Why do we teach algebra before calculus?

>no numbers used
>reiman integral explained

upload.wikimedia.org/wikipedia/commons/c/cd/Riemann_integral_irregular.gif

You don't need algebra but it comes really close. The pre-requisites for arithmethic are a weird pseudo-algebra.

When you define the axioms of arithmethic there are 3 main things you assume:
A set of numbers exist.
This set includes at the very least a first element
The addition law exists.

What is algebra? You define a set and some operations and make a structure out of them.

Arithmethic is just simpler algebra.

For calculus you need a field. A special type of field actually. For something to be a field, the set and the two operations must first be a ring.

Do you know anything about math?

shut the fuck up you stupid underage imbecile, you don't use any theory of algebra is what I clearly mean. you don't need to know what a group, ring or field is. saying you use algebra because one of the structures you use happens to be studied by algebra is retarded, just like if you said that you use differential geometry in calculus because you integrate surfaces, or measure theory because you happen to integrate

the moment you say a + b = b + a you are implicitly doing algebra.

The moment you start solving equations you are kinda implicitly doing algebra

The moment you state that a equation has a general solution you are definitely implicitly doing algebra

Do you even know what the words you speak mean?

I thought this was Veeky Forums so I expected you to know your shit.

>saying a+b = b+a is doing algebra
yeah I fucking guess when you learn your numbers in preschool you're doing number theory huh? you're fucking retarded

You know there is a reason that thing 6th graders do is ALSO called algebra right?

Because it is literally algebra without proofs.

a+b = b+a as a statement only makes sense in algebra and arithmethic (and I already explained how arithmethic is just pseudo algebra)

>Give me an example of something in calculus you can say without algebra

infinitesimals exist

your "explanation" is retarded: knowing what the number 3 is and how to do 1 + 1 = 2 is not studying number theory

retarded thread where op projects his insecurities of failing his middle school algebra class but learned calculus on khan academy with "flying colors" as a neet 20 years later

>the number 3

The number 3 is a number, you would just be doing set theory.

> how to do 1 + 1 = 2

1 + 1 = S(1) = 2

Given that in arithmethic the sucessor function can be proven mean S(x) = x + 1

You have a really weird view of math. You are not understanding how math is built up to calculus.

When you say that you want to teach calculus before algebra you are ignoring so many things that make calculus possible.

I'm reading through Serge Lang's Basic Mathematics to get a good grip on the foundations and how to prove things.

I came across this question and I do not know how to prove it.
Anyone know how to do this?
I mean, obviously both sides are the same but I do not know how to prove that using just associativity and commutativity

Meant to attach pic related

Full proof:

(a + b) + (c + d)
= a + b + (d + c)
= a + b + d + c
= a + (b + d) + c
= a + (d + b) + c
= a + d + b + c
= (a + d) + (b + c)

how you gon take a derivative without knowing what a variable is?

waste of paper waste of life

You're not proving anything. Restating axioms is not a proof, it's being a retard.

Not wrong assuming addition is taken as left-associative, but I would show all parentheses for clarity.

Thanks senpai

nice bait. I've self studied advanced physics up to 22 dimensional superstring M theory despite the fact that i failed algebra in high school and don't even know the quadratic formula. calculus is completely trivial once u get past all the needless algebraic obscurantism

The point where you became a retard was the point you took string theory as anything more than a mathematical curiosity.

Isn't this the first exercise?

Commutativity is that a+b=b+a, in other words you can rearrange the 2 integers without changing the solution. 2+5 will yield the same result as 5+2.

Associativity is (a+b)+c=a+(b+c), which means changing this particular order of operations does not change the solution. 1+2+3 can be 3+3 or 1+5, etc.

Use the definition of associativity and commutativity repeatedly on the left side till you get the right side it's a simple proof

I assumed that something like this for rule of associativity with 4 variables:
[(a + b ) + c] + d
Wasn't allowed

It's stupid to prove anything like this. Either associativity is axiomatically true or it's not true at all. So either (a + b) + (c + d) is (a + c) + ( b + c) or it isn't.

