How can we use irrational numbers in math if we don’t know the full number?
In: 232
For the irrational numbers you probably think about, the answer is that we work with it _symbolically_. √2 is not defined as an infinite string of digits, but as the (positive) number that squares to 2, hence by (√2)² = 2. You use that property and deduce everything else from it like in (√2+1)·(√2-1) = √2² – 1² = 1. Same with 𝜋, e, 1/7-th or really any number you encounter in your life.
For practical things, using a numerical representation such as ~3.1416 usually works, too. An engineer is already limited to a finite precision by how reality works. Hence why some system, usually decimal representations, was invented. It tells us exactly what they need to know, and some more.
The representation as a decimal is ultimately something arbitrary. For example, 1/3 is the infinitely long 0.3333… in decimal, yet in base 3 it is a simple 0.1 . It is totally possible to invent “base systems” where even some irrational numbers are finite or at least repeating. Potential bases include -3, √2, Fibonacci numbers (called Zeckendorf representation) and more.
Lastly, let me introduce the eldritch horror that are **non-computable numbers**:
When we talk about decimal representations or similar things, what we really mean is that we have some kind of description, pattern, machine, computer program or similar that actually tells us the digits. Most generally, we want a kind of algorithmic description of how the digits are formed! We can do that with 1/5 (output a “0”, then a dot “.”, then a “2”, then stop), 1/3 (output a “0”, then a dot .”, then keep outputting “3”s forever), but also √2 or pi, it just takes more complex programs.
However, there are only so many possible programs. Each of them has a finite length and only a finite range of symbols to code with. Most codes are not even valid, they fail to execute at all, or do not return numbers as we want it. In the end, a famous _diagonal argument_ by Georg Cantor shows that **there are vastly more real numbers as there are possible programs to describe them**!
Those many left we call _non-computable_. None of the numbers you likely have “seen” falls into that category; in some sense this is tautological because by their very definition we cannot describe their decimal digits in a proper way. All we can do is find purely symbolical descriptions and cope with it. We know they are out there, but no finite mind can truly grasp them.
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