Really basic stuff that was never proven in class

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Really basic stuff that was never proven in class

Postby mmmcannibalism » Fri Aug 20, 2010 3:06 pm UTC

Just a thread to discuss basic proofs for things that are often taught at some level in school without a proper proof. For instance, foil multiplication of polynomials is taught in algebra 1, but at least in my school it was never properly explained why(beyond you multiply each term in the first by each term of the last).
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Re: Really basic stuff that was never proven

Postby lemma » Fri Aug 20, 2010 3:37 pm UTC

Oh, that's awesome :D If only I taught the younger kids so that I could use it with them!

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Re: Really basic stuff that was never proven

Postby Dason » Fri Aug 20, 2010 3:39 pm UTC

Our teacher just showed us it as a basic result of the distributive property:

Code: Select all

(a+b)*(c+d)                  | (a+b) is just a number so we can distribute
(a+b)*c + (a+b)*d            | Now we can use the basic distributive property
a*c+b*c + a*d+b*d


rearrange if you feel the need. I think the picture your provide gives a good geometric interpretation of why it holds though.
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Re: Really basic stuff that was never proven

Postby mike-l » Fri Aug 20, 2010 3:46 pm UTC

Dason wrote:Our teacher just showed us it as a basic result of the distributive property:

Code: Select all

(a+b)*(c+d)                  | (a+b) is just a number so we can distribute
(a+b)*c + (a+b)*d            | Now we can use the basic distributive property
a*c+b*c + a*d+b*d


rearrange if you feel the need. I think the picture your provide gives a good geometric interpretation of why it holds though.


Yes, but why does the distributive property hold? (Same picture, just with only one row)
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Re: Really basic stuff that was never proven

Postby Kurushimi » Fri Aug 20, 2010 3:52 pm UTC

In order to answer that you have to define multiplication.

And the whole "repeated addition" thing isn't going to cut it. What about fractions? What are we doing when we multiply fractions?

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Re: Really basic stuff that was never proven

Postby mike-l » Fri Aug 20, 2010 3:59 pm UTC

Kurushimi wrote:In order to answer that you have to define multiplication.

And the whole "repeated addition" thing isn't going to cut it. What about fractions? What are we doing when we multiply fractions?


How do you define fractions without multiplication? You generally start with natural numbers, define multiplication as repeated addition, then define negatives and fractions as pairs with equivalence relations.
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Re: Really basic stuff that was never proven

Postby jestingrabbit » Fri Aug 20, 2010 4:18 pm UTC

mike-l wrote:Yes, but why does the distributive property hold? (Same picture, just with only one row)


I think we proved the rationals were a field at some point, but I don't think we ever made the final step to the reals (that step being "arithmetic operations are continuous"). Not teaching field axioms (and matrix arithmetic) is a real deficit of most advanced high school maths educations. You can also use them to prove that (-a)*(-b) = a*b for instance.

There's also a bit of a problem with the diagram if one of A and B is negative, but not the other, though you could rectify it with a case bash.

Probably most people see a proof of pythagoras and a proof of the volume formula for a sphere just not the best proofs of them.
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Re: Really basic stuff that was never proven

Postby Cleverbeans » Fri Aug 20, 2010 4:55 pm UTC

Pretty sure this is covered in Book I of Euclid's elements.
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Re: Really basic stuff that was never proven

Postby Aiwendil42 » Fri Aug 20, 2010 5:37 pm UTC

Pretty sure this is covered in Book I of Euclid's elements.


You've misunderstood the purpose of this thread. It's for basic things that posters were taught, but not given proofs for - not for basic things that have never been proven at all.

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Re: Really basic stuff that was never proven

Postby mmmcannibalism » Fri Aug 20, 2010 6:16 pm UTC

There's also a bit of a problem with the diagram if one of A and B is negative, but not the other, though you could rectify it with a case bash.


It's still easy to demonstrate with geometry

edit--yeah I messed up that picture when I thought about it, the top part is (b-a)(c+d)
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Re: Really basic stuff that was never proven

Postby squareroot » Fri Aug 20, 2010 9:06 pm UTC

Fundamental theorem of arithmetic. :evil: I'd asked why it was true when I was six, I never got a satisfactory response. I got things like "Well, can you think of any examples?" and crap I like that. I had to wait until I got a book of number theory before I actually saw a proof.
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Re: Really basic stuff that was never proven

Postby Seraph » Fri Aug 20, 2010 9:12 pm UTC

mmmcannibalism wrote:
There's also a bit of a problem with the diagram if one of A and B is negative, but not the other, though you could rectify it with a case bash.


