How do we assign rational value to numbers like pi and infinity, numbers that mathematically have no rational ending, if our brains cannot rationalize these numbers? The concept alone is as irrational as the numbers…
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> How do we assign rational value to numbers like pi…
For pi we can approximate.
For example, we can say pi is about 3.14159.
3.14159 is a rational number, and is close enough to pi to be good enough for almost any purpose we might need it.
Infinity is a different thing entirely; in normal number theory we cannot approximate that as a rational number or really do anything with it at all. Infinity is not an irrational number, it isn’t a number at all (depending on which area of maths we are in).
It is also worth remembering that “irrational” and “rational” are just labels, with specific mathematical meanings. Like “real” and “imaginary”, they are arbitrary terms that happen to have stuck through use. There is nothing irrational in the normal sense about “irrational numbers.”
> How do we assign rational value to numbers like pi
We don’t. When we say pi is equal to 22/7, we’re just approximating.
> and infinity
Infinity isn’t a number. It doesn’t have a value.
> if our brains cannot rationalize these numbers?
What does this mean?
You don’t. It very rarely matters in practice that pi has infinite digits. A few dozen will get you calculations accurate enough to be within a speck of dust sized error on a circle the size of our galaxy. If you just give it a name like pi, or x, or a, or whatever, you can just use that, the math will check out fine without you writing out the digits.
> The concept alone is as irrational as the numbers…
You better get used to it, because they outnumber rational numbers. In fact, in the set of real numbers (rational+irrational numbers), the rational numbers make up 0% of the total.
“Irrational” in mathematics doesn’t mean “impossible to rationalize”. It means a number that cannot be written as a *ratio* of two whole numbers (*integers*). For instance, the number 0.5 is a rational number: it’s 1/2.
Pi is an irrational number because it cannot be written this way. There is no pair of numbers *a* and *b* so that a/b = pi. Some pairs get close. For instance, 22/7 = 3.14285…, whereas pi = 3.14159… But you can never get it exactly, and this has been proven (but there’s no ELI5 explanation of this proof).
However, pi as a concept is perfectly reasonable. It is the ratio between the circumference of a circle and its diameter. (Wait, a *ratio*? Doesn’t that mean it *is* rational? Well, no, because to be rational it has to be a ratio of *integers -* not just a ratio of any two numbers. Any circle with an integer diameter (e.g. 4) does not have an integer circumference, and a circle that has an integer circumference does not have an integer diameter.) This ratio is fixed, and its value is what we call “pi”. And we can even compute this value to a high degree of precision – just not with infinite precision.
There are certainly concepts in math that are hard to wrap your head around. Infinity is one of them. However, in addition to the point I made at the start about the definition of “irrational”, you have to accept that just because *you* cannot fathom something, that doesn’t mean other people – specifically mathematicians – are equally uncomfortable with it, or that it is fundamentally impossible to think about logically.
we dont. **ratio**nal number means “a **ratio** of 2 integers” (like 1/2) not “a number we can rationalize about. all numbers can be rationalized about, thats kinda their thing, but some numbers like pi can never be expressed by any ratio of integers so we call them “ir**ratio**nal”.
We might use 355/113 or 3.141592654 as an approximation of pi, but pi its self can only truely be expressed as an infinite decimal. conveniently, numbers like pi and e are also able to be expressed as infinite but predictable sums (like Pi=4*(1-1/3+1/5-1/7+1/9-1/11….) ) that we can use to calculate them to any precision we feel like.
But this brings us on to infinity. infinity is NOT a number. when ever a mathmatician talks about infinity they are always cairful to say things like “as x approaches infinity” or “the number of elements is infinite”, but they never streight up “number equals infinity”
Rational numbers are defined as numbers that can be written as the ratio of two whole numbers. Some numbers cannot be written like that. These are irrationals. We need to define an irrational number with something different. Like an infinite sum that adds up to a certain value or a square root. We call them irrational because they are not rational, the name itself holds no more information. How do we assign rational values to numbers like pi? Its an approximation, any number can be approximated with a rational.
You don’t seem to understand what “irrational” means in the context of numbers.
It just means a number that cannot be expressed as a *ratio* of two integers. A side-effect of this is that if you try to write such a number as a decimal, it will never terminate or fall into a repeating pattern.
But this doesn’t mean the numbers aren’t real (both in the mathematic and the literal sense) or that they are infinite, e.g. pi is definitely within 3 and 4, and we can even describe where it is to an arbitrary level of precision, but we can’t write it in a neat closed form.
Pi is actually an irrational number!
In the mathematical sense “rational number” simply means “a number that you can express as the *ratio* of two integer numbers” (like 2/3, 1/100, etc.) while an “irrational number” is simply “a number that you cannot express as the *ratio* of two integer numbers”.
Once you understand the meanings of those definitions, you can see that “rational/irrational” has nothing to do with whether the number can be “rationalized” or “reasoned” with. In fact, a good many “irrational numbers” are “algebraic numbers” that represent the solutions to an equation like x*x-2=0 which produces the irrational algebraic number √2; these are numbers that we can very much work with in a logical manner, despite them being called “irrational”.
Irrational means that the number cannot be represented as a fraction, it does not mean that the numbers are harder to understand. π is the ratio between a circle’s circumference and diameter. This is true regardless of the size of the circle, but π cannot be represented in the form of x/y. √2 and e are also irrational. Both numbers are very important, and we know a lot about both of them, but neither can be represented as x/y
Irrational numbers effectively means you can’t represent the number with a finite number of digits using our decimal numeration system (not the literal definition). √2 is a perfectly acceptable representation for an irrational number, but you can’t represent it using decimal numeration. Is √2 “impossible for our brains to rationalize”?
If you really wanted, instead of using a base-10 number system we could use a base-√2 number system and then √2 would equal 10 exactly.
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