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2

You'll notice throughout the development of mathematics, and dependent branches such as physics, the use of mathematical notation increases more so as the field becomes more advanced. For instance, Isaac Newton's Principia is a renowned book on the early development of physics and mathematics, and yet you will find it hard to find someone who has actually ...


1

There are a couple of different ways to think about matrix multiplication. Here is what I usually do. To multiply two matrices $A$ and $B$, you should see $A$ as a stack of rows and $B$ as a sequence of columns. I.e. think in the following way: $$ A = \begin{pmatrix} a_1 \\ a_2 \\ \vdots \\a_n \end{pmatrix} ~~~~ B = \begin{pmatrix} b_1 & b_2 & ...


1

You certainly do not need calculus. You need some basic algebra. There are a few things in basic geometry that you show be thoroughly aware of: The number $\pi$ is the ratio of circumference to diameter of a circle. For example, a circle whose diamater is $1$ foot has a circumference of $\pi$ feet, i.e. about $3.14159\ldots$ feet. And $2\pi$ is the ...


1

Graph Theory: the complete graphs; the complete bipartite graphs; trees; the Petersen graph.


1

I suggest you study some Math history to see that people speculated about all sorts of things (they still do), and then they either lived with their speculations or they tried to prove them to be true. Even the most famous 'Mathematicians' have claimed things that were later proved false. I also suggest you ask yourself whether your claim that people ...


5

A typical way to find a formula is to start (and do practical work) with some (more or less) well known formula. Then you come into the unlucky situation that you have to extend it a bit, which often means hard work (brute force, laborious manual work, tricking a computer algebra system to help you). Then you do some guesswork based induction to get a ...


3

You might want to try the book "Birth of a Theorem: A Mathematical Adventure" by C├ędric Villani. There, he describes how he developed one of the greatest theorems ("formulas") of mathematics. In general, you can't generalize it. Nowadays, you can even use computers to find "formulas", such as the classic Four color theorem. You often have a goal you want to ...


23

How do mathematicians find formulas? Short answer: Observation, creativity, and hard work. A formula I personally came up with: As other answers have already addressed the triangle formula, I thought I would share a personal example of observing something, being creative, and then working hard to prove a formula I came up with (I'm sure others have ...


6

I thought fit to... explain in detail in the same book the peculiarity of a certain method, by which it will be possible... to investigate some of the problems in mathematics by means of mechanics. This procedure is... no less useful even for the proof of the theorems themselves; for certain things first became clear to me by a mechanical method, although ...


4

First note that the equation for a straight line is $$ f(x)= mx+b $$ For simplicity, let $b=0$ and let $mx\geq 0$. Note that now $f(x)$ passes through the origin and we have $$ f(x)=mx $$ Now let's find the area under $f(x)$ from $0$ to $k$, where the distance between $0$ and $k$ represents the base. Also note that $f(k)=mk$ gives us the height of the ...


2

This is what I personally think: They derive it and express them as a function of some variables and constants. How do they derive? By step-by-step calculations/operations from already known axioms, theorem, etc. or may be from the flow of thoughts which strike their minds and then they prove it later using already known axioms, theorems, etc. How was the ...


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There are many ways in which a mathematician or scientist can create a formula. Sometimes they use proportionality. For a triangle with a fixed base you can observe that the area of the shape is proportional to the height. In other words if you double the height you double the count of unit squares in it. You can follow the same argument by fixing ...


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For the particular case of the area of the triangle, here is one way to rationalise it: Decompose the triangle of interest into a purple piece and a red piece. Replicate the red piece, colour that replicate blue, and rotate it and attach it like above. Do the same to produce the green piece. Then, the area of (red piece + purple piece) is $\frac{1}{2}$ ...


5

Mathematicians are interested in combining postulates and axioms to yield new theorems or axioms. For instance, the area of a square is axiomatically the square of its length ( though this can be proven), and if we draw a diagonal across any square we will half the total area into two separate areas. And, these two separate areas are the areas of a right ...


1

Finite groups: The cyclic groups; the dihedral groups; the symmetric groups; the alternating groups; the quaternion group.


2

I agree that you'll be fine if you go straight into manifolds before curves and surfaces. But it is simply NOT true (as stated in another answer) that surface is of little interest in modern different geometry, in particular in area related to PDE and analysis. In case of curves, that is indeed not so interesting as all 1 dimensional objects are locally ...


8

You won't miss much by dropping curves and surfaces: every important article I studied, browsed or heard about published in the last 60 years in differential geometry by such luminaries as Thom, Milnor, Atiyah, Hirzebruch, Perelman,...contains little or no reference to curves and surfaces. On the other hand if you spend your time on Codazzi equations, ...


1

Honestly, for the most part you can even skip reading most of the book. Problems are millions of times more important. When you get a new book, don't even read it. It doesn't matter. Go directly to the problems, skipping everything in the book. Don't even glance at chapter titles. Go directly to the first problem. Obviously the problem probably won't make ...


0

Well, considering the fact that the limits of sequences of discrete objects - and one can very well define limits of graphs, of logical models, of sets and power sets of stuff, of groups and rings so on - are typically continuous if they have meaningful definitions and if they exist at all, then yes, you'll most probably need to have a reasonable command of ...


22

Eric's answer is quite thorough, but it is missing an important point: in mathematics, everything tends to be related and intertwined. You might think that doing abstract algebra will get you far from analysis, but you would need to restrict your scope a lot more to do that, as there are many things relating the two, just to name a few: Group theory and ...


25

To me, asking if you need to understand analysis is roughly* like asking "Is it necessary for one to understand how to operate a computer to pursue a career in mathematics?", in that the answer is technically no, but Everyone else does They'll assume that you do too There's no good reason not to know By not knowing, you are making things incredibly ...


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Yes. Analysis, as far as you have seen it, is training on mathematical reasoning, on the meaning and usage of quantifiers, on organizing more or less complex proofs, and such. You will not get far without all of this.


5

Don't just listen--engage in the conversation. If you're talking one-on-one, don't be afraid to say: "Can you explain that part again?" "So let me try to rephrase what you just said." "Can you give me another example?" "I didn't quite get that last point you made." "Hang on, give me a few seconds to ponder that." It's too easy to nod along when you should ...


2

Take a long walk if possible. I learnt that it helped Andrew Wiles while proving Fermat's last theorem. Hopefully it'll help you too.


3

Originally started as a comment, but I think it really serves as an answer: There is no Royal Road for learning mathematics. Of course, the above refers to what Euclid said to King Ptolemy when he (Ptolemy) requested advice for an easier way of learning mathematics, much as you are doing here. The point is that there is no shortcut for gaining ...



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