It's still easy to demonstrate with geometry

If it's easy then why did you screw it up?

Let B = 2, C = 3, D = 4, A = -1.
(A + B)*(C + D) = 1 * 7 = 7

Your claim:
(A + B)*(C + D) = BC + BD = 2*3 + 2*4 = 6 + 8 = 14

14 != 7

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Re: Really basic stuff that was never proven

Postby Dason » Fri Aug 20, 2010 9:27 pm UTC

Seraph wrote:
mmmcannibalism wrote:
There's also a bit of a problem with the diagram if one of A and B is negative, but not the other, though you could rectify it with a case bash.


It's still easy to demonstrate with geometry

If it's easy then why did you screw it up?

Let B = 2, C = 3, D = 4, A = -1.
(A + B)*(C + D) = 1 * 7 = 7

Your claim:
(A + B)*(C + D) = BC + BD = 2*3 + 2*4 = 6 + 8 = 14

14 != 7


To be fair the formula he has at the bottom just says area. The area of the whole thing would be ((b-a)+a)*(c+d) which does equal bc+bd. If you want to do (b-a)*(c+d) then you just sum the top two squares to get the desired result.
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Re: Really basic stuff that was never proven

Postby cristobal » Fri Aug 20, 2010 11:07 pm UTC

To be really pedantic, no one ever proved to me that the total area of a shape is equal to the sum of the areas of the partition. This is a non-trivial result.

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Re: Really basic stuff that was never proven

Postby dean.menezes » Fri Aug 20, 2010 11:31 pm UTC

squareroot wrote:Fundamental theorem of arithmetic. :evil: I'd asked why it was true when I was six, I never got a satisfactory response. I got things like "Well, can you think of any examples?" and crap I like that. I had to wait until I got a book of number theory before I actually saw a proof.


They were probably trying to get you to learn something.That theorem is absolutely trivial. ;)
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Re: Really basic stuff that was never proven

Postby TiglathPileser3 » Sat Aug 21, 2010 12:00 am UTC

Whenever people define pi to me they say it is "the ratio of a circle's circumference to its diameter." I always thought that it wasn't intuitively obvious that this is a constant.

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Re: Really basic stuff that was never proven

Postby dean.menezes » Sat Aug 21, 2010 12:16 am UTC

Divide the circle into [imath]n[/imath] sectors and make them into triangles. Then take the limit as the number of triangles approaches infinity.
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Re: Really basic stuff that was never proven

Postby letterX » Sat Aug 21, 2010 12:28 am UTC

TiglathPileser3 wrote:Whenever people define pi to me they say it is "the ratio of a circle's circumference to its diameter." I always thought that it wasn't intuitively obvious that this is a constant.


It's not. In particular, there is no such thing as 'pi' in non-euclidean geometries. The ratio of the circumference to the diameter will depend on how big your 'circle' is.

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Re: Really basic stuff that was never proven

Postby Kurushimi » Sat Aug 21, 2010 12:48 am UTC

TiglathPileser3 wrote:Whenever people define pi to me they say it is "the ratio of a circle's circumference to its diameter." I always thought that it wasn't intuitively obvious that this is a constant.


Well, it's intuitive if you remember that all circles are similar to each other and all parts of a similar shape are grown/shrank in the same proportion.

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Re: Really basic stuff that was never proven

Postby squareroot » Sat Aug 21, 2010 1:23 am UTC

They were probably trying to get you to learn something.That theorem is absolutely trivial.

Yeah, you go over to your nearest elementary school and try to prove that to them. And I think the fact that gcd(n,m)=1 -> (there exists r and s such that rn+sm=1) is much less elegant to show - if you can call the proof you gave elegant. Of course it seems trivial now, I just had trouble with the fact that (p does not divide n) && (p divides nm) -> (p divides m). It sound so simple it's silly, but that's just because we naturally think of integers' unique factorization. I actually got a bit of relief when I learned that unique factorization fails on some fields, and that I wasn't just being paranoid/stupid for those seven years of my life.
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Re: Really basic stuff that was never proven

Postby dean.menezes » Sat Aug 21, 2010 2:03 am UTC

squareroot wrote:Yeah, you go over to your nearest elementary school and try to prove that to them. And I think the fact that gcd(n,m)=1 -> (there exists r and s such that rn+sm=1) is much less elegant to show - if you can call the proof you gave elegant. Of course it seems trivial now, I just had trouble with the fact that (p does not divide n) && (p divides nm) -> (p divides m).

