How can we use irrational numbers in math if we don’t know the full number?
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It all depends on what you mean by “use.” We can manipulate irrational / transcendental numbers and prove things about them. That’s one sense of the word “use.”
The classic example is pi, the ratio of a circle’s circumference to its diameter: given such a ratio, we can prove certain properties about it and its relationship to other numbers.
It’s only irrational in base 10, meaning its decimal expansion is infinite. In base pi it would be written as “1”.
This is even true of uncomputable numbers which vastly outnumber computable reals. Like Chaitan’s constant, i.e. “the halting number.” Such a real does exist, and we can prove that if it were computable, a turning machine can decide the halting problem. Which means it must be an uncomputable number. So we’ve just proved something about this number. We can manipulate it and do math with it and prove things about it and its relationships to other mathematical objects.
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