Some ebooks, mostly from /u/lewisje's post

Just opened an online pdf on Spivak's calculus and a few minutes into scrolling I came to the property

For every number a, there is a number -a such that
a + (−a) = (−a) + a = 0

and wondered if I could prove −(−a) = a from that.

We know that for every number x, there’s a unique additive inverse y such that x + y = 0, meaning that y is the only value that satisfies the equation. The additive inverse of a is −a, since a + (−a) = 0. And because addition is commutative, we also have (−a) + a = 0, where a is the additive inverse of −a. By notation, the additive inverse of −a is written as −(−a). So we can write (−a) + [−(−a)] = 0

In the equation (−a) + y = 0, there can only be one possible value for y. But both a and −(−a) satisfy it. Since there can only be one additive inverse of −a, by uniqueness, those two values must be equal.

So therefore a = −(−a)

Please let me know if I got the proof right and if I could word it out any better!

I just can't... I don't understand it... No ELI5 helped me yet.

Here's an example with a brief explanation of what I know and where I get stuck.

Let R = Z[x] and I the set of all the polynomials from R whose coefficients add up to 0. Prove that I is the ideal of R. Is this ideal also the principal ideal? Prove that R / I is isomorphic to Z.

So, Z[x] is a set of polynomials with integers. I appears to be a specific subset of Z[x] and therefore should be a subset of R. By definition, a ring is an Abelian group for addition and a semigroup for multiplication. Next, a subring must have closure, addition/subtraction test and a multiplication of 2 elements within itself also stays inside.

An ideal further expands on the idea of a subring and only asks for 'external' multiplication by taking an element from the ring, multiplying it with an element from the subring and checking if the product remains in the subring.

Principal ideal is a specific ideal in which only 1 element can be used for multiplication and it generates the entire set. The common simple example of it I came up with is (Z/nZ), integers modulo n.

I would say I understand theory well enough (except the R / I which is the quotient ring concept I am not entirely familiar with) but I simply can't understand this or any similar task. I don't know how to solve it and literally need baby steps.

In the task itself I understand that coefficients adding up to 0 would probably mean that the 'free part' is always the answer so x - 1 would be a simple example of the kind of polynomial the ring is describing. How do prove the I is the ideal of R? I need to show it's a subring first, but I don't know how to do that either and even following the definition feels impossible to me.

Okay, let's take some 3x - 3 and 2x - 2 as examples. (3x - 3) - (2x - 2) = x - 1, so it checks out I guess. Next, multiplying with an element from the ring. Since this rule only applies to the subring I, I can take some x (the simplest polynomial from R) and then (3x - 3)x = 3x^2 - 3x, also checks out so is the ideal here done with?

Now, how would I show it's a principal ideal? Maybe it's generated by x - 1? You can add/subtract it as many times as you want and you'll get all the elements. The proof regarding the quotient ring is completely unknown to me and I would appreciate if someone could explain. Just so you guys know I had no solution to this task and was genuinely lost, now when writing it I came up with some examples which could demonstrate some things and my brain started working, hopefully it's all good.

Learning conversion factors between different units of measurement, but i want to clarify smthn to make sure im thinking correct

0.36mg x (1,000mg/1g) = 360g

(I accidentally changed the units while writing this, in my actual work i had the mg and g switched -> 0.36g=360mg)

The correct way to write the answer means it needs 2 sf, but 360 has an unknown number of sf. So we write 3.6 X 10^2mg

But if the first number was 0.360g, then 360mg would be correct, right? With no need for scientific notation

So, if i had a number like 50,000 but i needed something like 3 sf, would i write 5.00 X 10^4? Because there is 3sf in 5.00?

If you are 9-12th grader and struggling with Math, I can help you to find root causes and put you on a right track. I'm thinking 2-3 months per student, shorter or longer depending on your skills and goals. It's for free.

More about myself: I'm a Harvard grad, in my 40s, tutored a lot when I was in college / grad school. Ask me anything. DM is welcome.

Why free: I'm working on tools for math. I need to be teaching real students to know what's actually needed. You'd be helping me as much as I'm helping you.