That was mostly a parody of how mathematicians call proven theorems no matter how difficult it was to prove it in the first place, which is why I put the smiley.

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Re: Really basic stuff that was never proven

Postby Tirian » Sat Aug 21, 2010 2:14 am UTC

About a year ago, for lulz I undertook proving that multiplication of the natural numbers is commutative. Holy cow. That was a half-page of dense symbolic manipulation even with the assumption that multiplication is associative and addition is associative and commutative. I can appreciate why people take it for granted.

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Re: Really basic stuff that was never proven

Postby WarDaft » Sat Aug 21, 2010 5:09 am UTC

Very rarely (I would venture perhaps even never) has the proof been given that 1+1 does in fact equal 2 before more advanced topics are covered.
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Re: Really basic stuff that was never proven

Postby squareroot » Sat Aug 21, 2010 5:17 am UTC

dean.menezes wrote:
squareroot wrote:Yeah, you go over to your nearest elementary school and try to prove that to them. And I think the fact that gcd(n,m)=1 -> (there exists r and s such that rn+sm=1) is much less elegant to show - if you can call the proof you gave elegant. Of course it seems trivial now, I just had trouble with the fact that (p does not divide n) && (p divides nm) -> (p divides m).

That was mostly a parody of how mathematicians call proven theorems no matter how difficult it was to prove it in the first place, which is why I put the smiley.

Oh, I'm sorry. I was just feeling some old steam toward the people who never actually explained it to me. I didn't catch your humor.

For the record, addition's associativity and commutativity are trivial - inductive reasoning gives commutativity away almost for free, and associativity is also pretty simple.

To prove commutativity... Well, I think I would prove distributive property first.

a*(b+c) = (a-1)*(b+c) + b + c = (a-2)*(b+c) + b + c + b +c = (a-2)*(b+c) + 2*b + 2*c.... then use inductive reasoning and collect terms of b and c, with base cases (0)*(b+c) + a*b + a*c = a*b + a*c and (1)*(b+c) + (a-1)*b + (a-1)*c = a*b + a*c. Once you have that, proving commutativity is a lot easier.
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Re: Really basic stuff that was never proven

Postby silverhammermba » Sat Aug 21, 2010 5:30 am UTC

dean.menezes wrote:
squareroot wrote:Fundamental theorem of arithmetic. :evil: I'd asked why it was true when I was six, I never got a satisfactory response. I got things like "Well, can you think of any examples?" and crap I like that. I had to wait until I got a book of number theory before I actually saw a proof.


They were probably trying to get you to learn something.That theorem is absolutely trivial. ;)

Please tell me that the proof you posted is intentionally obfuscated. There are truly trivial proofs of the fundamental theorem of arithmetic that don't require all of that complexity.

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Re: Really basic stuff that was never proven

Postby antonfire » Sat Aug 21, 2010 6:54 am UTC

Give one.
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Re: Really basic stuff that was never proven

Postby dean.menezes » Sun Aug 22, 2010 9:04 pm UTC

Lemma 1

Let [imath]p[/imath] be prime and a positive [imath]<p[/imath]. Then no positive number [imath]b[/imath] can be found less than [imath]p[/imath] such that [imath]ab \equiv 0 \pmod{p}[/imath].

Proof

If the theorem is false, then we have numbers [imath]b, c, d, \ldots[/imath] all [imath]<p[/imath] such that [imath]ab \equiv 0[/imath], [imath]ac \equiv 0[/imath],
[imath]ad \equiv 0, \ldots[/imath].

Let [imath]b[/imath] be the smallest of all of these so that no number less than [imath]b[/imath] has this property. Clearly [imath]b>1[/imath], since otherwise [imath]ab = a < p[/imath]. Since [imath]p[/imath] is prime, it cannot be divided by b, but lies between two successive multiples of [imath]b[/imath]: [imath]mb[/imath] and [imath](m+1)b[/imath]. Let [imath]p - mb = b'[/imath] so that [imath]b'[/imath] is a positive number [imath]<b[/imath]. Since [imath]ab \equiv 0[/imath], [imath]mab \equiv 0[/imath] and so [imath]a(p - mb) = ab' \equiv 0[/imath]. This implies that [imath]b'[/imath] is one of the numbers [imath]b, c, d, \ldots[/imath] and it is smaller than the smallest of them QEA.

