CH 4: Sizing up Infinity
Most people are well acquainted with the word infinity and know it to mean a never-ending value, whether it happens to be the debut album, Infinite, by the rapper Eminem, or anytime you want to evoke a boundless value or idea.
In order to understand the idea of infinity strictly in terms of numbers it helps to first consider the notion of a set. In math, a set is a collection of objects, and most often these objects are numbers. We also can have no objects, in which case we call it the empty set. Get at the computer scientist rapper MC Plus+ and his
crew for more information on that.
We could have a small set like 1,5,9 or an infinite set like 2,4,6,8,10..... where the dots indicate that the numbers go on forever. So, consider the following infinite set, the set of even numbers, or to be more rigorous, any number that divided by two yields an integer value. Speaking of, now let’s consider another infinite set, the set of Natural numbers. This set is 1,2,3,4,5... and on we go for infinity. Now that you have an idea of those numbers marching down a never-ending line, consider the set of the Real Numbers. This set includes the set above of all the natural numbers, but also irrational numbers like π and e (Irrational numbers are numbers that cannot be expressed as the ratio of two integers).
Now what is interesting here is how it has been proven that the size of numbers between 0 and 1 of the Real numbers is greater than the entire set of Natural numbers. Part of the way in which infinities are measured has to do with the idea of correspondence or having a partner element in one set with a partner element in another set. Imagine you have two bags filled with marbles but are not sure which bag has more marbles. All that you need to do is take a count as you take out one marble from each bag, and if all the marbles in each bag are emptied simultaneously, then we know there is the same number of marbles in each bag. In this way of establishing correspondence, we do so similarly with the Natural numbers. The result is that there are more Reals than can find matches with the Naturals and thus, the infinity of the Reals is far greater than the infinity of Naturals.
To delve further into the nature of infinity, ask the World Wide Web about Cantors Transfinite Numbers.
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