Sure. Given any two real numbers a and b the number [imath]\dfrac{a + b}{2}[/imath] will be between them.There is a way to prove that between 2 irrational numbers there is another irrational number?
How do you define "the irrational part of a number"?Sure. Given any two real numbers a and b the number [imath]\dfrac{a + b}{2}[/imath] will be between them.
If a + b is irrational, then we are done because half of that is also irrational.
If a + b is rational then the irrational parts of a and b are negatives of each other. So then take [imath]a + \dfrac{b -a}{4}[/imath].
-Dan
There is a way to prove that between 2 irrational numbers there is another irrational number?
I did say that wrong, didn't I? I suppose a better concept would be to use the fractional part. The concept I'm reaching for is that the non-integer part of the irrational numbers add to a rational number (between 0 and 1, but that isn't important.) So if we have two rational numbers a and b such that a + b is rational then we know that {a} + {b} = rational. But I guess we really don't need to comment on the possibility. All we need to do is find a rational number around b - a and add it to a. (As Cubit is saying above.)How do you define "the irrational part of a number"?
I loved your method/ post, and I immediately knew the kind of thing you intended by "the irrational part of a number" but I couldn't tie it down more formally myselfI did say that wrong, didn't I? I suppose a better concept would be to use the fractional part. The concept I'm reaching for is that the non-integer part of the irrational numbers add to a rational number (between 0 and 1, but that isn't important.) So if we have two rational numbers a and b such that a + b is rational then we know that {a} + {b} = rational. But I guess we really don't need to comment on the possibility. All we need to do is find a rational number around b - a and add it to a. (As Cubit is saying above.)
-Dan