Theorem

If neither [imath]a[/imath] nor [imath]b[/imath] is divisible by [imath]p[/imath], then [imath]ab[/imath] is not divisible by p.

Proof

Let [imath]\alpha, \beta[/imath] be the least positive residues of [imath]a[/imath] and [imath]b[/imath]. By hypothesis, neither of them will be zero. Now if [imath]ab \equiv 0 \pmod{p}[/imath] then [imath]\alpha \beta \equiv 0[/imath] which contradicts the previous theorem.

The fundamental theorem of arithmetic follows by induction.

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Re: Really basic stuff that was never proven

Postby antonfire » Mon Aug 23, 2010 5:37 am UTC

That's just a (not terribly nice) way to redo two sentences in the original proof.

Yes, it's "the important" lemma.

Yes, all the nasty shit in the original proof is mostly to formalize just what "unique prime factorization" even means (something you didn't bother doing at all).

No, what you wrote is certainly not "truly trivial". It is, I would say, slightly more complex than the proof sketched out originally. It just brushes the boring details under the carpet.
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Re: Really basic stuff that was never proven

Postby Spambot5546 » Fri Sep 10, 2010 8:31 pm UTC

So i've been dicking around with some math stuff and i was looking at just why a negative * negative = positive, and negative * positive = negative.

I know that they do, and can prove it algebraically, but not when looking at the actual numbers.

Spoiler of algebraic proof
Spoiler:
[math]a + (-a) = 0[/math]
[math][a +(-a)]b = 0 \ast b[/math]
[math]ab + (-a)b = 0[/math]
Thus
[math](-a)b = -(ab)[/math]


But multiplication is really repetitive addition. a * b is a added to itself b times.
a + a +a...
-a*b is -a added to itself b times.
(-a) + (-a) + (-a)...
a * -b is a subtracted from itself b times
a - a - a...
and -a * -b is -a subtracted from itself b times
(-a)-(-a)-(-a)...

But those last two are wrong. They should be
-a - a - a... and -(-a)-(-a)-(-a)... respectively.
But i can't find any reason to justify why (besides, i guess, the fact that that matches the algebraic proof). If a * -b is a subtracted from itself b times then why must i start with -a? It doesn't seem to make sense. :-/
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Re: Really basic stuff that was never proven

Postby Dason » Fri Sep 10, 2010 8:44 pm UTC

Spambot5546 wrote:But multiplication is really repetitive addition. a * b is a added to itself b times.
a + a +a...
-a*b is -a added to itself b times.
(-a) + (-a) + (-a)...

ok
a * -b is a subtracted from itself b times
a - a - a...

Clearly this is wrong. Take a *(-1)
which in your way of going about it is
a - a = 0.
So we already knew this was wrong. But this just proves the point anymore. Your intuition is failing you here. What if we redefine your way of doing things as:
a*b is 'a' added to 0 'b' times. Notice that the a*b as "a added to itself b times" isn't really right because then 2*3 should be 2+(2+2+2).
Using the new method then we can consider a*(-b) to be 'a' subtracted from 0 'b' times. Then we also get (-a)*(-b) is (-a) subtracted from 0 b times. Play around with it.
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Re: Really basic stuff that was never proven

Postby Spambot5546 » Fri Sep 10, 2010 8:57 pm UTC

Dason wrote:What if we redefine your way of doing things as:
a*b is 'a' added to 0 'b' times.

That right there is exactly the kind of redefinition i was looking for.

With that a * -b would be 0 minus a series of "a"s and the whole thing works out. :D
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Re: Really basic stuff that was never proven

Postby Syrin » Fri Sep 10, 2010 10:05 pm UTC

Let a,b be be elements of a ring R. Let "-" denote additive inverse - that is, -a is the additive inverse of a, so a + -a = 0, where 0 is the additive identity of R. Then, (-a)*b = -(a*b), that is, a*b + (-a)*b = 0. This is a rather trivial consequence of the distributive property (and also that 0*x = 0 for all x, which is perhaps not immediately obvious), and so holds for any ring. It just remains to be seen what "negative" might actually mean in some cases. :P

antonfire wrote:That's just a (not terribly nice) way to redo two sentences in the original proof.

Yes, it's "the important" lemma.

Yes, all the nasty shit in the original proof is mostly to formalize just what "unique prime factorization" even means (something you didn't bother doing at all).

No, what you wrote is certainly not "truly trivial". It is, I would say, slightly more complex than the proof sketched out originally. It just brushes the boring details under the carpet.


