Quote Originally Posted by Dr Unne
1) In mathematical theory (which I believe is what were are dealing with), simplest terms are always important. You don't say 2A=2(pi)r^2. You don't say .77e=.77mc^2. You don't say a^2 + b^2 + 10=c^2 +10. You can, but you don't. Simplest terms are always used in mathematical theory. --SeedRankLou

I don't see any reason why you have to simplify everything. I've never heard this. Could you explain why this is true?

I'm fairly certain that the definition is specifying a ratio of two integers that is not also an integer itself, or the definition could also go: any real number that cannot be expressed as an integer (making fractions irrational number).

The definition of rational number is a ratio of two integers, where the denominator isn't 0. That's the only restriction. 1/1 is a rational number. 10/2 is a rational number. It doesn't matter if the rational number can be written in another form as an integer. Just a different way of representing the same value.
It's just math etiquette. Similar to writing etiquette. When you are just writing, you can write however you wish. But if you are writing a formal paper, there are rules you follow, like don't end a sentence with a preposition, use commas appropriately, ect. ect. When dealing with formal math (i.e. theory) there are rules you follow, and one of them is that all terms and equations must end as their simplest form. There are good reasons for this. One is irrelevence. In theory, extra terms are irrelevent. You are only dealing with what is at hand. Is .999....=1, not is .999....=1/1. For example, if you have an equation and you are solving for y and you get 10y=10x+20, in practicality that's fine, but in theory that's not fine. There is a factor 10 in that equation that in no way affect the value of y, and therefore the 10 is irrelevant and shouldn't be there. And to say 10y/10=(10x+20)/10 is also has irrelevent terms for the same reason. If people are going to be overly practical, then people could write something like 1(y+1-1)/1=1(x+2+1-1)/1. Yes, this is correct, but all of those 1s are irrelevent, they do nothing to the equation. Another reason is duplicity. Lets say I have the number 12. If I write 12/1, do I not still have 12? If I write 15-3, do I not still have 12? If I write (square root of 4)*(24/4), do I not still have 12? If I write log (base 2) 4096, do I not still have 12? If I write 24*(cos 45)^2, do I not still have 12? Numbers can be written in hundreds of differents fashions, but all of the above can be simplified to 12. Is that to say the number 12 is an integer, an arithmetic expression, a fraction, a compound fraction, a logarithm, and a trigonometric function, or is 12 just an integer? 12 can be expressed as all of the above things, but all of the above things can be expressed as 12. So which is right? Technically they're all right, but in mathematical theory, 12 is right. When numbers are in their simplest terms, they always lack duplicity, so the number is only what it is and nothing more, making a number concretely and inarguable what it is. If a number isn't in simplest terms, it can be something else, but if it is in simplest terms it is only what it is. That is probably the main reason for simplest terms. That and making equations as short as possible (i.e. laziness).

And your definition isn't entirely correct. A rational number is a number capable of being expressed as an integer or a ratio of two integers, excluding zero as the denominator. That distinction is made for a reason. A ratio of two integers is not itself suppose to represent an integer, or you would simply write integer and not the above phrase. Ratios in theory are in simplest terms, and if a 1 is in the denominator, then it is irrelevent.