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Definition ad infinitum5/6/2023 ![]() ![]() Binmore: Mathematical Analysis: A Straightforward Approach . (next): Chapter $1$: Real Numbers and Functions of a Real Variable: $1.1$ Real Numbers Stephenson: Mathematical Methods for Science Students (2nd ed.) . It was Georg Cantor in the $1870$s who finally made the bold step of positing the actual existence of infinite sets as mathematical objects which paved the way towards a proper understanding of infinity. The symbol $\infty$ for infinity was introduced by John Wallis in the $17$th century. The concept of infinity has bothered scientists, mathematicians and philosophers since the time of Plato (who accepted the concept as realisable) Aristotle (who did not). The term ad infinitum can often be found in early texts. The latter result seems wrong when you think of the rule that a negative number square equals a positive one, but remember that infinity is not exactly a number as such. $\forall n \in \Z: -\infty< n$ $\forall n \in \Z: -\infty n = -\infty$ $\forall n \in \Z: -\infty \times n = -\infty$ $\paren ^2 = -\infty$ Similarly, the quantity written as $-\infty$ is defined as having the following properties: $\forall n \in \Z: n < \infty$ $\forall n \in \Z: n \infty = \infty$ $\forall n \in \Z: n \times \infty = \infty$ $\infty^2 = \infty$ It is defined as having the following properties: However, outside of its formal use in the definition of limits its use is strongly discouraged until you know what you're talking about. ![]() The symbol $\infty$ (supposedly invented by John Wallis) is often used in this context to mean an infinite number. Informally, the term infinity is used to mean some infinite number, but this concept falls very far short of a usable definition. ![]()
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