What's the hardest mathematical thing you know?

[Scotty123]

Haha, I only sound like a pure mathematician because what's being discussed is very pure (and I have to do it in my degree - so I don't have a choice in the matter). I actually much prefer applied maths (I was going to do aerospace engineering, before I was stupid and accepted an offer to do straight maths)
.

I was doing straight maths originally, but missed the more applied aspect of it so switched to maths and physics. Was a good choice! Aside for experimental work in the first year. I've always enjoyed physics from a mathematical point of view and could appreciate the maths behind the physics that the pure physicists weren't really aware of - 'here is a formula, plug the numbers in and there you go!'.

Electromagnetism - vector calculus. Hamiltonian/Lagrangian mechanics - just variational calculus basically! It all tied in so well.

 

[lukeharvest]

I was doing straight maths originally, but missed the more applied aspect of it so switched to maths and physics. Was a good choice! Aside for experimental work in the first year. I've always enjoyed physics from a mathematical point of view and could appreciate the maths behind the physics that the pure physicists weren't really aware of - 'here is a formula, plug the numbers in and there you go!'.

Electromagnetism - vector calculus. Hamiltonian/Lagrangian mechanics - just variational calculus basically! It all tied in so well.

I have 50/50 split between pure and applied - next year I can make that more like 80/20 to applied (or pure). Pure is definitely easier if you enjoy spending hours going over theorems and proofs (which I definitely don't) whereas the struggle in applied is obviously understanding what on earth is going on, much more enjoyable IMO.

 

[Dave]

What is it you are doing then? I'm joint with the mechanical engineering department, although it's far from engineering really. The supervisor there is originally a mathematician (as is often the case in engineering actually, especially for fluid dynamics).

Fluid mechanics, mostly of the computational variety but also some theory development as well. Even though fluid mechanics has a very strong link to mathematics (especially at Cambridge, with the likes of Stokes, Lighthill etc. being former Lucasian professors), the academics in our department mostly have engineering backgrounds. It was a similar story at my old university (Sheffield) too - although as a chemical engineer we tend to attract more chemists and a broader range of academic fields than say electrical or mechanical.

 

[Scotty123]

I found pure more intellectually satisfying. I enjoyed applied more.

I think pure is more an 'art' and has a certain beauty to it. I also somehow did better in my exams than in applied (I class myself as an applied mathematician!).

Even the 'purest of pure maths' has an applied use really.

 

[Castiel]

I don't really get what you are talking about here, but a mathematical proof holds independently of the universe we exist in.

If there is a proof that 1+1=2 then that proof would hold even if the earth never formed, humans never evolved and philosophers never argued over semantics. That proof will always explain why the answer is what it is.


To give you a different example, the irrational constant PI exists in the universe and is completely independent of human discovery. All proofs and properties surround the use of PI are hold true in all worlds. Other intelligent aliens will have found the constant PI as well and will also have associated proofs, merely are numbering systems would differ. This is not something learned form experience or a fictitious idea derived from the human mind, it is a fundamental mathematical property.


People always seem to think that maths is some how related to the real world and you can simply translate 1+1 to be 1 orange + 1 orange. That reasoning is completely hopeless.

I was thinking more along the lines, not so much of why 1+1=2, but why it took so long to prove that it was and why did it take two of the greatest mathematicians so long to do so and does it not all depend on how you define 1 to begin with.....also some nutter mentioned Kurt Gödel's Incompleteness Theorem which he said demonstrated that sometimes there are propositions that cannot be proven one way or the other using axioms and rules.

Now I am not a mathematician, neither do I profess any great laymans knowledge either, but surely there is a way to explain this to a reasonable intelligent person without resorting to reams of maths or simply dismissing them.

 

[Scotty123]

Fluid mechanics, mostly of the computational variety but also some theory development as well. Even though fluid mechanics has a very strong link to mathematics (especially at Cambridge, with the likes of Stokes, Lighthill etc. being former Lucasian professors), the academics in our department mostly have engineering backgrounds. It was a similar story at my old university (Sheffield) too - although as a chemical engineer we tend to attract more chemists and a broader range of academic fields than say electrical or mechanical.

