delta (for functions) definition, and so could translate the logic of showing things about neighborhoods in to the terminology of (real analysis) limits. Limits, in the abstract, are a fairly simple concept: in the case of sequences, for any neighborhood of the limit, the entire tail of the sequence (past some point) is contained in the neighborhood; in the case of functions, for any neighborhood of the limit at f(x), there's a neighborhood around x, such that every point in that neighborhood maps to the neighborhood around the limit.

by rntz 10y ago

soVeryTired 10y ago

Limits use a construction that's pretty similar to an infinitesimal. The epsilon-delta definition of a limit is no joke for students.

alejohausner 10y ago
Oddly enough, I never understood the epsilon-delta description of limits until I read David Foster Wallace's book on infinity. All through my degree in math I was taught about things without learning the historical context that created those things.
- geoka9 10y ago
  Getting a degree in math without understanding epsilon-delta is quite an achievement! I am serious.
- SomeStupidPoint 10y ago
  I'm always super happy that I stumbled in to taking topology before real analysis. It meant that I understood the topological idea of limits before I had to do proofs using just the epsilon (for sequence) or epsilon-delta (for functions) definition, and so could translate the logic of showing things about neighborhoods in to the terminology of (real analysis) limits. Limits, in the abstract, are a fairly simple concept: in the case of sequences, for any neighborhood of the limit, the entire tail of the sequence (past some point) is contained in the neighborhood; in the case of functions, for any neighborhood of the limit at f(x), there's a neighborhood around x, such that every point in that neighborhood maps to the neighborhood around the limit.