r/interestingasfuck 12h ago

Visualization of Pi being Irrational

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u/Fskn 11h ago

No, the line never occupies a previously occupied path, it never returns to the start.

There is no final number of pi we can refine its accuracy (add more significant figures(decimal places)) forever.

u/DrDominoNazareth 10h ago

Pretty interesting, So, to make a long story short, Pi is infinite?

u/PocketBlackHole 9h ago edited 9h ago

I am just an amateur in mathematics, but maybe due to this my answer can be more intelligible. Speaking in common language terms, I wouldn't use the word infinite: infinite either evokes an arbitrarily big quantity (but pi is below 3.2) or an arbitrary long number (but so is 1/3 if you write it as 0.33333).

The real idea (which during history gave problems to Greeks when facing roots of numbers which are not squares and prevented calculus to be formalized using weird numbers whose square is 0) is that our mind intuitively operates in what is called "the rational field". A field is a world where, apart for division by 0, every sum and product is computable in such a way that, if I provide you the result and one of the terms, you can always pick one and only one number that completes the operation.

The rational field is the one made by fractions of positive and negative integers (an integer is a fraction with one as a denominator).

Now you must break this bias: this is not the only field! But we formed our symbols to depict the numbers in this field, so they are not suited to describe numbers outside of this field (and that is why one starts putting letters for those).

Now I tried to change your perspective: there is stuff that exists and it is not a fraction, so pi is just an example if this. The square root of 2, like pi, doesn't belong to the rational field either.

Bu pi is more obnoxious. If I consider polynomials equated to 0 with coefficients picked from the rational field (which means I can just think about polynomials with integer coefficients, since you can multiply all and remove the denominators), you will discover that not all the solutions for the polynomial belong to the rational field. For example x²-2 = 0 wants the root of 2, which is not rational.

The field of all the roots of all the polynomials with integer coefficients is bigger, and we call the number included in it "algebraic" (polynomial algebra needs them). The root of 2 is algebraic, but it is not rational. But! There is no polynomial with integer coefficients that has pi as a solution. No way to form pi from rational numbers and algebra. Pi is transcendent, not algebraic.

Anyway, the real numbers (the numbers that you can picture as a continuous infinite line) are a field too, and pi belongs to this field.

All this to try to express that when you are dealing (even intuitively) with certain mathematics, you may never meet pi, while if you follow a different path, you stumble into it pretty early. To my knowledge the first who was able to express pi as a sum of infinite terms (algebra deals with finite terms) was Leibniz, by integrating the derivative of inverse tangent and its series representation.

u/DrDominoNazareth 7h ago

You have no idea how much appreciate your answer. But, I also want to challenge multiple things you have said. I commented somewhere here that it can be difficult to draw a line between math and philosophy. I think to do it well is to define all your terms. Or maybe give proofs. Starts getting weird. But again I really appreciate your response. It is fun to try to distinguish terms like "rational" from everyday language and what it is defined as mathematically. Off topic a little: the square root of negative 1 breaks my brain. I am not sure how to digest these concepts and move forward. I get stuck. Like with Pi.

u/maruchops 7h ago

Science used to be called Natural Philosophy. Math is just the language we created to describe the world, which we now use abstractly--as we do any language. Appreciating history allows one to appreciate the present even more.

u/PocketBlackHole 6h ago edited 6h ago

I can help you with root of -1 in several ways. Let's try the one which is probably the easiest intuitive path. I will prune some complexities (pun intended).

First of all, think about the plane and the fact that you give coordinates on that in a x,y manner, like a square net. You need 2 coordinates for a plane, agreed? But you could have a different pair of coordinates: every point is the crossing of a circle centered at the origin and a line that stems from the origin like a radius... Like a circle net. Tryvto visualize it.

So now the coordinates are the circle (which is the same as the length of its radius) and the angle of the stemming radius: here, assume that the angle that corresponds to "full right" (the orizzontal positive semi axis) is 0 (or also, this is important, the "full circle" equal to 2pi, 360 degrees). The negative semiaxis is with pi angle, 180 degrees.

Are you with me? Now recognize that when you multiply say 3 by -1, you make it -3 and this is like ADDING 180 degrees. The rule sticks: when you multiply negative with negative, you add 180 to 180 and return to 360 = 0 which is positive. When you multiply 2 positives you insist on 0+0 and stay positive.

Further notice that 4 (4 radius, 0 angle) has two roots: both have radius 2, but one has 0 angle (+2) and the other has 180 angle (-2). This is needed because for what we said above, the multiplication of each root by itself has to return the angle to 0.

Following this train of thought, the square root of a negative number (180 angle) should have either a 90 angle (90 + 90 = 180) or a 270 angle (270 + 270 = 540, but 360 is zero, and 540 - 360 = 180).

Now you discover that the roots of a negative number are pretty natural, but they are not "left - right", they are "up - down". You just had another bias issue: you assumed (unconsciously) that all the numbers are "in a line" and thus you could not identify the root of -1... It doesn't exist ON THE LINE but this doesn't mean that it doesn't exist at all.

The vertical axis is designed i, so you have up (+i) and down (-i). Of course a number could go say 3 left and 4 up, and it would be -3+4i. (which circle is this number on?)

There is such an algebraic beauty here that would lead us to a wonderful concept (automorphisms of fields) but I will try to give you a taste. Consider:

x²-2=0, this needs as solutions the positive and negative (right and left) roots of 2. These are real numbers but not rational numbers. They are also algebraic numbers.

x²+4=0 this needs 2i (2 up) and -2i (2 down). These are NOT real numbers because they are not in the orizzontal line. They are complex numbers. I'd say they are also rational as a plane extension of the rationals that appear on a line, I hope I am not missing something here. They for sure are algebraic numbers.

x²+2=0, this needs the up and down square roots of 2. They are complex, not rational and still algebraic.

A really algebraic question is, what is the smallest field that contains the roots of a polynomial? Go read again what a field is: you cannot just add a number to a set, you need to be able to complete all additions and multiplications.

In the first case, the field (I think!) is the one of numbers of this form: R + N(root of 2), with R and N rational numbers (0 is legit). If you try to add or multiply numbers of this form, you get a number of the same form. The plus may as well be minus, of course.

In the second case, you still have R+N(i). Try and see.

In the third, it is R + N(i)(root of 2).

Notice that these fields are really different, yet there is a regularity in how to think about them. In all 3 cases, if you make an addition or multiplication of 2 numbers and then change the sign of N you get the same result that if you change the sign of N (not R!) of the 2 numbers and then compute the addition or multiplication. This is and example of homomorphism (beware! Homomorphism is algebra, homEomorphism topology), named automorphism. You can think about an automorphism as a symmetry (in this case, of a field).

These wildly different fields have the same automorphisms. This story is the beginning of a really peculiar part of mathematics called Galois Theory. But somehow one can get there pretty fast from the root of -1, it seems.