@elninovaquero This is the definition of "odd integer" we are working with. The even integers are integers of the form 2m for some other integer m, while the odd integers are integers of the form 2m+1 instead. (Equivalently, you can ask for remainder 1 when you divide by 2, but that is the same thing.)
@Bl4nkB0x The proof given shows that when n=2m+1, then n^4-1= 8(2m^4+4m^3+3m^2+m). To prove divisibility by 16, you need to show that 2m^4+4m^3+3m^2+m is even. This is obvious for the first two terms (2m^4+4m^3), so you are left with the problem of proving that, for all integers m, 3m^2+m is even. Again this is obvious if m is even. But if m is odd, then so is 3m^2, and so 3m^2+m is a sum of two odd integers, hence is even.
I'm hoping this course can help me, I'm doing for a math major and there's basically no concrete maths anymore, everything is proofs. I'm quite good at problem solving but I suck at proofs... I hope I will get better at them soon
let n=2m+1 n^4-1 = (n-1)(n+1)(n^2+1) = (2m)(2m+2)(4m^2+4m+2) = 2(m)2(m+1)2(m^2+4m+2) = 8(m)(m+1)(m^2+4m+2) = 8k, an integer divisible by 8. Furthermore, since m is even, m(m+1)(m^2+4m+2) is even and can be represented as 2l therefore, 8k = 8*2l = 16l, for l an even number Therefore, n^4-1 is divisible by 8 and 16.
+58VintageSunburst 'Hi there, most of that is correct, though in fact 4m^2+4m+2 = 2(2m^2 +2m +1). However, it isn’t necessarily true that m is even. For example, if n is 3, then m is 1, so m is odd in that case. But m(m+1) is still even, no matter what the integer m is.' - Dr Joel Feinstein
I proved question 2 using an expression similar to the solution from question 1, thus for (ii) f (n) = n^(2k+1) , g(n) = n^(2m+1) So fg(n)= n^2(k+m+1) and assuming n^2(x) will always be a even function , this proves that fg(-X) = X Will this expression help prove Question 2?
“Your argument will prove the special case where f(n), g(n) are suitable powers of n. However the notion of odd/even function is more general than this (although such monomials do provide good examples). Because the question asks about general odd functions, you should not assume that you are working with functions of a special form. There are plenty of other odd/even functions out there. (Think about the functions sin(x) and cos(x) for example.) Particular examples are useful if you are looking for a counterexample, and can also help with the intuition about what is going on in the general case. But the proof here has to work for ALL odd functions, not just some.” - Dr Joel Feinstein
Why do people who write proofs use confusing language like 'let', 'consider' (instead of 'if', 'look') etc.? Why do they write their proofs backwards, like they found it from thin air?
AnotherGuitarHero I think we might disagree about the meanings of "backwards" and "forwards". Mathematical reasoning often has a specific direction, and not all steps in the proof are reversible. Proofs are often discovered working backwards from the destination (almost like some mazes are easier to solve that way), but the logic of the final argument must point in the correct direction. If you try to prove something by making deductions from the desired conclusion, you won't have proved that conclusion unless all of your reasoning is reversible. (You can see some sample warnings about "backwards reasoning" in my Foundations of Pure Mathematics classes.) However, it is allowed to say "Y would follow if we could only prove X." and then prove X, as long as you don't use Y to prove X. So you can rewrite most proofs to fit more closely with the way they are discovered. On the other hand, sometimes, perhaps like in a game of chess, there are only as limited number of sensible options for what information or tool you might use next. Here experience and fluency plays a role: a strong chess player will usually focus quickly on a relatively small number of likely moves from what could appear to be a bewildering number of options. Correct use of "let" or "suppose" is a very important (and traditional) tool when you want to prove that something is true for ALL examples of a particular kind. You can think of it as an abbreviation for the following ideas. "We want to show that is true for ALL things of type A. So what we need to show is that, if we have something of type A then is true for that thing. As long as we only assume the thing is of type A, and nothing else, then our proof will be valid for all things of type A. So, let x be an arbitrary thing of type A. We'll show that just using the assumption that x is of type A, and no extra assumptions, will still allow us to show that is true for x. Because we made no other assumptions about x, our argument will show that is true for all things of type A." That is the traditional approach. But you could do a lot with "If", as in "If x is a thing of type A, then ...". But, at least to me, the important thing is to make sure that you really understand the structure of the proof you need. Using traditional language can help when you are using a traditional proof structure, but is not essential as long as the reasoning is correct.
I like this! Thanks for uploading. Great lectures. Hope I'm not being too persnickety but I think i) needs a little tweaking. Every Odd function f, has f(0) = 0 only if f(0). For example f(x) = 1/x is odd but f(0) not = 0.
@elninovaquero This is the definition of "odd integer" we are working with. The even integers are integers of the form 2m for some other integer m, while the odd integers are integers of the form 2m+1 instead. (Equivalently, you can ask for remainder 1 when you divide by 2, but that is the same thing.)
