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Reverted edits by George M'Harris (talk) to last revision by Aschlafly
== Similarities to zero ==
== Similarities to zero ==
Infinity has surprising similarities to [[zero]]. Neither can be diminished, and the multiplication or division of either cannot change it. While infinity and zero are opposites, neither is stronger or superior to the other: multiplication of infinity by zero yields an indeterminate result. As a pair they illustrate the logical truth that "the last shall be first, and the first shall be last" as taught by Jesus at the end of the parable of the vineyard workers, a parable that illustrates that infinity cannot be diminished.
[[Leonhard Euler]], the most prolific mathematician in history, famously said about infinity and zero:<ref>https://www.brainyquote.com/quotes/leonhard_euler_184025</ref>
{{cquote|To those who ask what the infinitely small quantity in mathematics is, we answer that it is actually zero. Hence there are not so many mysteries hidden in this concept as they are usually believed to be.}}
===Infinitesimal===
===Infinitesimal===
So infinity might arise in statements like these:
So infinity might arise in statements like these:
*<math>\frac{1}{0} = \infty\ \ </math> Mathematics does not allow division by zero, so this is wrong.
*<math>\frac{1}{0} = \infty\ \ </math> Mathematics does not allow division by zero, so this is technically not allowed unless does as a limit, as follows:
*<math>\lim_{x\to 0}\frac{1}{x} = \infty\ \ </math> Yes, one could say that "the limit is infinite," since that includes both positive and negative infinite values. But to say that the limit is "equal to infinity," one would have to say that this is the limit from the right. The limit from the left is "minus infinity," that is, unboundedly negative. This sort of statement, in terms of limits, is what was presumably meant by the incorrect statement above. This is extremely sketchy math, so just steer clear of division by 0.
*<math>\lim_{x\to 0}\frac{1}{x} = \infty\ \ </math> Yes, one could say that "the limit is infinite," since that includes both positive and negative infinite values. But to say that the limit is "equal to infinity," one would have to say that this is the limit from the right. The limit from the left is "minus infinity," that is, unboundedly negative. This sort of statement, in terms of limits, is what was presumably meant by the incorrect statement above.
*<math>\int_0^1\frac{1}{x} = \infty\ \ </math> Infinity has a special meaning for integrals.
*<math>\int_0^1\frac{1}{x} = \infty\ \ </math> Infinity has a special meaning for integrals.
*<math>\|\mathbb{Z}\| = \infty\ \ </math> The cardinality of the integers is infinite.
*<math>\|\mathbb{Z}\| = \infty\ \ </math> The cardinality of the integers is infinite.
In measure theory, one sometimes defines the "extended reals", allowing plus and minus infinity to be considered to be numbers, but this is a special construction, which makes certain arithmetical operations impossible. It can't be done in general.
In measure theory, one sometimes defines the "extended reals", allowing plus and minus infinity to be considered to be numbers, but this is a special construction, which makes certain arithmetical operations impossible. It can't be done in general.
Another place where infinity is considered acceptable is in [[Floating-point]] arithmetic in computers. The IEEE-754 standard (which all modern computers support) allows the values <math>\infty\,</math> and <math>- \infty\,</math>. But their treatment is very different from other numbers. To represent <math>\infty\,</math>, the exponent bits are all set to 1, and the mantissa bits are all set to 0, with the sign bit being set to 0 or 1 for <math>+\infty\,</math> and <math>-\infty\,</math> respectively.
Another place where infinity is considered acceptable is in [[Floating-point]] arithmetic in computers. The IEEE-754 standard (which all modern computers support) allows the values <math>\infty\,</math> and <math>- \infty\,</math>. But their treatment is very different from other numbers, and computer floating-point numbers don't faithfully model the real numbers in any case.
Also, in some non-standard models of Peano Arithmetic, <math>\infty\,</math> is treated as an actual number. But, once again, this is not standard mathematics.
Also, in some non-standard models of Peano Arithmetic, <math>\infty\,</math> is treated as an actual number. But, once again, this is not standard mathematics.