I'm 29, and have been unable to find anything to help me solve the long division problem I've got. 7 divided by 56. 7 can't go into 5, so that's a 0 above the five. Then you do subtraction right? Then 5-0 = 5. Then you bring down the 6 next to the 5, and we're once again back at 56. So how many times does 7 go into 6? 0. and I know 00 can't be right cuz the answer is 8. I was taught to break the problem down, to show work STEP BY STEP, not to keep multiplying the divisor till you get the dividend. So....anyone willing to help? :')

_____
7 | 56

https://imgur.com/a/jYknItf

i swear im having difficult to aops pre algebra more than with cal 1
where did (a+ab+b)'c'

how tf does it magically become the new x and y, im struggling with show this is commu and associa.

where did C come from???

is there like a youtube that can help me with this problem?????

im preparing for engineering i really like math but this book is pissing me off
i was finished with algebra 1, a book that i borrowed at a library. I struggled with algebra 1 and cal 1 sure, but compared to this aops its so frustrating im making mistake with basic arithmetic, i personally dont know how am I struggling with this book more than algebra 1 and cal 1

ps i want to be done with pre algebra before continuing with trigo

If I had an exponential function called:

g(x) = c^x + d

and a linear function called:

f(x) = ax+b

Besides just putting them into a graph, how would I find the points where they cross?

Hi all. I’m an undergrad accounting major and going through a math course sequence of Intermediate Algebra > Pre-Calculus > Calculus 1 and its Applications (business calc).

For context, I scraped by Algebra 2 and was decent at AP Stats in high school, took some gap time for an accounting job, and now it’s been a few years since I’ve studied math at all. I tried taking the intermediate algebra class last semester and had to withdraw because I didn’t know anything on day one. I feel like I’ve forgotten how to do all algebra and now I need to rebuild from basics. I’m going online this semester in hopes to crank out the intermediate algebra and pre-calc.

Does anybody have tips on resources or structure to follow in order to re-learn the foundations? I feel like I don’t even know HOW to study math, considering I’ve always gotten low grades in it. I really need to get through these classes to move to my upper division coursework and I feel so far behind that it’s overwhelming to know where to start. Any help would be appreciated!!

I recently started college as an electrical engineer major and I need to complete a placement exam in order to get into calc 1 (Aleks PPl, 1-100). When I took a practice exam without reviewing I got around a 50 (need 76 to get into calc) and after a few days of review and learning new material I didnt know existed i got it up to a 70. I can probably get the required 76 to take calc but I am unfamiliar with most trig, exponentials, and logarithms. I got told by a friend to stay in pre calc and set a good foundation because I will need it for my major. If I were to take that route my plan is to most of my GE this semester take calc 1 2nd semester along with physics. calc 2 I would take during the summer (I would work part time and only take that class).

I just wanted to hear some opinions from other people of which plan would be better, anything helps.

Like the title says, I am looking for solutions manual or just solutions to Tom Apostol's Calculus.

The question was this:
A cuboid has measurements- 70 cm length, 30 cm width, 40 cm hight
It is open from the top
Cylinder measurements- 20 cm length, 15 cm diameter

How many cylinders can be completely fit into the cuboid?

I got confused as there were 2 different paths to take in this,
1 was dividing the length by the radius and blah blah blah and finding out how many cylinders could actually be fit into the cuboid
The other was to divide the cuboid volume by cylinder volume which is wrong as there would be space between the cylinders which this method would not account for

Is it not possible that when in a hypothetical situation you take a cuboid with those respective dimensions and a few cylinders with its respective dimensions and start putting the cylinders in, by the time you get to the second layer of cylinders there would be space between the cylinders in the first layer resulting in the cylinders in the second layer dipping into that extra space and basically taking less hight and if that extra hight is enough you could maybe fit an additional layer of cylinders allowing a lot more (~4 more) cylinders to be fit into the cuboid?
I do not understand how to do those calculations to find out the extra hight from the dipping of the second layer of cylinders
If you understand what I am trying to say and know how to do these calculations please help me out here and tell me how many cylinders can be fit into the cuboid
Thank you

Like the title suggests. I am working with sullivan algebra book and get really stuck at the proof exercises how should i be going about them?

I was coding professionally before I learnt algebra at school, so a = b is forever first an assignment to me, and not a relation.

The problem is that computers compute; they evaluate arithmetic. They really are a form of "adding machine". [yes, you can code algorithms, and there's symbolic math computation (e.g. sympy), but most programming languages evaluate expressions]

Arithmetic vs relation (over the reals) is a bit like a point vs a line in geometry. A line is an infinite in length, and infinitesimal up close. It comprises infinite points in large and small scales.

A relation over the reals is a single object, with an infinite number of pairs at both scales.

So... I undsrstand this, but I really don't think about it correctly. My intuition is wrong. How can I fix this?

Maybe it is partly because it is just hard?

Hello everyone, I am someone who hated Math, because I never learned or memorized mutiplication. So when, I went to college. I always avoided it thinking math is hard, but now, I regret it cause I think I can hardly solved any complicated problem and I hate it. Even today basic math is hard for me. If I want to start where y’all think I should start?

