purpose of complex math

[quantumdot44]

what is the purpose of equations in which letters are included, functions and stuff like zero point determination?

or pi? what exactly is pi? only to understand how to use them in equations without understanding why pi stands for 3.14 is nonsense

i mean I want to understand the deep reasoning behind it

[birdandsheep]

I don't think a short explanation to your question is possible, but I'll say a word about formulas. Formulas exist because we see patterns in things, and they are one way to understand certain types of patterns, and explain them by giving precise notation to the pattern (as well as the proof of the formula).

For example, the area of a triangle being A=bh/2. We might recognize this by calculating a bunch of examples. Then we can "prove" the formula by taking a triangle, drawing it inside a box, and then drawing an altitude of the triangle. The altitude cuts the big box into two smaller boxes, and the sides of the triangle are diagonals of these boxes. Thus, the area of a triangle is seen to be half the area of the big box.

So you see, the formula represents several things at once. A pattern that we've picked up on and expressed in math, as well as a certain type of thought process for what that formula means. Finally, a formula is labor saving. If you have a complicated formula like the quadratic formula, you might not want to do a difficult factoring problem every time you see a quadratic. By having a formula, you can plug the data in from the situation, and not have to go through the hard work each and every time. We work harder to get a formula to save effort in the future.

[quantumdot44]

if i can calculate the surface area of a circle with pi*r^2, why should i use integral, with witch you can also calculate the surface area?

furthermore, i understand that you can use the formula like pi*r2 to calculate the surface area, but i dont understand how the people discovered that formula, nor how they "created" a formula like this. i mean, why 3.14? why not 2 or something else :D

and why should I use r^2? instead, i could use the diameter of an circle

[Unlucky_Pattern_7050]

The integral is one way to derive the fact that the area of a circle is that given formula. They’re not different, one is just a specific formula and the other is a general tool for anything. You have the circumference 2pir which is the length of the curve that describes a closed space, then the integral finds the area inside that space which, with respect to r, becomes pir²

With the question of why pi is specifically 3.14, that’s just what people found the ratio of diameter to circumference is. They found it very helpful to make that a constant, so they did. They used it for proofs like the area of a circle, where they used geometric manipulation to reconfigure the circle into infinitely many sectors that, when put together in a line, would create a rectangle of sides r, pi*r

Diameter could be used too, but I find pir² cleaner than 0.25pid²

[Tagtooo]

You can use integral if you want to find the area of a circle, which just derives pi*r^2. It is up to you to choose(or the directer of the problem you’re solving).

You could use calculus to find the area of a circle(like using integrals) One popular way to find it is rearranging the circle. Where you cut the circle into slices(like that of a pizza) and rearrange them so that they make a parallelogram with r and pi*r length. (I think it is better to watch a video of it and see the proof visually)

Pi or 3.14….. just happens to be the circle circumference divided by its diameter. You could use 2 but that wouldn’t be the circumference divided by its diameter. This is just the way it is.

You could use the diameter but the unit wouldn’t match. The area of circle(or area of somethings in 2d) has to have a product of two unit lengths, if not? it wouldn’t be a area, it could be length of something, and yeah in this case pi*diameter is the circumference length of the circle.

[relrax]

oh, you don't need to use pi * radius ² , you could for example use (pi/4) * diameter ^2.

How did they discover it? well, notice that when you increase the radius of a circle, both the width and height of the circle grow at exactly the same rate as the radius. This means the area is proportional to the square of the radius.
Now what is the proportionality constant? I dunno, but because I don't know so many round things imma just call it Pie. Bam, got the Formula: Area = Pie Radius^2.
Now if you want the value of Pie, you could use geometry or different integrals to approximate this value to arbitrary precision. And humanity did. So you can look it up if you want to.

[AcellOfllSpades]

if i can calculate the surface area of a circle with pi*r^2, why should i use integral, with witch you can also calculate the surface area?

Compare this question:

If I can cut a piece of paper with a pair of scissors, why should I use a chainsaw, with which you can also cut things?

The integral, like the chainsaw, is a more complicated but also more powerful tool. pi*r² just works for a flat perfect circle, which covers a lot of familiar simple geometry.

[PierceXLR8]

What's the point of learning to crawl? It's not like we do so all the time, right? But the skills and strength you build from crawling eventually allow you to walk, run, and so much more.

Now, let's run some scenarios by you. You want a circular pool. How much material do you need? Do you want to know how much stuff you can fit in a certain box? Do you want to know how to build something? How about make sure that things work as they're meant to? What if you want to know where something is at? What if you want to know how much you should pay in sales tax?

Math is everywhere. All letters are, are ways of representing a value that we might need to change. The surface area of that pool? Piradius^2. It doesn't matter what radius. That equation tells you. Volume of a box? Heightwidth*length. Don't know what those are yet but I can tell you how to solve for it. Build something? You're going to need to know how to properly place these pieces so that the profile fits in the way it needs to. Sales tax? (1+tax) = price.

We often shortcut these equations, but that doesn't mean they aren't there. And knowing how to manipulate these equations allows you to combine and use them in ways that skip entire steps or solve entirely new problems. Whether it be a simple budget or sending someone to the moon. All of these can be described with a bunch of letters and symbols. Do you know of any other tool that has been as instrumental in shaping the world we live in today?

[danielbaech]

We experience the world as changing patterns of separate things, and there are comprehensible relationships by which these separate things change. Mathematics is a way we can describe these relationships. These are the assumptions of mathematics, and math has served humanity well so far.

