How do I know if basic algebra works?

This may be a very silly question, but I've been wanting to ask it for a while so here goes. I've been reviewing basic high school math in preparation for analysis, and I find myself questioning a lot of things - especially basic algebra - to the point I'm going a little mad.

For instance, polar coordinates can be related to rectangular coordinates like so: $$x=r \cos\theta$$, $$y=r \sin\theta$$. So, $$x^2+y^2=1$$ can be rewritten as $$r^2=1$$, so $$r=1$$. And surely enough, it's the same graph. But why does this work? I know it algebraically makes sense but it just feels a bit odd to me. I guess a broader version of this question is: I've been taught that if something is algebraically derived/true, it must produce the right answer. But how do I know that other than saying, "if you broke zero algebraic rules, it holds." The manipulation of these variables is just odd to me. Maybe this feeling will pass.

Another example:

$$\iint 1-x^2-y^2\,dx\,dy = \iint 1-r^2\,rdr\,d\theta$$

This algebraically holds since $$x^2+y^2=r^2$$ but how do I know it works other than saying "algebra works"?

I'm definitely overthinking this but it keeps nagging me.

• – J. W. Tanner Jul 21 at 22:24
• I did a short video on this sort of idea. I don't know if it will help. If it does, I can write it up as an answer. It basically shows how to do Algebra without numbers. youtube.com/… – johnnyb Jul 21 at 23:33
• If you're not sure why two representations of the same thing are equivalent, it may be because you don't know what thing they're representing. Your first example uses the Pythagorean equation to identify all right triangles with hypotenuse 1 and a non-right corner at the origin, modulo a rotation to fix one side on the x-axis; the other equation shows that the other non-right corner must lie on a circle of radius 1 and centred at the origin. Try actually drawing the triangles and then the circle over top. More generally. – Nij Jul 21 at 23:44
• The way mathematics is historically taught (at least in the US) focuses on developing familiarity with basic concepts and techniques of arithmetic, algebra, and geometry. It isn't until undergraduate level mathematics that emphasis is placed on rigor and proof. You may benefit from "Calculus" by Spivak, or Apostal, or an equivalent text. These books begin the process of logically justifying all the steps you've been taking that you're now unsure about. Your Analysis class will hopefully also address this... – Ben Jul 22 at 0:04
• In short, it's good you feel this way. Get ready for much more satisfying, pleasurable, fascinating mathematics. Get ready for fun! – Ben Jul 22 at 0:05

Your question is not silly. Some aspect of it drove hard-core mathematicians totally crazy in the 20'th century. People like Hilbert, one of the greatest, never quite got over it; a guy called Gödel actually proved that it is impossible to create a "perfect" kind of maths and was the big destroyer of dreams. You can read up on the fascinating history of that, maybe starting with Wikipedia's article on the Foundations of Mathematics.

But back to your current situation of being at the end of school education and wondering whether it all really works.

In maths, this happens the same as in all other complicated fields:

First, you are learning some aspects which are probably at least partially, more probably massively simplified, and not deduced from first principles in a thorough manner. This does not give you a complete insight into the topic, and basically summarizes school-level education.

This kind of introduction gives you a good idea what the topic is about, and may make you familiar with some very basic concepts. And with basic I do mean basic. This is the same as talking about a few of Einsteins famous thought experiments - this gives you an idea what Relativity is about, but gives you zero insight on how it actually works (i.e., the maths behind it).

Also, this gives you a chance to detect whether maths, in this case, interests you and whether you want to go deeper. One may argue that this could be the main function of school-level education; you probably do not need your integrals for everyday life, ever. It certainly does not give you a chance of really knowing that it all is correct, there is a lot of belief involved.

When actually studying maths at the uni level, at least when I did it some decades ago, the education process is totally different. Back then, we started from first principles (e.g., with the construction of numbers with the Peano Axioms; with formal ZF set theory; with Rings and Groups and such). Starting from literally nothing (all of these basically start out with the empty set as the only object), you build up the whole of maths.

This is founded in formal logic (i.e., concepts like "for all", "there exists", "and", "or" and so on). In my experience, this was introduced relatively intuitively in my maths courses - the symbols were displayed and quickly explained, and that was that; I also studied computer science, where you formalize it excessively in the context of "lambda calculus" and similar areas; and it is also a big topic in classical philosophy.

All the little steps in these deductions are, individually, dead simple, very easy to understand, require only the axioms and every previous deduction, and are in themselves "atomic". So much so, that when learning every little piece for the first time, you constantly ask yourself what this madness is all about.

The benefit is that as it is so simple, there is never a point where you really doubt that it is "true" and "works" (for some definition of truth and "working", which is not an easy topic at all) - it is better to think in terms of consistency. That is, is all of what you're doing and learning consistent with the basic axioms you found your maths on (which there can be different sets you can choose, but usually it's all founded on the Peano Axioms and ZF(C) set theory).

All operations you do with your mathematical objects have been proven by some mathematician, based on simpler operations. If you so wish, you can buy volumes of books where you can trace back all your questions right back to the original few axioms in excruciating detail.