Note: the person who made the proof is not the person who said that there was a trivial proof. In case you maybe wanted to bottle up your vitriol for another time.

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Re: Really basic stuff that was never proven

Postby Quaternia » Mon Sep 13, 2010 4:08 am UTC

antonfire wrote:Give one.

Just thought I'd add this.
It's far from a trivial proof, but there is an elementary proof that shows that the sequence P consisting of unity and all the prime numbers is a basis of the sequence of natural numbers. "Three Pearls of Number Theory" by A. Y. Khinchin talks about it. The proof is by L. G. Schnirelmann.
(the fact this was always assumed in high school and first year university also hurt my heart, but I guessed it was just too hard to be covered).
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Re: Really basic stuff that was never proven

Postby Kurushimi » Mon Sep 13, 2010 11:14 am UTC

I've been trying to prove that that for an ellipse with a semimajor axis of a and eccentricity of e the distance between the center and the focus is ea and the distance between the directrix and center is a/e but I'm kinda having trouble. Any hints on what path I should take?

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Re: Really basic stuff that was never proven

Postby benhowt » Sun Sep 19, 2010 11:55 am UTC

WarDaft wrote:Very rarely (I would venture perhaps even never) has the proof been given that 1+1 does in fact equal 2 before more advanced topics are covered.


Refer you to Bertrand Russell's Principia Mathematica, volume II, page 87. I haven't read the proof, but suffice to say that by the time it is done you're at page 200 and something. Also note that many other laws are discussed previously (in Volume I)

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Re: Really basic stuff that was never proven

Postby Eebster the Great » Mon Sep 20, 2010 1:43 am UTC

WarDaft wrote:Very rarely (I would venture perhaps even never) has the proof been given that 1+1 does in fact equal 2 before more advanced topics are covered.

Isn't that essentially the definition of 2, though? I mean 2 is usually defined as something like the successor of one (let's write it S(1)) or as {0,1} = {{},{{}}}, which are essentially equivalent. Given 2 = S(1) and 1 = S(0) and a + S(b) = S(a + b) and a + 0 = a, we know 1 + 1 = 1 + S(0) (by definition of 1) = S(1 + 0) (by definition of addition) = S(1) (by definition of addition) = 2 (by definition of 2).

So that one truly is a trivial proof.

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Re: Really basic stuff that was never proven

Postby Spambot5546 » Mon Sep 20, 2010 5:13 am UTC

Yeah, at that point you might as well ask for a proof that 1 = 1. That addition functions the way it does on our fingers, and things like the properties of equality, have to be accepted a priori because there's really no way to prove them and nothing in math works without them.

To put it another way, hold up one (1) finger. Now hold up one (1) other finger. Note that you have held up two sets of one (1) finger. Careful observation and measurement will demonstrate that you are now holding up two (2) fingers. Ergo 1 + 1 = 2. QED.
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Re: Really basic stuff that was never proven

Postby Eebster the Great » Mon Sep 20, 2010 5:57 am UTC

Spambot5546 wrote:Yeah, at that point you might as well ask for a proof that 1 = 1. That addition functions the way it does on our fingers, and things like the properties of equality, have to be accepted a priori because there's really no way to prove them and nothing in math works without them.

To put it another way, hold up one (1) finger. Now hold up one (1) other finger. Note that you have held up two sets of one (1) finger. Careful observation and measurement will demonstrate that you are now holding up two (2) fingers. Ergo 1 + 1 = 2. QED.

Technically once you bring fingers into the picture, you are talking about physics, not pure math. However, if we assume that we can model your process of finger-addition or whatever as an operation, we could easily prove that there is an isomorphism between the group of natural numbers with addition and the group of fingers with finger-addition.

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Re: Really basic stuff that was never proven

Postby gmalivuk » Mon Sep 20, 2010 2:37 pm UTC

Any "proof" that 1+1=2 isn't done because we have any doubt that 1+1=2, but rather because we need to make sure that the system we're building implies the mathematical results we expect. If Russell's proof ended up showing 1+1=3, of course it wouldn't mean that we'd been wrong all these years. It would just mean that whatever Russell started with turned out not to be the correct way to build math on solid logical foundations.
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Re: Really basic stuff that was never proven

Postby voidPtr » Wed Sep 22, 2010 9:58 am UTC

One thng I don't ever remember being proved in grade-school maths is that the angles in every triangle add up to 180 degrees.

Actually, I remember a lot more of what wasn't proved than what actually was proved, some of it for good reason, some not.


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