Arithmetical Puns

But arithmetic puns make me number

Algebra puns are too linear, arithmetic puns are too basic, trigonometry puns are too graphic, calculus puns are all derivatives. Only the statistic puns are the occasional outlier.

I came across what looks like an inconsistent behavior when using integers of different precisions in operations.

From what I had understood, the integer with the smallest precision is implicitely cast to the same precision as the other for the time of the operation and the result is returned with this new common precision.

But then is "=" an operation ?

I tried this:

uint16_t a,b;
uint32_t p;
a=50000;
b=40000;
p=a*b;

After the operation, p is 2000000000. So, I figured, the equal sign also plays a role, and both source integers are casted to the destination precision before the operation to be able to return a result with the correct precision. A bit dangerous, but convenient.

But then I tried this :

uint32_t a,b;
uint64_t p;
a=1000000000;
b=2000000000;
p=a*b;

But here, after the operation, the result is 1321730048. So, this time, the operation has been made with the operands' precision, returned an overflowed result and then cast the 32 bits result into the 64 bits variable.

This is closer to what I would have expected, but I don't really get why it doesn't act the same way in both situations. Does anybody have a better understanding of the rules? Is it just an undocumented, compiler specific thing?

I'm only personally familiar with SVO languages, so until recently I had never really thought about whether the infix notation I use in English or Spanish is a result of those languages' grammar.

To clarify, what I'm talking about is the placement of operators (+, -, ÷, etc) in relation to operands (numbers, variables, etc). In infix notation, this would be exemplified by the expression '2 + 2'. Alternatively, prefix notation, it would be expressed as '+ 2 2', while in postfix it would be '2 2 +'.

As far as I can tell, most languages, regardless of their word order, would use infix notation (at least in written form), but do any languages verbally use a notation corresponding to their word order? For example, would any SOV language say something like "Two two plus" rather than "two plus two"? Or are there any other interesting ways different languages would express arithmetic expressions?

I am trying to learn myself a little rust and played around with iterators:

struct NumberGenerator<T: Copy> {
    cur: T,
    func: Box<dyn Fn(T) -> T>,
}

impl<T: Copy> NumberGenerator<T> {
    fn new(func: Box<dyn Fn(T) -> T>, cur: T) -> Self {
        NumberGenerator { cur, func }
    }
}

impl<T: Copy> Iterator for NumberGenerator<T> {
    type Item = T;
    fn next(&mut self) -> Option<Self::Item> {
        self.cur = self.func.as_ref()(self.cur);
        Some(self.cur)
    }
}

fn main() {
    let mut odd_gen = NumberGenerator::new(Box::new(|x: f64| (1.0 + 1.0 / x).powf(x)), 0.0);

    println!("approximation of e:");
    for _ in 0..10 {
        println!("{}", odd_gen.cur);
        odd_gen.next();
    }

    // also usable with str
    let mut bla = NumberGenerator::new(Box::new(|x| x), "asf");
    bla.next();
}

As you can see, i restrict the generic type T to Copy. But i am still able to use my struct with str. So my question is: Is there a trait for arithmetic types, so i can use the struct with u8, f64... but not with str?

Something like Ord, but for +,*,/,-?

That's pretty much it. I may be answering a bit later in the day because of chores and whatnot, but I will answer everything! AMA!

Has anyone else experienced this in the latest experimental build. I'm not sure how to report a bug in the game but I know of at least 3 people who have experienced this so far in the latest experimental build. It's happened to me at least 4 times now. The only occurrence that I can confirm it happening with is a can of orange soda spawned on the floor of a house. Someone else had it happen with a Sealed Jar of Peanut Butter.

This error doesn't seem to occur with items inside of fridges, only on the floor of buildings.

(Product image)

Expressio it's an 'Automatic' Parser and Solver of custom provided arithmetical expressions.

Based onto the chosen implementation, it supports:

• Constants: numeric string literals, formatted as: #.#####;
• Variables: custom-defined string literals not followed by parentheses that hold values;
• Operators: set of defined common operators like +, -, *, / etc; (Supported operators)
• Functions: a default extensible and ovveridable set of functions to call within an expression;
• Expressions/Arrays: mix and match of the above elements.

