## Discovering Mathematics

• 40
Hi,

I didn’t do too great at school. Found it boring and was removed at 14. Tough upbringing. Managed to educate myself quite well but avoided mathematics. Now I’ve always understood that mathematics is a thing of beauty and I’d love to get started. Any advice where to start theoretically and practically?
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Youtube videos on math topics. Don't expect to understand them entirely; enjoy them, and see where they take you. Also, likely you can find a used college textbook on algebra for not a lot of money - maybe 500 pages. Good to have as a reference and educational tool, and if it has good problem sets, entertaining as well. Two such books will likely provide excellent triangulation on ideas that in one book may seem opaque.

Get started, then go with it. In the process you will learn some math, but more important at the moment, about mathematics. You will learn something about the possibilities.
• 131
Nobody knows the extent to which a population of consenting and willing people can be taught mathematics. It's a mystery! However, I do feel that a person's response to the exercise below might be a good indicator of whether they will get on well with maths. If you can't do it then it might be that you just need to be shown the answers to a few examples. If you still don't "get" it then maths is not for you right now!

Exercise: Bernice is three times as heavy as Sandip, but if Sandip was 60 kg heavier then they would be of equal weight. Write down two equations that express this information (no need to solve them even!)

A correct answer is in the next post....

.
• 131
B = 3 * S
B = S + 60
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:)
• 131
Thanks @Wallows!
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• 40

Ok. It’s not about the athrimetic is it, almost, it’s similiar to deductive logic. Ha the funny thing was I (kind of) pictured the equation. I didn’t quite visualise it. But I had it in my mind abstractly. It existed and it looked similar. In fact it’s much more like logic than arithmetic and now I look at it i can see how exciting it is. What do you think. Should I persevere? Is studying Euclid the answer as Banno suggests?
• 40

I also love the example and thanks very much for that.
• 40

At first I was like

60- ? Then trying to add up
Bernice’ fat arse... lol thanks dude.
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60- ? Then trying to add up
Bernice’ fat arse.

B-60 = S will do just as well for the second equation (second piece of information)!

B-60 = S is the same information as B = S+60.

Notice how you can take -60 to the "other side" as long as you change its sign (in this case from minus to plus). But that is all about the rules of manipulating equations, which become intuitive after a while. The fun is to be had in generating the information in a mathematical form in the first place. I personally don't think it is worth putting people through learning how to manipulate and solve equations if they can't generate this information from situations; also I suspect they would struggle with manipulation anyway.

Ok. It’s not about the athrimetic is it, almost, it’s similiar to deductive logic.
Yes, I think you're right. Let me know if you would like to see some more examples similar to the first :)
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Is studying Euclid the answer as Banno suggests?
Whatever you "get" and enjoy!
• 40

I’d like some more equations.
• 131
OK!
All one equation only and let's cut straight to algebraic "names" as follows. A person's name isXXX. Some assumed/stated value about them will appear as X . So if I say XXX is one year older than YYY, and X and Y represent the current ages of XXX and YYY, then the equation is

X = Y + 1.

Note that the exact form of the equation will depend upon the assumed meaning of the algebraic letter. Sorry about that little complication but you need to know that you need to be clear what each variable represents before using it to declare information.

a)
assume M is MMM's age now
assume J is JJJ's age now
In 5 years' time MMM will be three years older than JJJ

b)
assume M is MMM's age now
assume J is JJJ's age now
In 5 years' time MMM will be twice as old as JJJ

c)
assume M is MMM's age now
assume J is JJJ's age now
4 years' ago MMM was 3 years' younger than JJJ

d)
assume M is MMM's age now
assume J is JJJ's age now
4 years' ago MMM was twice as old as JJJ

e1)
assume M is MMM's age now
assume J is JJJ's age now
In 7 years' time MMM will be 20 years older than JJJ is now

e2)
assume M is MMM's age now
assume J is JJJ's age in 7 years' time
In 7 years' time MMM will be 20 years older than JJJ is now

f1)
assume E is EEE's age now
assume X is XXX's age now
2 years' ago EEE was 3 years younger than XXX is now

f2)
assume EEE is EEE's age now
assume X is XXX's age 2 years' ago
2 years' ago EEE was 3 years younger than XXX is now

f3)
assume E is EEE's age 2 years' ago
assume X is XXX's age now
2 years' ago EEE was 3 years younger than XXX is now

f4)
assume E is EEE's age 2 years' ago
assume X is XXX's age 2 years' ago
2 years' ago EEE was 3 years younger than XXX is now

f5)
assume E is EEE's age 4 years' ago
assume X is XXX's age 11 years' in the future (yeah I know wtf??!!!!!)
2 years' ago EEE was 3 years older than XXX is now

g1)
assume A is AAA's age now
assume B is BBB's age now
5 years ago AAA was 3 times older than BBB will be in 2 years' time

g2)
assume A is AAA's age 5 years ago
assume B is BBB's age in two years' time
5 years ago AAA was 3 times older than BBB will be in 2 years' time

Answers in next post, Good luck!
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OK here goes .. hope I've not made many mistakes (easily done if you lose concentration!)

a) M+5 = J+5+3

(equivalent to M = J+3, making explicit the fact that once 3 years older, always 3 years older!
not so with respect to ratios though . You are only twice as old as your younger sibling once in your life)

b) M+5 = 2 * (J+5)

c) M-4 = J-4-3

d) M-4 = 2 * (J-4)

e1) M+7 = J+20

e2) M+7 =J-7+20

f1) E-2 = X-3

f2) E = X-2-3
in case you are having difficulty at this point , that equation was generated by breaking down the information as follows
2)2 years ago EEE was aged E
2)2 years'ago XXX was aged X-2

f3) E-2 = X-2-3

f4) E = X -3

f5) E+2 = X-11+2

g1) A-5 = 3 * (B+2)

g2) A = 3 * B
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This is amazing!
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Yep. Best toy ever. But I'm stuck on Eta1: sum of the area of two squares.

