I have just come across the concept that there are different types of infinite numbers (whole integers, irrational numbers etc.) and the concept that for example, the amount of irrational numbers between 0 and 1 is higher than the amount of whole numbers from 1 to infinity.
I guess I just don’t understand how an infinite amount of something can be bigger/smaller than an infinite amount of something else…
Please un-f**k my brain 😀
In: 4
Infinity isn’t usually compared directly the same way you compare regular numbers. One way mathematicians compare them is by making maps between different infinitely sized sets.
For example, set of all integers is the same cardinality as set of all even integers because there exists a bijective (meaning one is to one) mapping/pairing between the two sets: x <-> 2x.
However between integers and real numbers, there’s no such mapping (for a full proof, see Cantor’s Diagonalization argument). An intuitive, non rigorous perspective is that integers are “countable” – you can count them off in a straight line 0,1,-1,2,-2,etc, in such a way that every integer will be hit if you go on forever (and I can tell you exactly when any particular number will be hit). No such enumeration exists for the real numbers – again rigorous proof left for Google but I invite you to try to find such an enumeration.
So one is countable and the other isn’t – so we say the set of real numbers is “larger”. Integers and rationals are both countable (meaning they are the same cardinality); irrationals and reals are uncountable, and it turns out these two are also the same cardinality (but both larger than any countable set).
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