It is really easy to setup. and even easier to use!

You can find a 'Lite' (Demo) version of the product here: zzatomiic.gitlab.io/expressio

Otherwise, go check it on the Unity Asset Store: assetstore.unity.com/packages/expressio

What do you think of it?

If you end up using it in a project/demo, post an example of your work with Expressio here in the comments!

I am looking for a book that contains a bunch of brain-dead exercises that train your algebraic manipulation and arithmetical computation skills. I find that lacking of such a technical foundation is a major cause for why students find Mathematics at the high-school level difficult.

Does anyone know of any such resources? I don't want any verbose, "fat" material. The book should be very lean - just the exercises and the answers.

Edit: Thanks for all the replies! So far, I have found two very useful resources that suits my purposes.

1. The pair of books suggested by /u/likecharges
2. The book "Essential Mathematics" by Franco Vivaldini which was written to solve a similar issue at the university level. Very handy!

Also, what exactly is an "arithmetical statement”?

I've seen second-order arithmetic denoted as Z2 (Z with '2' subscripted, e.g., here), so if first-order arithmetic is Z1 there is presumably an infinite set of arithmetic theories Z3, Z4, ... . Is there a name for this set?

And is there a good resource to find theorems relating elements of this set? For example, Buss' proof of Gödel's speedup theorem uses the fact that for any Zi, you can prove the consistency of Zi from within Z(i+1), but not from within Zi. What other convenient relations does Zi have to Z(i+1)?

Apologies if this is a very basic question, I'm working through the Incompleteness theorems for the first time.

By arithmetisation of strings, formulae, and proofs of the language of arithmetic (LA), and by the representability of recursive functions in LA, we can form these interesting formulae in LA like "Formula(x)" or "Provable(x)". These formulae have one free variable, x, and are true of a natural number n iff n is the Gödel number of a formula or a provable sentence or whatever.

Gödel uses these facts to produce a formula, "Unprovable(x)" which is true of a natural number n iff n is the Gödel number of an unprovable sentence of LA. Then, by substituting for x the Gödel number of "Unprovable(x)" itself, we get a sentence that "says of itself that it's unprovable", and so on.

My question is: what sort of sentence in LA does this produce? If g is the Gödel number of "Unprovable(x)", what arithmetical claim does "Unproveable(g)" make? For example, is it an equation, or an inequality, or a quantified formula?

Many thanks in advance and please feel free to correct me if I've made any expository mistakes above.

cause math can have fun too

Alright, so I get the basic stuff, how to calculate the sum, the nth term or whatever, what I'd like to ask you is what's the logic behind solving excerises like these:

1. Let a^n, where n >= 1, be an arithmetical progression, for which a^3 + a^19 = 10. Find the sum of a^6 + 1^16.

Like I can write the terms in the other forms like this:

a^3 = a^1 + 2d

a^19 = a^1 + 18d

So 2a^1 + 20d = 10. That means that a^1 + 10d, or a^11 = 5. And then what?

2. Let a^n, where n >= 1, be an arithmetical progression, for which S^n is equal to the sum of the firs n terms.

If a^1 + a^4 = 100, a^n-3 + a^n = 200 and S^n = 600, find n.

3. a, b, c are integers. We know that the sum of the three numbers is an even number, find out whether a, b or c are even or not.

Sorry if it's a long post, but.. it's a sort of more difficult subject and having a shit math teacher.. yeah..

EDIT: I have no idea how to type like.. subscripts in reddit so I went with exponents. They're supposed to be down under, you get it.

EDIT 2: Just found the "Custom superscript and subscripts" part. I'm kinda dumb.

Hi All

I am looking for a proof that the arithmetical hierarchy does not 'collapse' to some finite level, that is, mathematical utility of each new set of arithmetical statements included as the number of alternations of ∀ and ∃ quantifiers are included/permitted, right up to a countable infinity of alternations.

Note that I am not looking for you to prove this to me as a reply, but rather direct me to a textbook/paper or internet resources where this proof is done (preferably the former) and which I can cite in a thesis I am writing about this stuff.

Thanks in advance

Reg.

Also, what exactly is an "arithmetical statement” in this context?