Ah same as me then. I am doing detonation, so it's essentially the Euler equations with the addition of reaction rates. I'm doing numerical stuff. Currently writing a paper at the moment. Might be finished in 2020 by this rate .

 

[lukeharvest]

I was thinking more along the lines, not so much of why 1+1=2, but why it took so long to prove that it was and why did it take two of the greatest mathematicians so long to do so and does it not all depend on how you define 1 to begin with.....also some nutter mentioned Kurt Gödel's Incompleteness Theorem which he said demonstrated that sometimes there are propositions that cannot be proven one way or the other using axioms and rules.

Now I am not a mathematician, neither do I profess any great laymans knowledge either, but surely there is a way to explain this to a reasonable intelligent person without resorting to reams of maths or simply dismissing them.

Maths is maths, from the first line of any proof - to the end. It's rather inexplicable in any other language than maths really.

 

[Castiel]

I'm not assuming anything, someone who thinks 1 orange + 1 orange = 2 oranges is a proof of 1+1=2 does not know what they're talking about.

I don't think any such thing. I was not asking to explain why it wasn't a proof, I am not an idiot, I understand what a mathematical proof is, I was careful to make sure that was clear in my post....All I wanted was an explanation regarding the proof available and why it is considered such and what assumptions are made.

I can understand your frustration, I can empathise somewhat when trying to explain complex etymologies relating to my own field to some people, especially on a forum, but I don't dismiss them so haughtily, I try to give them some sense of what it means and give them some source material if they want it. In all fairness you probably do not intend to come across that way, and I know that I can do so also on occasion, just don't assume that someone will not understand something, and give it a shot, or point them in the direction of someone that can....some peole may just surprise you, they do me all the time.

Maths is maths, from the first line of any proof - to the end. It's rather inexplicable in any other language than maths really.

D.P has given me a link anyway, and I have found several sources that give a reasonably easy explanation:

The proof starts from the Peano Postulates, which define the natural
numbers N. N is the smallest set satisfying these postulates:

P1. 1 is in N.
P2. If x is in N, then its "successor" x' is in N.
P3. There is no x such that x' = 1.
P4. If x isn't 1, then there is a y in N such that y' = x.
P5. If S is a subset of N, 1 is in S, and the implication
(x in S => x' in S) holds, then S = N.

Then you have to define addition recursively:
Def: Let a and b be in N. If b = 1, then define a + b = a'
(using P1 and P2). If b isn't 1, then let c' = b, with c in N
(using P4), and define a + b = (a + c)'.

Then you have to define 2:
Def: 2 = 1'

2 is in N by P1, P2, and the definition of 2.

Theorem: 1 + 1 = 2

Proof: Use the first part of the definition of + with a = b = 1.
Then 1 + 1 = 1' = 2 Q.E.D.

Note: There is an alternate formulation of the Peano Postulates which
replaces 1 with 0 in P1, P3, P4, and P5. Then you have to change the
definition of addition to this:
Def: Let a and b be in N. If b = 0, then define a + b = a.
If b isn't 0, then let c' = b, with c in N, and define
a + b = (a + c)'.

You also have to define 1 = 0', and 2 = 1'. Then the proof of the
Theorem above is a little different:

Proof: Use the second part of the definition of + first:
1 + 1 = (1 + 0)'
Now use the first part of the definition of + on the sum in
parentheses: 1 + 1 = (1)' = 1' = 2 Q.E.D.

I assume that is accurate? It seems pretty straight forward if it is......

 

[rz30]

Did a degree in maths years ago, embarrassed to say i can't remember much of it, i think number theory was quite difficult though.

 

[Rroff]

I'm not particularly good at maths but as a programmer with a keen interest in video game development I've become quite capable at handling vector maths (getting my head around dot products especially took some doing), using matrices, quaternions, basic trigonometry, basic ballistic maths, etc. for me this is quite an achievement as it doesn't come naturally.

 

[D.P.]