How to get an intuitive understanding, which is required to work forward (i.e., to find new proofs, or even new areas of maths) is a totally different beast, and probably best left for another question...

I know your feel, just remember that for all of us who learn mathematics, we've had encountered concepts that are difficult to understand. I would say that if you are not sure if your reasoning is correct, then it is not because you don't understand what's going on (even though your proof might be correct "by accident"). This feeling will pass as you manipulate the concepts and do exercises, you'll earn an intuition of what's going on and you will know if your proof is correct. For instance the formula $$\iint(1-x^2-y^2)dxdy=\iint (1-r^2)rdrd\vartheta$$ is a particular case of $$\iint f(x,y)dxdy=\iint f(r\cos\vartheta,r\sin\vartheta)r dr d\vartheta$$ It is simply a change of cartesian coordinates to polar coordinates, $$dxdy$$ being change to $$rdrd\vartheta$$ (you can have a look at physics books with drawings to understand why it works). Anyway, the main advice I'd give to you is : let time do its work and do your exercices.

You've got two good answers. Here are some points about polar coordinates that seems not to be addressed, but that consistently bother at least some students in ways that sound similar to your doubts.

The equations $$x = r\cos\theta$$ and $$y = r\sin\theta$$ define a mapping from the plane to itself, which we might express $$P(r,\theta) = (x, y) = (r\cos\theta, r\sin\theta).$$ Often we speak of mapping "the $$(r, \theta)$$ plane" to "the $$(x, y)$$ plane".

These coordinates are not intrinsically associated to the planes, but in the context of working with polar coordinates it's efficient to fix the meanings of all four variables. Note carefully that $$(r, \theta)$$ are Cartesian coordinates for the domain of $$P$$, and are polar coordinates for the $$(x, y)$$ plane.

The mapping $$P$$ is surjective or onto: For every point $$(x, y)$$, there exists a pair $$(r, \theta)$$ such that $$(x, y) = P(r, \theta)$$. Loosely, every point in the plane has a set of polar coordinates.

By contrast, the mapping $$P$$ is not injective, or not one-to-one: There exist distinct sets of polar coordinates that map to (or "represent") the same $$(x, y)$$. It's a good exercise to work out precisely when two points in the $$(r, \theta)$$ plane map to the same $$(x, y)$$, partly to review trigonometry, and partly because I think this underlies the unease motivating the question.

We have $$P(0, \theta) = (0, 0)$$ for all real $$\theta$$, while if $$r \neq 0$$, then $$P(r, \theta) \neq (0, 0)$$ for all real $$\theta$$. For every integer $$k$$ and every real $$r$$, we have $$P(r, \theta) = P(r, \theta + 2k\pi)$$ and $$P(r, \theta) = P(-r, \theta + (2k+1)\pi)$$. Conversely, if $$P(r_{1}, \theta_{1}) = P(r_{2}, \theta_{2})$$, then either $$r_{1} = r_{2}$$ and $$\theta_{2} - \theta_{1} = 2k\pi$$ for some integer $$k$$, or $$r_{1} = -r_{2}$$ and $$\theta_{2} - \theta_{1} = (2k+1)\pi$$ for some integer $$k$$.

Why should this be bothersome? For one thing, the equations $$r^{2} = 1$$ and $$r = 1$$ are not equivalent for real numbers $$r$$. Well then, why should they define the same locus, the unit circle $$x^{2} + y^{2} = 1$$, in the $$(x, y)$$ plane? Because without constraints on the angle $$\theta$$, each of the lines $$r = 1$$ and $$r = -1$$ "traces out under $$P$$" the entire circle. (To be more formal, we can express this using images of sets under a mapping.)

It may be instructive to sketch the images under $$P$$ of segments of the form $$(r, \theta)$$ with $$r = 1$$ and $$0 \leq \theta \leq \pi/2$$, or with $$r = -1$$ and $$0 \leq \theta \leq \pi/2$$. We get visibly-different curves: It's only when we let the angle range over an interval of length at least $$2\pi$$ that the difference ceases to be visual, and we recover the entire locus $$x^{2} + y^{2} = 1$$.

Another worrisome property of polar coordinates arises if we ask, "What are the polar coordinates of a point $$(x, y)$$?" This is a sneaky trap: By using the article the, the question implicitly suggests each point $$(x, y)$$ has unique polar coordinates, when in reality every point $$(x, y)$$ has infinitely many distinct sets of polar coordinates.

My suspicion is, if you (beginner, or future reader) closely examine your unease with polar coordinates, "sketchy" claims will originate with different sets in the $$(r, \theta)$$ plane that map to the same set in the $$(x, y)$$ plane when you're trying to discern something by looking only at the $$(x, y)$$ plane.

More generally, understanding the polar coordinates mapping in detail clarifies abstract concepts, including images and preimages of sets, surjectivity and injectivity of mappings, and one-sided inverses; this example justifies giving these properties special names and expending time on them in analysis and elsewhere. (In fact, the humble polar coordinates mapping concretely touches on more subtle concepts, such as covering maps and factoring one mapping through another, lifting and sections of fibre bundles, simple-connectedness, and de Rham cohomology!)

• Great answer. Thanks so much! – beginner Jul 22 at 18:26