Question 482707: I still don't understand irrational, real, whole, natural, rational, and integer.
Can you give me simpler definitions and examples for each term.
Answer by tinbar(133)   (Show Source):
You can put this solution on YOUR website! in my explanation a
Now you can treat all the various numbers you are thinking of this way.
whole numbers/natural numbers < integer < rational < irrational <= real
So firstly I think the whole numbers and natural numbers refer to the same set. Maybe one of them contains the number 0 and the other doesn't? Not sure, doesn't really matter. They contain 1,2,3....possibly with 0 as well, no negatives, no decimals. Next, let's say we want to include the negatives, well, in that case, if we add all the negative versions of the numbers in the natural set, so now we have -3,-2,-1,0,1,2,3...we call these the Integers, and still we have no decimals/fractions. Now say we want those as well, we can take the set of Integers, and for each p,q in the Integers, we also add p/q and q/p. So let's say we are considering p=2,q=3, then 2/3, and 3/2 become rational numbers.
Wow, that's a lot of numbers, are we done? Actually no, we are not, there are still numbers that exist that cannot be expressed as what we already have (natural, integer, rational). sqrt(2) is an example of an irrational number, which mathematically means that there is no p,q from the integers such that sqrt(2) = p/q (Google or email me for a short proof of this if you are curious). So for all these numbers like sqrt(2) we need a higher-level set, which we call the irrationals, and we get the irrationals by taking the rational set and adding things like sqrt(2)(In general, the sqrt(prime number) is irrational, so we can add all prime square roots). I'm not sure about the level of your course, but unless you are dealing with 'complex' numbers, you can consider the real set as not being any different from the irrational one, though in future courses this will not be the case.
So basically, start with the lowest set, then for each 'kind' of number you don't have, we add those kinds and 'upgrade' the set.
I hope this has helped you.
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