4 ms·

Not exactly. In math, epsilon is basically "it's not zero, except when it is." When you're dividing, then it's close to zero, but still qualitatively not zero.

by lmkg 3y ago

Not exactly. In math, epsilon is basically "it's not zero, except when it is." When you're dividing, then it's close to zero, but still qualitatively not zero. But when you're multiplying or adding, then we say it's close enough that we'll treat it as exactly zero. So it's not a specific number, it's something where we can handwave its qualitative size depending on what's convenient. Limits are a formalization of this concept. It's not a specific value, it's very much a variable that "approaches" zero. This establishes rules about how much bullshit you can actually do it. It's not zero, you do some arithmetic, then it becomes zero and you do the rest of the arithmetic. But when you're writing a program, epsilon is indeed often just a small number. The trick is choosing the right "small."

Someone 3y ago
> In math, epsilon is basically "it's not zero, except when it is." In the math I know, it’s not necessarily small, but it is larger than zero. https://mathworld.wolfram.com/Epsilon-DeltaDefinition.html https://mathworld.wolfram.com/Epsilon-DeltaDefinition.html: “An epsilon-delta definition is a mathematical definition in which a statement on a real function of one variable f having, for example, the form "for all neighborhoods U of y₀ there is a neighborhood V of x₀ such that, whenever x in V, then f(x) in U" is rephrased as "for all ε > 0 there is δ > 0 such that, whenever 0 < |x - x₀| < δ, then |f(x) - y₀| < ε.” (Also used in proofs. https://mathworld.wolfram.com/Epsilon-DeltaProof.html https://mathworld.wolfram.com/Epsilon-DeltaProof.html) https://en.m.wikipedia.org/wiki/Limit_of_a_function#%7F'%22%60UNIQ--postMath-00000007-QINU%60%22'%7F-definition_of_limit https://en.m.wikipedia.org/wiki/Limit_of_a_function#%7F'%22%...: “For every real ε > 0, there exists a real δ > 0 such that for all real x, 0 < |x − p| < δ implies |f(x) − L| < ε.” > it's something where we can handwave its qualitative size depending on what's convenient. Aha! You’re doing physics, not math. Mathematicians only handwave when they can prove it doesn’t affect results ;-)
- iamevn 3y ago
  The paper, on the epsilon-delta: > This has nothing to do with the subject of the paper
  - deleted 3y ago
    [deleted]

by lmkg 3y ago

Someone 3y ago

> In math, epsilon is basically "it's not zero, except when it is." In the math I know, it’s not necessarily small, but it is larger than zero. https://mathworld.wolfram.com/Epsilon-DeltaDefinition.html https://mathworld.wolfram.com/Epsilon-DeltaDefinition.html: “An epsilon-delta definition is a mathematical definition in which a statement on a real function of one variable f having, for example, the form "for all neighborhoods U of y₀ there is a neighborhood V of x₀ such that, whenever x in V, then f(x) in U" is rephrased as "for all ε > 0 there is δ > 0 such that, whenever 0 < |x - x₀| < δ, then |f(x) - y₀| < ε.” (Also used in proofs. https://mathworld.wolfram.com/Epsilon-DeltaProof.html https://mathworld.wolfram.com/Epsilon-DeltaProof.html) https://en.m.wikipedia.org/wiki/Limit_of_a_function#%7F'%22%60UNIQ--postMath-00000007-QINU%60%22'%7F-definition_of_limit https://en.m.wikipedia.org/wiki/Limit_of_a_function#%7F'%22%...: “For every real ε > 0, there exists a real δ > 0 such that for all real x, 0 < |x − p| < δ implies |f(x) − L| < ε.” > it's something where we can handwave its qualitative size depending on what's convenient. Aha! You’re doing physics, not math. Mathematicians only handwave when they can prove it doesn’t affect results ;-)

iamevn 3y ago
The paper, on the epsilon-delta: > This has nothing to do with the subject of the paper
- deleted 3y ago
  [deleted]