I don't think any such thing. I was not asking to explain why it wasn't a proof, I am not an idiot, I understand what a mathematical proof is, I was careful to make sure that was clear in my post....All I wanted was an explanation regarding the proof available and why it is considered such and what assumptions are made.

I can understand your frustration, I can empathise somewhat when trying to explain complex etymologies relating to my own field to some people, especially on a forum, but I don't dismiss them so haughtily, I try to give them some sense of what it means and give them some source material if they want it. In all fairness you probably do not intend to come across that way, and I know that I can do so also on occasion, just don't assume that someone will not understand something, and give it a shot, or point them in the direction of someone that can....some peole may just surprise you, they do me all the time.



D.P has given me a link anyway, and I have found several sources that give a reasonably easy explanation:



I assume that is accurate? It seems pretty straight forward if it is......

Such a proof using Peano arithmetic is good for laymen (it is also incredibly useful in other theories) and is the limit of my mathematical ability to prove something like 1+1, but it is still a relatovely high level of maths and relies on the Peano axioms holding true. The lower level proofs in various set theory paradigms etc. will have more rigorous proofs on why some of the axioms holds. Peano axioms only really define the natural numbers and Even then, only if the axioms are consistent. Reading a little history on the topic, it was Poincare who provided this caution, and Hilbert who formalized the problem for proofing their consistency. Godel famously showed that proving this consistency was impossible only with Peano arithmetic, part of the so called incompleteness theorem.

Godel's incompleteness theorem is of critical important in maths and computer science ( it sets limits of information processing). But it is often misunderstood. It is more related to the existence of paradoxes. Godel famously said "This sentence is false", which is a paradox and unprovable as true or false. Another very important example is with Cantor's power sets of sets, given a set that contained absolutely everything then the power set of that set contains every permutation of elements, except by definition that power set must be a subset of the set of everything, otherwise it wouldn't be a a set of everything. A similar concept is the Barber of Seville, who shaves every man who does not shave themselves- does the barber shave himself or not, it is a paradox.

 

[Nitefly]

Does 1+1 always =2? one orange + one orange=two oranges.....but one drop of water + one drop of water = one drop of water albeit a larger one.

But that's because there's no uniformality in what a droplet of water is, size or otherwise... Surely?

Was this a late night 'I must argue about something!' set of posts?

 

[Greebo]

1+1 = 2

Proving this.

Well I used to be able to when I did Maths at Uni, couldn't do it nowadays though.............

 

[anything I don't mind]

I also did XY graphs with algabra and basic statistics, not sure if that is before or after algebra and fractions on geometry.

 

[Mason-]

Differential equations. Proofs matricies and 3D vectors. Not really that hard I know, just up to the level of Maths + Further Maths A-Level. I have forgotten a lot of it because I now do a non-maths related degree.

 

[TommyV]

1kg = 2.204lb

Probably the most complicated thing I know mathematically.

 

[Col_M]

I never fully got to grips with Fourier & LaPlace transforms - but that's more because of the way we were taught them which was in a pure mathematical sense, with no concept of real-world application being passed on to us as learners.

Similar issue with me, I just didn't click with the the way my lecturer taught us.

 

[Castiel]

But that's because there's no uniformality in what a droplet of water is, size or otherwise... Surely?

That was kind of what I was implying, that anything depends on a certain application of assumed criteria to define them.

Was this a late night 'I must argue about something!' set of posts?

Not really, I was just curious why it was the case, I was a bit put out about the nature of the responses at first (I don't think that 'you don't or wouldn't understand is really constructive, and is a little insulting) but DP came good in the end and along with a bit of google foo, I now have a better understanding of the why and so hopefully do some other members......

 

[oli356]

Well I was doing a Cisco test yesterday and had to convert 231 to binary. There were 5 possible answers, my answer was different. I ended up working out the answers in decimal.

I came home and tried the conversion again... I found my problem,
I couldn't do: 103-64= I kept getting 49 -.-
Words could not describe how I felt lol... On the brighter note I did pass the test about subnetting and NAT

So the hardest mathematical thing I know is probably 1+1!

 

[Stinger53]

i always found Differential equations and intergratoin fairly simple. i always hated stats cuz of the way they worded the questions in exams