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Out of curiosity I was watching this video from njwildberger on youtube: https://www.youtube.com/watch?v=4DNlEq0ZrTo
Where he says that you can't define associativity between irrational numbers because they are only an approximation. Later he says that when you put letters on the decimals of pi then you can generate the whole history of every single particle in the universe, which is ridiculous. Then how are the real numbers defined that they make sense?
I realize this is a bit late, but let me try to provide a bit of context:
Numbers are of course one of the main building blocks of mathematics, but during the history of mathematics it wasn't always clear what exactly "numbers" are. The Egyptians for example knew enough math to build the breathtaking pyramids but they seemingly didn't care much about the underlying philosophical issues. For them it was enough that they for example "knew" (in the sense of: having seen enough evidence) that the ratio of a circle's circumference to its diameter (which we today call $\pi$) is always the same and that they could approximate it if needed.
The Greeks were different, though, because they cared. For the Pythagoreans, numbers even had religious status. However, numbers for them were only what we now call the "natural" numbers (without zero) and they also dealt with ratios (what we would nowadays call positive rational numbers), but that was it. It then came as a big shock for them when they realized that the length of the diagonal of a unit square (our $\sqrt2$) could not be expressed as a ratio. (This is of course also true of $\pi$ but was not proved before the 18th century.)
This meant that the "continuum" (the geometrical line) was a pretty mysterious thing because there were points on it that you couldn't "reach" with "numbers". The result was that they made a distinction between (their concept of) numbers and "magnitudes" which is something they could "measure" (with a compass) in geometry but refused to compute with (in the sense of arithmetic). And that's also the reason why geometry always had the highest status in their view of mathematics - a view that dominated math for centuries to come.
It was only when the calculus came up in the 17th century and revolutionized most of mathematics (and physics) that it became more and more apparent that it would be much more convenient and consistent if entities like $\sqrt2$ and $\pi$ were treated like "normal" numbers. But it turned out that to make this work in a way acceptable to the majority of mathematicians you had to perform a "leap of faith" and begin to deal with infinite entities:
It is one thing to say that $\sum_{k=1}^n \frac6{k^2}$ provides an algorithm to approximate a certain value you're interested in and it's another thing to imagine that this approximation will ever "finish" and to call "the result" $\sum_{k=1}^{\color{red}\infty} \frac6{k^2}=\pi^2$. The good news is that this procedure of giving irrationals the same ontological status as rational numbers will provide you with a working model of the continuum "without gaps" (namely what we now call "$\mathbb R$"). The bad news is that this opens up a whole new can of worms (e.g. that $\mathbb R$ is an uncountably infinite set).
Anyway, some mathematicians are not willing to go this route and they only want to accept in math objects that can at least in principle be computed in a finite number of steps. (Indeed, even if you "believe" in $\mathbb R$ you can prove that almost all - in a precise sense - real numbers aren't computable.) Wildberger is one of them and I think this point of view is perfectly valid. I only have a problem if these views are presented in a way that tries to disparage other views as obviously misled.
For an in-depth view of how the concepts of "number" and "continuum" evolved, see Philosophie der Mathematik by Bedürftig and Murawski. See also my answer here including the book recommendation.
The question in the title is, "How do you multiply infinite quantities?" The question in the body is, "...how are real numbers defined [so] that they make sense?" These two questions seem to be related in the view of OP, but I don't see the relation, so I'll ignore the woolly one in the title, and speak to the one in the body.
There is a standard way to define the reals, and it makes perfect sense. I'll assume you're happy with defining the rationals so they make sense; then the reals are defined as equivalence classes of Cauchy sequences of rationals. It is not hard to extend to the reals, so defined, the operations of addition and multiplication from the rationals, and to show that the reals form a field under those operations, and then to prove the other familiar properties of the reals. I'm not going to write out the details, since you can find them in dozens of textbooks at your local university library. Or you can have a look at this.
My friend Norman would reject this definition of the reals, but most mathematicians have no objection to it.