*thanks everyone and I am not an English native so please have a mercy on my grammar

Hello, I'm a uni student who is relearning algebra, and during my time studying and practicing and et cetera, I run into this issue in my mind where I know how to do something, and yet I find it hard to reason out in my brain why I'm doing it.

I can understand the roles that each operation or sign plays in an equation, I know that an exponent is just multiplying a number by itself or dividing something is just seeing how many times a number can 'go into' another number, but I don't know how exactly this is important or why this leads me to the correct answer, y'know?

I do it all correctly, I'm right in the end and the equation is correct, but if you asked me to translate a question into real world terms, I couldn't, obviously math is incredibly important but I myself just can't visualise or intuit in my head what I'm actually doing besides following steps mentally over and over.

Is this normal? Is this what people normally think when doing math? Will this hurt me when I progress to calculus?

Hi, we are trying to improve Wikipedia's list of vital articles, which are meant to be the highest priority articles for Wikipedia to improve. In particular we are trying to improve the math section, which should list 1200 articles (currently has 1211).

You can help by suggesting the best topics to add or remove to improve the list, taking into account the quota of 1200. It would be especially helpful if you can make proposals ON THE TALK PAGE to add, remove, or swap articles, make general suggestions, or vote on existing proposals, to create a balanced selection of the highest priority topics.

Here is the page, click the talk link to go to the talk page.

https://en.wikipedia.org/wiki/Wikipedia:Vital_articles/Level_5/Mathematics

Hei! :) Jeg bestilte denne boken på Finn, den tar litt tid før den kommer. Kan noen sende meg side 15–22? Jeg må gjøre ferdig leksene mine innen klokken 17:00 i dag.

Hi! :) I ordered this book called Sinus R1, it will take a more days to arrive. Can someone send me pages 15–22? I have to finish my homework by 5:00 PM today.
🫶

I'm 16, in my last year of high school. Over the last few years, I've been working on mathematics and physics on my own because I love them so much. Since I'm nearly graduating from high school, I was wondering what's really holding me back from doing research. I'm on track to finish Calc 1, 2, and 3 by the time I graduate (late 17s), but what comes after that? I'd like to study just about as much as I can from different branches, but once I'm decent in a variety of them, I'll research number theory. My question may not be very straight with how I'm wording this post, but I'm only really wondering what material realistically stops me from studying number theory at an 'advanced' level (as for the other branches, I'll pick up some books and go through them slowly). Any resources are helpful, assuming a basis of Calc 1 + fraction of Calc 2, basic number theory such as modular arithmetic, Fermat's little theorem, Euler's theorem, and other (light) theorems.

> When I said advanced: 'Advanced' is subjective, but to me, that's being able to write a worthy master's thesis in the field.

And of course I expect this to be a years-long challenge (an enjoyable one, at that) that I'm willing to take on

Recently I'm self-studying wedge product and Exterior Algebra in the term of Linear Algebra, and I'm quite curios to the Shoelace Formula that how is that be rigorously correct since first learn.

Shoelace stated that

for any v_i=(x_ie_1, y_ie_2)∈ R\^2 , 2≤i≤n

then x=(x_1,x_2... x_n), y=(y_1, y_2... y_n)∈ R\^n

Area bounded by these points is

1/2| ∑(v_i ∧v_i+1)+v_n ∧ v_1| , i=1,2..n-1

v_i ∧v_i+1 ∈Λ\^2(R\^2)

If we compute the area by using wedge product, x ∧ y ∈ Λ\^2(R\^n), we will get C(n,2) terms

My question is: where are those v_i ∧v_j , for j-i>1 such as v_1 ∧v_3?

I tried looking this up on Google but only AI math websites came up.
Is there a website where I can create Questions and create answers and solutions?
One of my friends struggles extremely with algebra (And math dyslexia), and unfortunately they are in college and I work full time. They LOVE practicing math, but do not understand how to get the answers very well, which I used to just explain to them (this has worked for years as I know how to explain it in a way they understand, but they also have had a head injury so they forget easily).
I love him, and I know he won’t ask anyone else for help due to feeling ashamed.

I only do what i know, and what i know is not enough at all, i forgot how to math anything almost and stuck with patterns only.

How to start from scratch and basics?

Hiii , I am really bad at maths let it be calculus , trigonometry , arithmetic or even fast calculation without the need of calculator. And the reason being I always hated maths but now it's kinda growing on me . I like how maths work and now I regret not learning it when it was the time for the basics .

The thing is I always learned maths just enough to pass for each class / standard .

But now I realise I have missed out on many things . Currently I am pursuing engineering in electronics and I am interested in software . Specifically AI and ML which requires a significant amount of maths .

So plz guys help me ... Tell me how can I learn maths from the basics and become good at it by the time I graduate and get a job .