How can we use irrational numbers in math if we don’t know the full number?
In: 232
we actually do. we just cant properly represent it in decimal form so we either terminate the number at some point (normally 2-3 decimals in) or just use the symbol. like take pi for example. its 3.14158926535 etc etc whatever. you cant respresent all of pi faithfully with decimals lol. so you just write the symbol, or use something like 3.14 instead
Most fields do not require a fully exact value to be used. For example, for most applications that use Pi, they only need like 5-10 digits of it (and for rough calculation, just the “3.14” part is often enough).
Besides, there’s a separate but related concept called “precision”. In actual physical measurements there’s always some degree of error, which “precision” is the measure of. Mathematical approximations of irrational numbers should be a few digits more exact than the imprecise measures, but any more than that is really not necessary because it introduces false precision into the result.
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.
We can use many things without “fully knowing” them. And actually, we know so much about irrational numbers.
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.
Much of mathematics is based on approximations and statistical likenesses. As such, many times in math, “pretty damn close” is often times “good enough” in most cases.
Reference: see the delta/epsilon proof laying the foundational framework for calculus. Me: BS mathematics.
NB: and yes, I know approximations are often times not good enough, but pi to sixteen places is pretty close to pi at six million.
Computers can only compute to a fixed amount of digits. Humans have proven approximations that can produce these digits infinitely, so even if we don’t have a precise decimal representation, we can always reproduce it to the highest extent of our computational capabilities.
Two things. First, “1.1616” and “√2” and “*e*” are all perfectly valid symbols for real numbers. You’re bothered that *you* can’t recite the number in its entirety, or that it’s not an Arabic numeral, but the choice of symbol and number of digits doesn’t make a number less real. It only affects how you express them in math problems. “3 pi” is just as valid as 3*e* or 300.
Second, things we measure have a significant number of digits, a number of decimal places that we “trust” depending on the tools we measure with. ALL measurements are rounded. All of them. You have “1” ruler, because that’s counted. But your ruler is approximately 12 inches, rounded up from 11.9-something inches. That’s measured.
Let’s say we want to know the circumference of a circle, and the formula is “diameter x π (pi)”. You might measure a circle to be 3.5 inches in diameter with a ruler. Your answer has to be rounded to 2 digits. A physicist with a laser circle-ometer might measure it as 3.4779236 inches. That answer will be rounded to 8 digits. Each of you has some instrument where the number’s on the left side of the line but not exactly on the line, and you’re both rounding. So it doesn’t matter that pi goes on forever; you can’t measure the width of a circle accurately enough to need forever digits. But when you grow up and invent a circle-ometer that can measure to trillionths of an inch, there will be digits of pi waiting for you to use them.
This may be on a bit of a tangent, but let’s take Pi in a more practical application.
Now if we contextualise Pi being used on a large scale, where more decimals become important.
Using only 37 digits, we can very precisely calculate the circumference of the visible universe. Not just accurately. But down to the size of a hydrogen atom. We could never really need more digits, and only 37 was used to get here.
We do know the full number, it’s just not really possible to explain in it decimal numbers.
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