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The InfinitiesNo discussion of zero is complete without discussing its distant cousins - infinity and infinitesimal. Infinity is the term we commonly use for uncountable things or immeasurably large quantities. Infinitesimal is the inverse - infinitely small but greater than zero. There are at least three distinct types of infinities, although they are commonly treated indiscriminately. To differentiate between these let's call them Type 1, Type 2 and Type 3.
Infinity Type 1Our "Type 1" infinity is simply an unbounded quantity with any value other than zero. We can define "infinitesimal" to be as arbitrarily large or as small as required, as long as it is not equal to zero. This is the same whether zero is viewed as { 0 } or { }. Example:
Infinity Type 2Our "Type 2" infinity is generated only when zero is viewed as { 0 }. It is generated by division by zero and only by division by zero. Example:
The logic behind it is this: Subtract zero from one, then if the result is not zero, repeat.
Infinity Type 3The "Type 3" infinity can be best described as "the set of never-ending things". Never-ending distances and infinite loops also fall into this category. This is much the same whether zero is viewed as { 0 } or { }. Example:
The Same but DifferentThe various types of infinities have many things in common and some subtle differences. For example, here we differentiate Types 1 and 2, which are the result of division, from Type 3 which can be described as a simple endless loop. Also, contrast a Type 1 infinity with a Type 2 infinity. A Type 2 infinity is infinitely larger than a Type 1 infinity, as zero is infinitely smaller than infinitesimal. In calculations, Type 1 and Type 3 infinities can generally be replaced with the words "any number". Type 2 infinities do not follow this rule - they are the results of a division by zero.
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