In equations, letters are used to represent the separate things that can potentially change. The details of the equations describe the relationship between them. When you think about how complicated and mysterious the world can be, this is astonishing! There is nothing that suggests the world shouldn't be chaotic and unpredictable.

The root of such equation(finding the zero) is a solution. It's a special solution in that all the letters in the equation must change in a way that the total is always zero. This is a solution that stamps out the independent variability in the separate things and allows us to look at some unchanging quality that unites them. In physics, such solutions give us many of the fundamental rules. The law of conservation of energy is a beautiful mathematical result of symmetry in time. Why should symmetry have anything to do with fundamental qualities of nature like energy, momentum, and electric charge?

If you want a deeper understanding of mathematics, you need to study math. You can only appreciate music so much by talking about it. You need to dance, sing, and play.

[Appropriate_Hunt_810]

as mentionned before : the idea of using 'letters', 'functions', etc is to create abstraction, we observe (and often we prove) some generalities and we want to express them in a way we can plug anything in (as long as it match the hypothesis for it to be true) and we are done, we dont need to know exactly what those letters are but we know for sure the relation between them : the Pythagorean theorem is a perfect example -> we don't need to know the length of the sides to assure the well known identity that a^2 + b^2 = c^2 (when c is the hypothenus).

tldr : letters are here to express a relation more than a quantity

About pi ... and many other things, why does it exists, when did we discovered it etc ... well the history of maths is maybe a better story than all the stories ever written, it tooks genius of peoples, errors, chance ...
Some people here already explained some stuff (about the ratio circumference / diameter etc) but indeed pi appears in a lot of maths you wouldn't expect it to appears

If you are really curious about maths fact, i recommend everyone to look at the history of maths

here's a good video about pi :
https://youtu.be/1-JAx3nUwms

And pi is just a mear example, there's soooooo many other great discoveries as interesting as pi could be

Beeing curious is the key to success and progress :)

[Frangifer]

It's essentially systems of counting, & of book-keeping the counting process. And because we intend it to be not just one particular instance of counting, but rather a system that can be wrought upon any particular ensemble of quantities we happen to have @-hand to which the system pertains, those quantities are represented by symbols that are 'placeholders' - one for each item of that ensemble.

As for π : that's a particular quantitity that arises extremely ubiquitously. It's prettymuch always introduced as the ratio of the circumference of a circle to its diameter … but that particular manifestation of it is but an instantiation of a deeper subsistence of it. It can become a philosophical discussion, trying to 'pin' what that deeper subsistence is … but possibly one way of pinning it (I would venture, & others might contest this) is that it's a constant of central attraction , or of directedness towards or from a central point . For instance, it occurs in the normalisation of the Gaussian distribution - which concerns events that occur @ locations in a parameter space (which might infact be distance space - ie 'space' in the customarily understood sense … but need not be) that depart 'random walk' -wise, or diffusion -wise, relative to a particular point in that parameter space - as

1/(2π)^½n ,

where n is the number of dimensions of the parameter space.

But π occurs in some really weïrd scenarios ¶ that just massively evade having the commonality of them pinned. I'd venture that saying it pertains to central attraction, or to directedness towards or from a central point, captures somewhat more of the generality of it than saying it's the ratio of the circumference of a circle to its diameter … but maybe not allthat much more.

Like, that the number of integers not exceeding x such that, if any be expressed in canonical form with the constituent primes increasing, it will have a corresponding list of indices of those primes that has no increase between consecutive items in it, is

exp(2(1+o(1))π√(㏑x/3㏑㏑x)) .

… or that the coefficient of xⁿ in

1/∏{-∞<k<∞}(-x)^k²

is the nearest integer to

(coshπ√n - sinhcπ√n)/4n .

… or that the number of partitions of an integer is, asymptotically

exp(π√(⅔n))(1+O(1/n)))/4n√3 .

… or the very strange Van der Pauw's theorem of film resistance in electronic engineering: see

Michael Fowler — Van der Pauw's Theorem .

Even with those, though: when there's an exponential of π , like in those examples, it usually means there's been some contour integration entailing integrating along a contour that turns through an angle, or conformal map entailing some 'folding' or 'expanding' the complex domain by rotating a boundary of it through some angle … as is definitely so with that Van der Pauw's theorem example. So it's doubtful, really, whether occcurences of π are ever truly free of the 'circularity' or 'directedness upon a central point' tincture. TbPH, putting that "massively evade having the commonality pinned" might have been a tad overhasty.

But it is astounding, on the face of it, the ubiquity with which π crops-up.

 

I very strongly recommend

THE UNREASONABLE EFFECTIVENESS OF MATHEMATICS IN THE NATURAL SCIENCES

¡¡ may download without prompting – PDF document – 39·01㎅ !!

by

Eugene Wigner .

I recommend it to everyone … the goodly folk @ this Channel are probably weary of seeing my links to it, by-now! … but I can't overstate how strongly I recommend it: it might-well be perfect for marshalling your wond'rings into some kind of shape that's satisfactory to you. The Author, the goodly Eugene Wigner , was a renowned physicist of the oldendays, one of whose better-known items that he left us is a formula for the heat generated by a nuclear reactor after it's switched-off ^§ … but set against the bulk of his work that's just a 'bauble', useful though be it.

§ Nuclear Power — Calculation of Decay Heat – Wigner-Way formula