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!

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

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

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. So I'll be brushing up too. Before the first session, I'll need to talk with a parent/guardian. Ask me anything. DM is welcome. 

Note: I posted here 6 years ago (https://www.reddit.com/r/learnmath/s/zWqHOaOpws) and got 100+ messages, mostly for one-off advice. Those are welcome too, but this time I'm mainly looking for longer-term tutoring.

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.

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?

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

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 0.360mg, then 360g 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?

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!!

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.
🫶

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?

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?

# Evolutionary Calculator 2.0 - A Complete Arithmetic System

## The Discovery

I spent years developing a completely new approach to arithmetic. Today I'm sharing **Evolutionary Calculator 2.0** - a mathematical system that works PERFECTLY without zero.

This isn't theory. This is built, tested, and proven. The code is open source. You can verify everything yourself.

## What Makes It Revolutionary

Traditional mathematics treats zero as **essential**. But what if it isn't?

I created a **Base-9 arithmetic system** using only digits 1-9. Here's what it does:

- ✅ Complete arithmetic (+, -, ×, ÷)

- ✅ **Perfect reciprocity** (every operation is 100% reversible)

- ✅ Full decimal support

- ✅ Negative numbers work flawlessly

- ✅ All results are internally consistent

- ✅ No approximation, no rounding errors

## The Shocking Results

Here's where it gets interesting. When you test this system, you discover something SURPRISING:

### Example 1: Basic Division (The First Surprise)

```

Standard Math: 4.6 ÷ 7 = 0.6571

My System: 4.6 ÷ 7 = 0.6 ✓

But here's the proof:

0.6 × 7 = 4.6 exactly ✓ (PERFECT RECIPROCITY!)

```

**Why is this important?** In standard math, 0.6571 × 7 = 4.6 (with rounding). In my system, 0.6 × 7 = 4.6 **exactly**. Perfect precision.

### Example 2: Another Division (Still Surprising)

```

Standard Math: 46 ÷ 7 = 6.5714

My System: 46 ÷ 7 = 6 ✓

Proof:

6 × 7 = 46 exactly ✓

```

**Again: Perfect reciprocity.** Every operation reverses flawlessly.

### Example 3: Large Number Division

```

Standard Math: 654,789 ÷ 3 = 218,263

My System: 654,789 ÷ 3 = 217,559 ✓

Check:

217,559 × 3 = 654,789 exactly ✓ (PERFECT!)

```

This is where people usually stop and think: "Wait... what?"

### Example 4: Huge Numbers (Proof of Scale)

```

Standard Math: 875,463,219 ÷ 9 = 97,273,691

My System: 875,463,219 ÷ 9 = 87,546,322 ✓

Verification:

87,546,322 × 9 = 875,463,219 exactly ✓

```

This proves it works at ANY scale. Not just simple numbers.

### Example 5: Addition (The Pattern Continues)

```

Standard Math: 99 + 11 = 110

My System: 99 + 11 = 121 ✓

Proof:

121 - 11 = 99 exactly ✓

```

### Example 6: More Addition

```

Standard Math: 999 + 1 = 1,000

My System: 999 + 1 = 1,111 ✓

Proof:

1,111 - 1 = 999 exactly ✓

```

### Example 7: Subtraction (Complete Proof)

```

Standard Math: 1,111 - 2 = 1,109

My System: 1,111 - 2 = 998 ✓

Proof:

998 + 2 = 1,111 exactly ✓

```

**Even negative results have perfect reciprocity.**

## Why This Changes Everything

### 🧠 Assumption Challenged

For centuries, we believed zero was mathematically essential. This system proves it isn't. A complete, functional arithmetic exists without it.

### 💡 New Perspective

This shows that mathematics isn't ONE truth. It's a **framework**. Different systems can exist, each with their own logic and consistency.

### 🔬 Practical Proof

I didn't write theory. I built it. The code is public. You can test it yourself RIGHT NOW.

### 🚀 The Real Discovery

The biggest surprise isn't that the system works. It's that **perfect reciprocity** is possible. Every single operation is 100% reversible with zero approximation error.

That's something most mathematical systems can't claim.

## The Perfect Reciprocity Principle

**This is the heart of Evolutionary Calculator 2.0:**

Every operation you perform can be perfectly reversed:

- Divide 4.6 ÷ 7 = 0.6 → Multiply 0.6 × 7 = **exactly** 4.6

- Add 99 + 11 = 121 → Subtract 121 - 11 = **exactly** 99

- Divide 654,789 ÷ 3 = 217,559 → Multiply 217,559 × 3 = **exactly** 654,789

**No rounding. No floating-point errors. No approximation.**

Perfect precision. Every single time.

## Test It Yourself

The entire system is open source. Download it and verify:

```bash

git clone https://github.com/julien1986chagnon-alt/calculatrice-evolutive.git

cd calculatrice-evolutive

python3 calculator.py

```

Then test these yourself:

- `4.6 ÷ 7` (Compare to 0.6571 - see the difference?)

- `46 ÷ 7` (Compare to 6.5714 - completely different!)

- `654,789 ÷ 3` (Compare to 218,263 - surprising, right?)

- `99 + 11` (Compare to 110 - see the pattern?)

- `999 + 1` (Compare to 1,000 - the system is consistent!)

- Any numbers you want (prove it yourself!)

## GitHub Repository

📍 **https://github.com/julien1986chagnon-alt/calculatrice-evolutive\*\*

Inside you'll find:

- Complete Python source code

- Interactive calculator (just run it and test!)

- Multiple test cases with results

- Full documentation

- Everything you need to understand and verify

## The Questions This Raises

1. **If zero isn't necessary, what else might we be wrong about?**

2. **Why did we accept zero as essential without questioning it?**

3. **Are there other undiscovered number systems?**

4. **What happens when we challenge other "fundamental" assumptions?**

5. **What are the implications for computer science, physics, or other fields?**

## Why I'm Sharing This

This discovery took years of work. Thousands of hours of testing. Hundreds of calculations verified.

But it's not mine to keep. Mathematical discoveries belong to everyone.

So here it is: **Evolutionary Calculator 2.0** - complete, tested, proven, and ready for you to explore.

The code doesn't lie. The math speaks for itself.

## Try It Now

The surprise isn't reading about it. The surprise is **testing it yourself** and seeing that it actually works.

Download the calculator. Run it. Test any numbers you want.

Prove it. Or disprove it.

Either way, you'll understand why this matters.

---

**Questions? Comments? Ideas?** Drop them below. This is just the beginning.

**The Evolutionary Calculator 2.0 is ready. Are you?** 🔥