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21

Yes, mathematicians do a lot more besides writing proofs! While some already pointed out that “they also have lunch” I assume that you are more interested into the intellectual processes involved in the mathematical activity. When I¹ examine my intellectual processes, I can see that some exercise my intuition and some exercise my rationality. My intuition ...


17

One motivation, if you can call it that, is that $i^2=-1$ does not define $i$, because $-i$ also satisfies that equation. So, there are two elements that could be $i$ and there is no algebraic reason for choosing one over the other. In other words, $\pm i$ are interchangeable, hence conjugation. Technically, interchangeable means that there is an $\mathbb ...


16

Literally all math you've ever done are "proofs". You just get more rigorous about it. "Solve the equation $2x+5=0$" really means "prove that there exists a solution to $2x+5=0$ and give an expression for it". Doing math is just doing logical reasoning with certain rules. In higher math, it is more explicitly logical and rigorous but honestly IMO the ...


12

Mathematics is about understanding things. Proofs are one part of this. The bulk of the time is spent asking questions, working out examples or doing calculations, figuring out what others have already done, positing conjectures/hypotheses, and thinking about ideas for why something should or should not be true. There are many famous unsolved problems ...


12

A calculus exam is designed to test whether you know calculus, not its prerequisites. Your professor probably assumes you are well-versed enough in arithmetic and algebra that you will not make substantial "simple" errors. If you know the calculus,[*] then, but you're consistently making errors in arithmetic and algebra on the exam, then I'd suggest you need ...


8

It's mostly the latter: the study of number theory from an algebraic viewpoint, just as analytic number theory is the study of number theory from the viewpoint of analysis. With algebraic number theory, it is often easier to solve equations that would be more difficult if not impossible with elementary methods. Algebraic number theory often deals with these ...


8

That depends. If by "realistic" you mean something that has to do with physical reality, then I defy you to come up with a set which has exactly $200^{200^{200}}$ elements. If by "realistic" you mean something which comes up naturally in mathematics, then $\Bbb R$ is an uncountable set. As for explaining the difference between them? That's not very easy, ...


7

(The following is not meant to be serious mathematics.) In decimal there are $1000$ three place numbers. The probability that at the $N^{\rm th}$ decimal place of $\pi$ the last three figures enounce exactly the number $N$ therefore is ${1\over1000}$, and the probability that this does not happen is ${999\over1000}$. Assuming independence of the involved ...


6

I must disagree with claims that "Algebraic Number Theory" is an algebraic study of anything-whatsoever, possibly including number theory, or, possibly "numbers", whatever the reference may be. That is, in genuine practice, it is "the theory of algebraic numbers", including "algebraic integers", including $p$-adic methods, including complex variables ...


6

Always show enough work on your exam so that it is clear, when you make an error, what sort of error it is. Writing clearly makes it easier to grade, but it also makes it easier for you to check later. At heart, if you are making a lot of stupid mistakes, either (1) you don't really understand the steps, and are just applying rote thinking, or (2) you ...


5

1) "Maybe because this area is very recent?" No, algebraic geometry goes back at least to Descartes and Fermat and is essentially as old as calculus. 2) "Because there aren't many buyers to buy algebraic geometry books?" No, that's irrelevant. Authors don't care, since they know very well that any book beyond the level of linear algebra or calculus won't ...


5

It is the study of number theory from an algebraic viewpoint. The methods of algebraic number theory are used to solve many problems in number theory. For example, the study of Gaussian integers sheds light on problem of which prime numbers are the sum of two squares.


5

I'm going to start with your example and work towards a more abstract notion of structure throughout this writing. So let's see, the bijection you give is a function $f:A\rightarrow B$. But all we have are the sets $A,B$. No other information is given. So what does the bijection encode? Well, both sets have $3$ elements. Perhaps that is what we should look ...


5

Talking from my own experience (about 3 years of mathematics at university). I'd like to make few separate points, which somehow connect together, but the connection might be different for different people. High school I was (sometimes am) the kind of person, to whom proofs were something boring (at the high school). Somehow, math has that kind of ...


