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We will now look at how numbers are classified and break them down into other parts. While it might be a pain to memorize every little thing, it would be beneficial to know as much as possible about numbers. The more you learn, the better off you’ll be in whatever subject matter is discussed.

Our previous article talked about cardinality, defined as any positive whole number greater than 0. In this definition, there were no negative or imaginary numbers involved, so we will only go over the integers and real numbers for this lesson. We start with integers since they are very easy once you get through the concept of negative numbers – no pun intended – who…just caught me from making an awful joke.

## So now, let’s go over the integers, which are essentially whole numbers.

• I feel like I can make a little more puns with integers since there is no “negative” (no pun intended) in them like there is with negative integers. However, this is not my struggle; it’s yours, says Aron Govil.
• The positive integers start at one and go by one number after another to infinity; then, you would have to say they switch from counting up to down if you want to hold onto that terminology. Of course, we all know that negative numbers do not exist as they might be assumed from the name but remember that when we talk about positive numbers, they are considered any number than zero. So really, they are any number than negative numbers.
• The positive integers go by the formula of 1, 2, 3, 4, 5…n (where n is infinity) and when it comes to the negative integers, what you have to remember about them is that you subtract x from 0 for every x in the set; the same thing but opposite. So instead of getting 6 because 6-0=6; 6 becomes -1 (because 6 – 0 = 6) then becomes -2 (because -1-0 = -1), then you get -3 …and so on. The only possible problem here might be remembering whether it starts at 0 or 1; well, since there isn’t yet a negative one, it’s 0.
• It gets a little more difficult for the real numbers, but I’ll try to make this as easy as possible to understand. There are three different categories of real numbers: rational, irrational, and transcendental. Transcendental means that you cannot find an n-th root for any given positive number, where n can be any number. The first type of real number we will talk about is the rational ones, which are any number can be expressed as a fraction or terminating decimal (the sequence goes on forever with no repeating pattern). A fraction is very simple if you know how to already; think 1/n for whatever integer you might have up until infinity since there aren’t any fractions larger than 1. So 1/2=1/4, 1/3 would equal 1/6, then on down to infinity. There are also terminating decimals that go on forever like the fractions; however, they repeat a pattern at some point on the line (example: 0.3333…, where there is an endless sequence of 3s).
• Irrational numbers are any numbers that cannot be anything besides numbers or their square root (like pi), which you could say is their length (which makes sense since it represents distance). So if I asked you what comes after pi? Well, it’s 2pi because once you get to pi, you have to go back over one complete cycle, so two times pi equals going all the way around. Another example would be to ask you what comes after the square root of 2? Well, it’s the square root of 4…which is irrational because if you were asked what was next, there could not be a sequence since it breaks from the pattern.
• Calculus [the infinitely small] and all that other math stuff will probably kick in here soon, but for now, I hope this has been helpful. If not, then hopefully, you’re smart enough to understand another way since I’m no genius with numbers. Also, if you do have any questions about these subjects or ones in general, feel free to ask! P.S., all real numbers =/= rational or irrational; instead, they can fall under either one of these two categories, so don’t assume anything by looking at one example. Thanks!
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