Now I know there is a simple way of seeing the answer, but my intuition or imagination or whatever has failed to see it so far.

That's the core of mathematical thinking.
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I breezed up to doubling an angle. Now I'm stumped.

I'd add that sometimes the 'simple way'; the guiding intuition; isn't always accurate, as it's quite common to discover that real structural symmetries or logical relationships between mathematical objects don't mirror the imaginative background you have about them - which usually translates to either assuming too much or to little.

Two examples I have of this are: you want to prove that a proper subset is always smaller than its superset, is the nesting of two circles in a Venn diagram a proof?
spoilers
It's not. One assumption is that the sets lay in the plane, generic sets do not. Another is that you have two specific sets, rather than generic ones. However, trying to figure out what additional assumptions on the types of shape you can 'nest' for which the possibility of applying this procedure of nesting one within the other is a proof is an interesting journey if you've done some real analysis or topology
.

The second is is that when teaching introductory topology, you generally require that the student has seen elementary real analysis (continuity and convergence of functions and sequences) so that they understand metric spaces (with their metric topology), which are the go-to intuition to provide for a topological space.
spoilers
There's an interesting connection between the 'imaginative background' for sets suggested by Venn diagrams and the fact that metric topological spaces aren't generic topological spaces. This intuition misleads in the first example but is useful in getting a feel for the second, before breaking down that feel under the boots of qualitatively different structures and pathologies.
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the guiding intuition; isn't always accurate,

Yes, and I think the answer to the area of squares will have to come from a bit of algebra. But it feels like cheating.

SO I might broaden my point to say that the appreciation of mathematical aesthetics is clearer in geometry. I don't mean that the diagrams are pretty, Although they are. I mean the feeling of being perplexed and then the joy of seeing the solution.

Much like ducks and rabbits, I think.

So for @Dan84, the lesson might be just to enjoy mathematics.
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Yes, and I think the answer to the area of squares will have to come from a bit of algebra. But it feels like cheating.

Interesting, I just got the double angle one by applying the 'equal arc length swept over by a radius => equal area swept out in the circle' calculus intuiton, which also felt like cheating.
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the joy of seeing the solution.
Without algebraic/written proof, you could be seeing a mirage ....
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Not so with Euclidean geometry puzzles! You end up thinking in much different terms from algebra operations; the choices in constructions you can make are from a much broader space, and the constraints of the problems have to be cottoned onto in a different way. What algebraic operation will tell you that bisecting an angle is a poor initial step in dropping a perpendicular from a point to a line?

With algebra, you preserve the numerical equality of the sides of the equation - whatever moves that do that are allowed. With Euclidean geometry proofs, you're not conserving the equality of sides of an equation, you're trying to make an ordered sequence of interacting (properties of) shapes which exhibit a target property (or shape). The 'exhibition' occurs within the drawn picture, rather than in the last line of x =...
• 131

Fair enough, there are steps to take and blind alleys to go down. It's not so much that one "sees" the solution but that one arrives at it - albeit with insights on the way. My point is that most solutions to most maths problems are not intuitive overall, and geometry is no "beautiful" exception!
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Indeed. It all requires work, and different areas require developing different competences. Intuitions at one stage can become theorems at another; just as what once was a theorem can become an intuition. Moreover, intuitions can impede as well as guide the resolution of a problem, just as much as they can facilitate or stymie the development of precise conceptual problems. From this vantage point, learning mathematics is riddled with joyous revelation and shameful folly in equal measure, the same as learning any other skill.
• 7.2k
Now I’ve always understood that mathematics is a thing of beauty and I’d love to get started. Any advice where to start theoretically and practically?

Consider the OP.
• 3.4k

I'd never let a being on topic get in the way of a fun post to write.

Cool videos on calculus, loads of animations to give a good picture rather than lots of proofs and manipulating equations.

Huge repository of lecture series going from basic arithmetic to advanced calculus, including tangents on applied mathematics

Channel devoted to exam style questions for later year high-school math (16-18), Khan Academy (second link) usually has the same content as this. I chose PatrickJMT as his videos are really clear and concise, and excellent exam prep.

With regard to actually learning mathematics, I'd suggest Khan Academy as a primary resource. The videos typically have a 'general explanation, specific problem' format. Often with more than one example problem. The best way I've found to use their videos is to watch the concept being introduced, follow through their first example in detail - do so until you think you understand the idea -, then try the second example given. If the video doesn't have a second example, Google usually provides one by using the key words used in the introduction, the title of the video or both at once.
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Consider the OP.
The OP expressed an interest in finding out about the "beauty" of maths as well as maths itself.

Graph theory might be a branch of maths that might be interesting to a novice - R J Wilson's book is a very readable introduction (he is Harold Wilson's son don't you know...)
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Graph theory
??
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In my experience graph theory concepts are very difficult to explain to applied researchers and very numerate people. They only seem simple in principle when you've grokked the idea of an arbitrary binary relation.
• 131
Wilson claims his book is readable by non mathematicians. They could skip lots out but still get an appreciation.

Zone filling games on phones lend themselves to a nodal description. Graph theory can be used creatively to describe what's going on.

So @Dan84 how did you get on?

I would love to set those exercises to the adult population at large to see what percentage get it. I'm thinking 10-20% at a guess. And then, could it be intensively and successively taught? I tend to think not. But I would love to know. The results should be used to determine curriculum policy in schools. "No getty no mathsy" ... I'm not being elitist, I just think many young people could spend their time more productively. Also they may "get" it later in life, especially if they haven't been cruelly put off when younger.
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