Defining Real Numbers Between Two Integers

[Gear300]

If we were to take any two integers on a real number line and mark a point (a number) halfway between the two, do the same in the range between the halfway point and each of the two numbers, and continue the process, would we be able to define all real numbers between the two integers (including irrational numbers) as the limit of a rational expression or a series of rational expressions?

 

[Guest]

I think not. Because whatever you define would not be countable.

 

[Office_Shredder]

Consider in between 0 and 1. You'll end up getting every rational number that has a 2^^k in the denominator. These points are dense, so every number between 0 and 1 is a limit of a sequence of them. Then given any two integers a and b, the map f(t)=a(1-t)+bt is a bijection between the intervals [0,1] and [a,b] that preserves your construction (so 1/2 is mapped to halfway between a and b, 1/4 is mapped to one quarter of the way between a and b) and so your construction is dense on any interval between two integers

 

[mikeph]

I think the answer is no because the halfway point between any two rational numbers is another rational number. You can't obtain irrational numbers using this method.

 

[CellCoree]

Yes, this process is known as the "Bisection Method" and it is a common method used in mathematics to approximate the values of real numbers. It is based on the principle of dividing a given interval (in this case, between two integers) into smaller and smaller subintervals, and taking the midpoint of each subinterval as an approximation for the value of a real number. This method can be used to approximate any real number, including irrational numbers, as the limit of a rational expression or a series of rational expressions. However, it should be noted that this method may not provide an exact value for the real number, but rather an approximation. Therefore, it is important to consider the level of precision needed for a particular application when using this method.