4

The reason is also to acquire inverses and do division $$\frac{z}{q}=\frac{z\bar{q}}{q\bar{q}}=\frac{z\bar{q}}{|q|^2}$$ Where you get $(a+bi)(a-bi)=a^2-(bi)^2=a^2+b^2$


3

If $f(x)$ is a polynomial with real coefficients, and $z \in \mathbb C$ is a root of $f$, then $\overline{z}$ is also a root of $f$; in other words complex conjugation acts on the roots of $f$, and we can separate the roots of $f$ into orbits according to this action. An orbit is either a root with $z = \overline{z}$, i.e. a real root, or a pair $\{z, ...


3

Consider the Wikipedia page for Algebraic Number Theory in other languages: Théorie algébrique des nombres Algebraische Zahlentheorie Teoria algebrica dei numeri Teoria algébrica dos números The exception that proves the rule is Spanish: Teoría de números algebraicos which starts by acknowledging the other form: "La teoría de números algebraicos o ...


3

Not everyone was taught what you say. I was not, for example. I was never taught how to write expressions with exponents in-line, so I never found out what the canonic meaning of x^a+b actually is. What I was taught is that whenever there is some confusion and there may exist two ways of interpreting an expression, I should use parentheses. And that is ...


3

In university departments, mathematics is usually grouped with the physical or natural sciences. The physical or natural sciences usually include physics, chemistry, mathematics, statistics, computer science, etc.


2

For Group theory, I can highly recommend "Visual Group Theory" by Nathan Carter. He does a great job delivering a good intuition not only of the concept of a group, but also of different group-related definitions, different groups... (Like the title suggests; in a visual way) http://web.bentley.edu/empl/c/ncarter/vgt/gallery.html For Ring theory, what ...


2

To better grasp the behavior of complex functions, in the first place you must master the behavior of the classical ones, like the powers ($x^d$, for all $d\in\mathbb R$), the polynomials, the exponential, the trigonometric functions and their inverses. You must know by heart their range and their domain, their behavior at $\pm\infty$, their asymptotes and ...


1

It's about not taking theorems for granted. I'm not a mathematician, but it was quite a realization for me when it hit me that a lot of the things we "found" in high school were... complete magic at that point. For example, the Rational Root Theorem and Descartes' Rule of Signs... did you ever wonder why those work? Or did you--like me--just think "Cool!" ...


1

Very intuitive proof that: $$\sum_{n=1}^\infty {(-1)^{n-1}}\frac{1}{3^{n-1}} = \frac{3}{4}$$ (from this web page)


1

Not only are the answers from the Mathematics StackExchange community "more intuitive and instructive," they're much more valid than anything you will find on Wikipedia. Although it's true that a lot of the "community" here are also active on Wikipedia, their talents and insights are mostly wasted over there. There is much tighter control here than there. ...


1

For full disclosure, I am hardly an expert in this area (indeed, I'm learning from Fulton, just like you), but I thought I'd throw in my two cents. First, a recommendation for a "really beginner book:" Ideals, Varieties, and Algorithms by Cox, Little, and O'Shea. This is one of the friendliest math books I've ever read, and the emphasis on computation and ...


1

Well I'm not a programmer, but I'm pretty sure that you need math to program like the air you breathe. Especially if you want to become very, very good at it; I mean if you want to understand algorithm running time, do analysis on the complexity of a specified algorithm or trying to find a new one who performs better, you really need to be confident in ...


1

We don't know if space is infinitely divisible, but if it is, then it has uncountably many points, because if $(x_n)$ is any infinite sequence of points, and if a sequence of regions of space $(S_n)$ is constructed recursively so that $S_{n+1} \subset S_n \setminus \{x_n\}$ ($n = 0, 1, 2, \ldots$), then this nested sequence contains at least one point that ...


1

If you accept that one can form infinite sequences from say the set of symbols $\{a,b\}$, then I would say that the set of all such sequences gives a fairly nice example of an uncountable set; as shown by Cantor's diagonal argument. This may be simpler than the uncountability of the reals, or of the interval $[0,1]$ as you don't have to worry about some ...


1

My professor always called them the shoe-sock theorems or the likes, the idea basically being that you put on your socks first then the shoes, but to do the reverse you must get off the shoes first and then the socks.


1

Considering the dimensions of the various matrices shows that reversing the order is necessary. If A is $m \times p$ and B is $p \times n$, AB is $m \times n$, (AB)$^T$ is $n \times m$ A$^T$ is $p \times m$ and B$^T$ is $n \times p$ Thus B$^T$A$^T$ has the same dimension as(AB)$^T$



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