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I am confused about the density and size of $\mathbb{Q}$ and $\mathbb{R}$; between any two rationals there is an irrational number and between any two irrationals there is a rational number. Does from this not follow that rationals and irrationals alternate? That cannot be the case however, as $|\mathbb{Q}| < |\mathbb{R}|$ as provable through a diagonal proof on the digits of the irrationals and if they were alternating they could be paired up and thus would be equal in quantity. So, how can there be a rational between any two irrationals and vice versa yet irrationals and rationals not be alternating?

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    `between any two rationals there is an irrational number and between any two irrationals there is a rational number` May be easier to rationalize if you think of it as "*between any two rationals there is an (uncountable) infinity of irrational numbers, and between any two irrationals there is a (countable) infinity of rational numbers*". – 2017-01-08
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    Another way to look at it: you're thinking about elements (numbers) being next to each other, but you can't think of a continuum of real numbers in such a way. For example, there's no such thing as "two neighboring rationals" -- between any two rationals we have infinitely many other rationals, as well as infinitely many irrationals (as @dxiv already pointed out). – 2017-01-08
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    There is no meaning to the word "alternate" here because you cannot list all of the real numbers. – 2017-01-08
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    If you pour a cup of wine into a barrel of sewage, you have a barrel of sewage. If you pour a cup of sewage into a barrel of wine, you have a barrel of sewage. Even if the amount of sewage has measure 0. – 2017-01-08

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This is a copy-paste of dxiv's comment and should serve as an answer too.
Between any two rationals there is an irrational number and between any two irrationals there is a rational number. This may be easier to rationalize if you think of it as "between any two rationals there is an (uncountable) infinity of irrational numbers, and between any two irrationals there is a (countable) infinity of rational numbers".