How can we use irrational numbers in math if we don’t know the full number?
In: 232
A flip side to many of the good answers that have been given here
One angle to consider is in what ways do we use irrational numbers?
A lot of the answers here are discussing it’s use in theoretical, or pure maths, an area that likes exact definitions. If you want to know more about that, and why something like pi is usable without the decimal representation, you’re looking for “Group Theory”
In essence, we can use these irrational numbers because the way we express them has defined rules of mathematics that work the same as the rules that we have for decimal numbers.
The flip side is when we want to use maths practically. Say you’re an engineer, and you want to build a bridge. How many decimal places of pi should you use? If you use too few, will the bridge be at risk of breaking?
With only 40 decimal places of pi, you can calculate the circumference of the universe to within the width of an atom.
So in terms of application, our estimates do not need to be exact, since a difference of 0.0000000000000000000000000000000000000001 meters, doesn’t make that much difference
Decimals are only one way to represent a number. They aren’t even a good way of doing it, most of the time.
If we don’t know all the decimal places of a number..m that’s just because decimals are stupid. Representation the number a different way instead.
We do know the ‘full number’, and using them for practical purposes just requires us to get it ‘close enough’ to be within tolerances.
To calculate the circumference of the universe down to the accuracy of the diameter of a hydrogen atom?
37 decimal places of pi is MORE than enough
https://www.jpl.nasa.gov/edu/news/2016/3/16/how-many-decimals-of-pi-do-we-really-need/
So while we don’t know the EXACT number, we know that we don’t need it to be exact for any practical work
This applies to all irrational numbers, there becomes a point where it doesn’t really matter. We’re within the width of a hydrogen atom, why do we need more?
1/3 is an irrational number. We know what 1/3 is. This is simply a limit of having a base 10 system. Nature isn’t base 10; it doesn’t always fit into this arbitrary system that we created.
We usually can get rid of them later eg. squaring square roots, dividing by the same irrational number etc. You can think of it as letters in your equations.
They also represent various truths in geometry like the Pythagorean theorem, area and volume equations, pi etc. Diagonal of a square is a√2 and area of a circle is πr². It just is. We can leave it as is or calculate an approximations.
There’s two answers here:
**Arithmetic**: When you’re calculating something use traditional digits (1, 2, 3, etc), you don’t need to be perfectly precise.
Even NASA only needs about 15 digits of Pi to put a rover on Mars. It doesn’t take very many digits to have *basically* perfect precision.
**Mathematics**: For most irrational numbers, we do *know* the full number. We know how to calculate the exact number to any digit into infinity. We just cannot represent that full number in decimal form. This is a limitation of how we write numbers, not a limitation to how we *understand* numbers.
π (aka Pi) is the notation for the exactly precise number of Pi. You might lose precision when writing it out as digits (3.1415…). Losing precision doesn’t mean we don’t know the number.
It’s kind of like if you have to draw your own face. You *do know your face*. You recognize it, you would immediately recognize an imposter, even if it were very close to your face. But some methods of representing your face (a picture) are much more accurate than others (a drawing).
For many, many applications, you don’t need to have a decimal form of an irrational number in order to use it.
Here’s an easy example. I have Circle A with a radius of 1. I want to know what is the radius of a Circle B with a diameter exactly twice as big.
When I go through the math of that, I’ll find I don’t need a single digit of Pi. To double the circumference, I double the radius (Pi cancels out).
I don’t understand this subreddit lmao I thought you were supposed to explain things to people like they’re 5 but all the replies are in depth answers using big words that’s not how you’d explain things to a 5 year old lol
In general math very rarely makes statements about individual numbers. Instead it works with “lets say we have a set of objects that have this and that property and we show that some member of this set now has this and that property.”
When we do have to work with individual numbers we can give them a name and then make use of the fact, that each real number can be characterised by at least one sequence of rational numbers converging onto it. We just give whatever this sequence is converging to a name.
For example to find PI you can use something like [https://en.wikipedia.org/wiki/Leibniz_formula_for_%CF%80](https://en.wikipedia.org/wiki/Leibniz_formula_for_%CF%80) to get arbitrarily close approximations for it.
If you do numerical calculations you can then in general find some rule that goes like this: “If we use this and that approximation by rational numbers our result will be less them (some other rational number) apart from the the actual result. And this is enough in virtually all case math is put to a practical use.
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