It's very simple to give examples of but very difficult to explain, I know that the BTW community is wrought with people who are not only very skilled in mathematics but also willing to offer assistance.
My issue is this:
Why is it that the sum, the difference, and the product of two rational numbers are rational?
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given A and B being real rational numbers
A+B
A-B
A*B
Will always produce the rational number C
However this is not always true for the arithmetic operations of irrational numbers which can produce rational numbers in some instances.
I can understand it but not explain it unfortunately, if someone could help me out I'd be very appreciative.
Math theory wasn't my forte in school, but I do believe it has something to do with the fact that rational number can always be displayed in a fractional form. As such, you can't take two fractions together in addition, subtraction, and multiplication and get a non-fraction.
Also, I can't think of a single system where A*B, where as A and B are different irrationals, gives a rational number. Do tell, if you know of any off hand.
First of all, note that you can even make integers by multiplying/dividing irrational numbers; for example, if sqrt(a) denotes the square root of a, then 3/2 sqrt(2) multiplied by 2 sqrt(2) is equal to 3.
As for the rational numbers, I'll just go through each of the operators. From here on, Q represents the set of rational numbers and Z the set of integers. Let p,q in Q. Then p and q can be written as a/b with a,c in Z and b,d in Z\{0}. We have:
a/b + c/d = (ad + bc)/bd
a/b - c/d = (ad - bc)/bd
a/b * c/d = ac/bd
and if q is not zero:
(a/b) / (c/d) = ad/bc.
In all cases, the resulting number can be written as a fraction i/j, with i in Z and j in Z\{0}. This is sufficient to prove that these numbers are indeed rational if you accept that the rational numbers and the operations on them exist. So basically, what KWilt said. A more formal explanation would be to state the properties you want Q to have and construct a set that satisfies those properties.
So why are the irrational numbers different? Well, this is because a number ceases to be irrational as soon as it can be written as a fraction (ie, is a rational number), whereas a rational number is still called rational if it is also an integer. If we would look at the rational numbers that are not an integer, we would see that that's not closed under the forementioned operators. For example, 2/5 + 3/5 = 1, which is an integer.
