The number of numbers.

The number of rational fractions between any two consecutive numbers.

The number of irrational fractions between any two rational fractions, however close.

Any number that can be measured, regardless how big or small, is called a "finite" number. When determining a value not easily found, we often begin by identifying an upper bound (translation: a number for which the answer cannot be "more than").

For example, what is the positive square root of 63? Difficult to pin down exactly (it is an irrational number), but we easily determine that it is less than 8 (which is the square root of 64),

Therefore, 8 is said to be an "upper bound" for the square root of 63.

If a value has an upper bound (and a lower bound, same idea below the value), then that value is said to be "finite" because any measurement, starting from zero, will "finish" at some point.

If there is no such finite number to bound the value, then the value is infinite (translation = not finite).

The limit of 1/x, as x goes to zero (gets closer and closer) cannot be bounded by any finite number. It is said to be "infinite". The limit is said to tend to infinity.

There is no actual value called "infinity" when dealing with real numbers. Infinity is not a "well-defined" number.

That is why trying to do normal arithmetic with "infinities" leads to strange results.

For example, consider the line of non-negative integers.

Z = {0, 1, 2, 3, 4, 5, 6, 7, 8...}

How many numbers? An "infinity" since we cannot find a bound for "a highest number".

Multiply each number by two, and call this new set "even".

even = {0, 2, 4, 6, 8, 10, 12, 14, 16...}

By construction, the number of numbers in "even" is exactly the same as in "Z".

Now, add 1 to every single number of "even" and call that set "odd"

odd = {1, 3, 5, 7, 9, 11, 13, 15, 17...}

By construction, there is exactly the same number of numbers, in "odd" as there are in "even" and in "Z".

Also, it is possible to show that there are no common element in the sets "even" and "odd" (no number is simultaneously even AND odd).

If you create a new set by combining "even" and "odd", should you expect to find twice as many numbers in this combo as you had in the original "Z"?

new = even + odd = {0, 1, 2, 3, 4, 5, 6, 7, 8...}

infinity + infinity = 2 * infinity ?

If so, what is different between this new combo and the original set "Z"?