>Restating axioms is not a proof, it's being a retard.

How did this thread get filled with people who don't know math?

Commutativity and associativity are only axioms in algebra..

In arithmethic commutativity and associativity are theorems for both operations and their primitive proofs are done using only two numbers because these proofs have to be done by induction and you can't really do induction over many variables unless you are god.

After a + b = b + a you can use this to prove a + b + c = a + c + b, for example and if you want to get fancy you can also prove that no matter how many terms you have, it holds.

Does anyone in this board actually study math?

>It's stupid to prove anything like this.

It is not stupid at all.

One of the greatest achievements of meta-mathematics is the fact that you can prove associativity and commutativity hold with only very limited axioms and definitions in our natural arithmethic.

I'm going to suppose that most of the dumb posts in this thread have been written by the same person, who is also the OP. In particular, I'm guessing that all of the posts

, which frequently contain wrong, confused ideas, are by the same person, who has no understanding of mathematics.

For the record, I have not written anything in this thread until this post. This is my first post ITT.

This post seems to be redundant with the one below it, and so at least one of the two must have been written by some other dumbo who is not the OP, according to my theory. Neither of these two people have considered that there exist certain elementary arithmetic operations which are neither associative nor commutative, for example, much less that there are slightly intermediate objects whose multiplication, say, is not commutative. When one is aware that there exist such operations and objects, and almost at once, then the point of proving that the properties at least hold good in the cases of addition and multiplication becomes clear.

--- that is, in the cases of the addition and multiplication of real numbers, of course. :^)

Well, what do you expect. When the premise of the thread is that you can do calculus without algebra you know that it smells like retardation

It is like doing prime factorization without numbers.

way to say absolutely nothing and quote 1000 posts just to look good in a fucking anonymous board
stop cluttering the board with your nonsense

>I don't like their opinions so they don't know algebra
fuck off, retarded fucking baboon

Most people in this thread do show complete ignorance of algebra and calculus.

OP was saying that you don't need algebra for calculus because you don't need to know what a field is but what are you doing in calc BUT studying the field of real numbers.

It is like saying that if you learn physics but just never call it physics then you are not really doing physics.

Take it easy OP. What's going on here is not so much, "I don't like your opinions" as it is "the content of what I presume are all your posts demonstrates that you don't understand algebra, and this can be verified independently of me." For example, the other user is correct in his first line of . Just because something happens to be very simple algebra, does not mean that it is not algebra. Your negation of this general idea for other areas of math is one of your basic mistakes throughout the thread.

Calculus, like algebra and math in general, is inevitably concerned with quantity, among other things. True, some concepts (for example, the derivative of one graph looking something-like another graph) can be conveyed pictorially, qualitatively, etc. But you simply can't /do/ calculus as-such without quantity, symbol manipulation, which inevitably entail some versions of arithmetic and/or algebra, even if we dress them up in different (same) isomorphic-clothing.

I agree with you. Although I think there should be at least algebra 1 and geometry taught before calculus. Plus I really don't know why precalculus is a thing. Taking derivative and integrals and understanding limits really isn't that hard. I may be a genius but I know when something is simple when I see it. Plus wouldn't it make sense if we had the option to take a calculus or algebra route in mathematics in high school. Then senior year would be linear algebra or multivariable calculus.

Algebra doesnt require graphical analysis.

Neither does calculus.

>wouldn't it make sense if we had the option to take a calculus or algebra route

You cannot do calculus without first doing algebra.

You know what HS algebra entails right? Algebraic properties of real numbers, solving equations involving real numbers and real functions, simplification of algebraic expressions.

Tell me how are you going to do calculus without knowing how to solve 'hard' equations? Just to get any meaningul information out of a derivative you need to solve the its roots. Just how are you going to do limits without knowing first how you can simplify and otherwise change expressions into equivalent ones?

What do you think algebra even is?

Yes, Calculus requires an understanding of the function, graphically.