@Bl4nkB0x The proof given shows that when n=2m+1, then n^4-1= 8(2m^4+4m^3+3m^2+m). To prove divisibility by 16, you need to show that 2m^4+4m^3+3m^2+m is even. This is obvious for the first two terms (2m^4+4m^3), so you are left with the problem of proving that, for all integers m, 3m^2+m is even. Again this is obvious if m is even. But if m is odd, then so is 3m^2, and so 3m^2+m is a sum of two odd integers, hence is even.
I'm hoping this course can help me, I'm doing for a math major and there's basically no concrete maths anymore, everything is proofs. I'm quite good at problem solving but I suck at proofs... I hope I will get better at them soon
for the first you could use induction.
like n^4 -1 = 8m for some integer m.
and then prove for k+1
let n=2m+1
n^4-1 = (n-1)(n+1)(n^2+1) =
(2m)(2m+2)(4m^2+4m+2) =
2(m)2(m+1)2(m^2+4m+2) =
8(m)(m+1)(m^2+4m+2) =
8k, an integer divisible by 8.
Furthermore, since m is even, m(m+1)(m^2+4m+2) is even and can be represented as 2l
therefore,
8k = 8*2l = 16l, for l an even number
Therefore,
n^4-1 is divisible by 8 and 16.
+58VintageSunburst 'Hi there, most of that is correct, though in fact 4m^2+4m+2 = 2(2m^2 +2m +1). However, it isn’t necessarily true that m is even. For example, if n is 3, then m is 1, so m is odd in that case. But m(m+1) is still even, no matter what the integer m is.' - Dr Joel Feinstein
Hi,
1^4-1 = 0, and 0 is certainly divisible by 16.
wouldn't be able to work with 1 (which is a odd integer) so it doesn't apply to all odd integers?
1^4 = 0
I proved question 2 using an expression similar to the solution from question 1, thus for (ii)
f (n) = n^(2k+1) , g(n) = n^(2m+1)
So
fg(n)= n^2(k+m+1) and assuming n^2(x) will always be a even function , this proves that fg(-X) = X
Will this expression help prove Question 2?
“Your argument will prove the special case where f(n), g(n) are suitable powers of n.
However the notion of odd/even function is more general than this (although such monomials do provide good examples). Because the question asks about general odd functions, you should not assume that you are working with functions of a special form. There are plenty of other odd/even functions out there. (Think about the functions sin(x) and cos(x) for example.) Particular examples are useful if you are looking for a counterexample, and can also help with the intuition about what is going on in the general case. But the proof here has to work for ALL odd functions, not just some.” - Dr Joel Feinstein
Why do people who write proofs use confusing language like 'let', 'consider' (instead of 'if', 'look') etc.? Why do they write their proofs backwards, like they found it from thin air?
AnotherGuitarHero
I think we might disagree about the meanings of "backwards" and "forwards". Mathematical reasoning often has a specific direction, and not all steps in the proof are reversible. Proofs are often discovered working backwards from the destination (almost like some mazes are easier to solve that way), but the logic of the final argument must point in the correct direction. If you try to prove something by making deductions from the desired conclusion, you won't have proved that conclusion unless all of your reasoning is reversible. (You can see some sample warnings about "backwards reasoning" in my Foundations of Pure Mathematics classes.) However, it is allowed to say "Y would follow if we could only prove X." and then prove X, as long as you don't use Y to prove X. So you can rewrite most proofs to fit more closely with the way they are discovered.
On the other hand, sometimes, perhaps like in a game of chess, there are only as limited number of sensible options for what information or tool you might use next. Here experience and fluency plays a role: a strong chess player will usually focus quickly on a relatively small number of likely moves from what could appear to be a bewildering number of options.
Correct use of "let" or "suppose" is a very important (and traditional) tool when you want to prove that something is true for ALL examples of a particular kind. You can think of it as an abbreviation for the following ideas.
"We want to show that is true for ALL things of type A. So what we need to show is that, if we have something of type A then is true for that thing. As long as we only assume the thing is of type A, and nothing else, then our proof will be valid for all things of type A. So, let x be an arbitrary thing of type A. We'll show that just using the assumption that x is of type A, and no extra assumptions, will still allow us to show that is true for x. Because we made no other assumptions about x, our argument will show that is true for all things of type A."
That is the traditional approach. But you could do a lot with "If", as in "If x is a thing of type A, then ...". But, at least to me, the important thing is to make sure that you really understand the structure of the proof you need. Using traditional language can help when you are using a traditional proof structure, but is not essential as long as the reasoning is correct.
I like this! Thanks for uploading. Great lectures. Hope I'm not being too persnickety but I think i) needs a little tweaking.
Every Odd function f, has f(0) = 0 only if f(0). For example f(x) = 1/x is odd but f(0) not = 0.
....oops; should have said ".. only if f(0) exists.
im only in 8th and I'm going to learn this next year.
I forgot the binomial theorem. :-(