Domains which equals to each other (1) has any significance?

[MathematicalPhysicist]

between 0-1 there are infinite number of rational numbers now between 1-2 there are also infinite number of rational numbers, how can we proove that the number of rational numbers between 0-1 equals to those between 1-2?

does the difference of the domains which equals to each other (1) has any significance?

 

[master_coda]

To show that the number of rationals on [0,1] is the same as the number of rationals on [1,2] you just need to find a bijection from [0,1] to [1,2].

Clearly f(x)=x+1 is a suitable bijection.

 

[MathematicalPhysicist]

what is x represnts in this context? (the number of rational numbers?).

 

[Guybrush Threepwood]

x represents any number in [0, 1]

 

[MathematicalPhysicist]

Originally posted by master_coda
To show that the number of rationals on [0,1] is the same as the number of rationals on [1,2] you just need to find a bijection from [0,1] to [1,2].

Clearly f(x)=x+1 is a suitable bijection.

let me see if i understand, x is in [0,1] then f(x) is in [1,2] therefore f:x->f(x) therefore the number of rationals in [0,1] equals to [1,2].

 

[Hurkyl]

Also, using the function f(x)=2x, one can prove the number of rationals in [0, 1] is the same as the number of rationals in [0, 2]