Does anyone have any hints on showing that f(x)=x^4+x^3+1 over Z_2, the field with 16 elements, that somehow x is a generator of this field?

is this the same as Z_2[x] mod x^4+x^3+1?

what's a practical way to show that a set of orthogonal functions $f_n$ is complete over a bounded interval if I don't have Sturm-Liouville?

@Maximus I do not understand this question

What is the field?

@Maximus This is a field of 16 elements, yes

it's GF(16), I figured out what they were asking, thank you1

im arguing with my friend and he says that 9.99... is not a number cuz he cannot accept the fact that even tho there is infinite amount of nines , it is still a number. I showed him bunch of proofs from wolfram alpha clearly saying its a number etc. , but he is still going his way. You guys have any idea ?

get better friends

ahahahah i agree

It is a number, it's 10

Yeah I know it is , but I can't find a clever way to show him that . I've tried using the basic definition , but it would be too abstract for him obviously.

I agree, 10 is a number

I even tried using x=9.999... , 10x=99.999... , 10x-x=90 , x= 10 . Still didn't work ahah

@Krijn I'm glad you're not an ultraultrafinitist accepting only number from 1 to 7 :P

@AlessandroCodenotti I only use the numbers 1,4, 6 and 21 up to 25

@Elsa It depends on what you mean by a number. =) Maybe you want to use a particular construction of the real number system, but that would involve too much mathematics.

@Elsa That itself already assumes that these objects can be multiplied in this way.

Does your friend have issues with $\pi$? Infinitely many 9s sounds more believable than infinitely many never repeating digits

He thinks that infinite decimals are just a way of approximation of these actual number and that are not itself a number

Perhaps at a high school level, there is no point arguing about these things.

Yeah , you're probably right. I was just thinking there is some clever way of showing him things like these.

High school mathematics isn't meant to be too precise.

One would be surprised at the machinery involved in making simple mathematical objects precise.

I was actually realllly suprised in my first year of college how some simple things in highschool require precise and sometimes hard definitions to understand

The real world is not mathematics. In the real world, a point does not exist, because something cannot be infinitesimally small. But in mathematics, we can talk about a point.

I was surprised by how little I understood high school mathematics when I graduated and how little I was aware of that at the time

@Elsa This sounds weird, but maybe your friend is actually a very smart guy who senses that something is mysterious about infinite decimals and so he cannot accept it. =)

I used to ask a lot of questions about basic things, and everyone around me thought I was stupid. Happens everywhere, in school, at work, etc.

@JasperLoy you have some news about leptons you'd like to share?

@JasperLoy that might be true , but one would accept the truth after so many proofs given to him i figure

Asking questions about basic things is some kind of brilliance

@Elsa For a detailed construction of the number systems, I recommend Mendelson's Number Systems and the Foundations of Analysis. It is a cheap Dover book.

Good evening chat

@JasperLoy thanks for advice :)

@Elsa Depends on whether they are actually proofs or not. Some proofs are not real proofs. =)

What is the topic ? :)

Whether 9.999.... is a number @Astyx

@JasperLoy I thought after showing him the picture of wolfram alpha "saying" 9.999.. is a number would do the trick haha

Isn't that more a question of a convention ?

@Elsa Strangely, I have never used Wolfram Alpha in my entire life. =)

@JasperLoy seriously ? Mind if I ask you what is your degree in mathematics atm?

The real numbers can be constructed in 3 main ways: Dedekind cuts, Cauchy sequences, infinite decimal expansions, to summarize.

@Elsa Oh I graduated long ago but haven't done any math since then. =) I do not know much math at the moment.

@Astyx Also, long time no see. =)

Indeed

I didn't even recognize you before looking at your name with that profile picture :p

How have you been ?

@JasperLoy I would agree even more now that he might be smarter than I think he is. Not agreeing with a definition that doesn't seem very natural , especially if you're not familiar with infinity is not strange at all.

@Astyx Yes, I go by various usernames such as Jason Bourne and Will Hunting. Too bad I am not Matt Damon himself, lol.

@Elsa Yes, if you want to study infinity, you have to study set theory where there are different sizes of infinity as well.

@Astyx Still sick, still struggling.

@Elsa is 1/3 a number for your friend? infinite amount of 3s, and three times that ...

@Nicolas Are you the Nick I spoke to long ago?

@Elsa I think it's more about accepting what people mean by 9.9999.. than acknoledging a proof. What we mean when we write a number with infinite decimals is a limit. It is not about the number existing or anything, we actually define 9.999... as the limit of the sequence

@JasperLoy no relation

@Nicolas 1/3 is a number for him , but 0.3333... is not . He is confused by 3 dots.

@Astyx wouldn't be a help for a highschool student. I'm sure he's not familiar with sequences at all.

@Elsa Are you studying math in university now?

The argument still holds, 9.9999.. is just a notation for something. We give that notation a meaning through a definition. There can be no agreeing or disagreeing with the existence of the number if we define it to be a number

@Elsa god bless

@JasperLoy starting 3rd year this semester :D

Although I get that this might be hard to understand at first sight

@Astyx I think he has every right at this point to not agree with such abstract definitions .

Yeah, this might be too deep into the "what is maths" question. If he doesn't know about axiomatisation and stuff it would be a hard concept to grasp

@Astyx I would say that he is actually showing a great potential to be involved with maths in his future. Doubting should be the first sign if you asked me.

Sure ! It is such questions that lead to the understanding of why we need axiomatisation (is that even a word in english?) and stuff :)

Now I feel that I should have treated him like " who's a smart boi " instead of trying to prove him wrong haha

@Elsa Yes, like I used to say, questioning all things is the beginning of wisdom. =)

@Astyx Yes that is a word. =)