Guide Representations of Real Numbers by Infinite Series

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Notice that the terms in the series are alternately positive and negative. Such series have the nice property that if you stop adding at any particular term, the finite approximation that results differs from the sum of the series by no more than the next term. For the series above, if we want two decimal places of accuracy, then we need to add terms until the denominators are less than ; that means 50 terms.

Granted, a computer can do that in the blink of an eye, but trying to do it by hand would take a while.

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To get the first trillion digits correctly? That would take half a trillion terms. That's a lot. Let's try some heavier math.

Implementing infinite precision arithmetic - IEEE Conference Publication

Calculus got us a series expansion for the arctangent function, but maybe we can do better. But if this isn't fancy enough for you, then we can always simply appeal to the genius of Ramanujan and use his famous formula. It turns out there is a way, and I'll get to it in the next installment. Stay tuned. I am a professor of mathematics at the University of Florida with research interests in various areas in topology, including topological data analysis.

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In the Pixar film Toy Story, the gung ho space action figure Buzz Lightyear tirelessly incants his catchphrase: "To infinity … and beyond! That assumption, however, is not entirely sound.

Are Prime Numbers Made Up? - Infinite Series - PBS Digital Studios

As German mathematician Georg Cantor demonstrated in the late 19th century, there exists a variety of infinities—and some are simply larger than others. Take, for instance, the so-called natural numbers: 1, 2, 3 and so on.

These numbers are unbounded, and so the collection, or set, of all the natural numbers is infinite in size. But just how infinite is it? Cantor used an elegant argument to show that the naturals, although infinitely numerous, are actually less numerous than another common family of numbers, the "reals. In fact, Cantor showed, there are more real numbers packed in between zero and one than there are numbers in the entire range of naturals. He did this by contradiction, logically: He assumes that these infinite sets are the same size, then follows a series of logical steps to find a flaw that undermines that assumption.

He reasons that the naturals and this zero-to-one subset of the reals having equally many members implies that the two sets can be put into a one-to-one correspondence.

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That is, the two sets can be paired so that every element in each set has one—and only one—"partner" in the other set. Think of it this way: even in the absence of numerical counting, one-to-one correspondences can be used to measure relative sizes.

Imagine two crates of unknown sizes, one of apples and one of oranges. Withdrawing one apple and one orange at a time thus partners the two sets into apple-orange pairs. If the contents of the two crates are emptied simultaneously, they are equally numerous; if one crate is exhausted before the other, the one with remaining fruit is more plentiful.

Cantor thus assumes that the naturals and the reals from zero to one have been put into such a correspondence. Every natural number n thus has a real partner r n.

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The reals can then be listed in order of their corresponding naturals: r 1 , r 2 , r 3 , and so on. Then Cantor's wily side begins to show. He creates a real number, called p, by the following rule: make the digit n places after the decimal point in p something other than the digit in that same decimal place in r n.

A simple method would be: choose 3 when the digit in question is 4; otherwise, choose 4.