Art of Problem Solving - Pre-Algebra

Eternity · August 15, 2025, 3:27pm

Project Summary

Goal

What’s your goal? Why do you have that goal? How will you judge success and failure? What bigger picture goals or values are you pursuing? How is this relevant to CF?

My goal is to finish Art of Problem Solving’s Pre-Algebra textbook. I want to work on some basic math stuff. I will have succeeded at this project when I have read all chapters in the textbook and completed all the problems. This is related to getting better at basic math skills for philosophy. Learning math is relevant to CF.

Plan

What’s your plan? How big is the project? What resources do you expect it to require and what have you allocated for it? How confident are you about succeeding? What sort of errors or error rate do you expect and how will you deal with that? Got any error correction mechanisms? What are the risks of not finishing the project or failing and do you have any plan to address those risks?

Spend at least 15 minutes a day with a two day buffer. I do intend to do more than that, probably an hour a day, but I just want to make the goal small. I’ve gone through a lot of this textbook before. Part of this is review and part of this is getting feedback from stuff I previously just self-studied.

I don’t know how big the project is. I don’t think its that big. I remember going through it in the past pretty quickly while feeling like I understood things okay. I don’t have any projects to compare to (of my own I mean).

I’ve finally got my schedule shortened and gotten shorter shifts on many days. So I have more time available to put towards CF.

I’m fairly confident about succeeding. I nearly finished this math textbook in the past while being a bit busier with a poorer lifestyle.

I expect some days to be too tired to do stuff. So I gave some buffer. I also made the goal short so that I will do a bit everyday and (probably) end up doing more.

On the math side of things one error I expect to run into is rushing things. Idk how to address this other than I intend to be more methodical in doing my math problems.

I’m unsure of any risks of failing this project. Mmm. I have a habit of dropping stuff like this and revisiting stuff later. I think thats a risk. Though this is only the second time I’m touching this textbook, with other stuff in my life I have re-done the basics a bunch because of dropping stuff. That seems like a waste of time. Idk how to address that risk directly. I just kinda aimed to make this a manageable project so I don’t end up dropping it. I also think finishing my latest college math class gave me some confidence.

Other People

What help are you asking from others? What value are you offering to others? Will you complete the project independently if no one else participates? Why are you sharing this with others? What sort of criticism do you want?

Feedback on some stuff I don’t understand.

Idk if there’s any value here to others (this is partially just my tutoring stuff done as a project, I thought it would be neater this way). Mmm. I think there’s some value in seeing how others learn and think about stuff.

Context

What’s the context? What’s your relevant background and track record? Why are you prioritizing this over alternative projects? Why are you doing it right now? What have you already done?’

I like math. I liked the math class (more-or-less) I just recently finished. I’m trying to build a record of doing stuff and I feel like doing something like math, which I really like, would help with that.

Eternity · August 15, 2025, 3:32pm

Some notes I took recently on Chapter 1.1 Why Start with Arithmetic?

Not everybody agrees on what “prealgebra” means, but we (the writers of this book) like to think of prealgebra as the bridge between arithmetic and algebra.

Hmm. I guess that makes sense, but I wonder: why is it hard (I’m assuming here) to define the math between arithmetic and algebra?

Arithmetic refers to the basics of adding, subtracting, multiplying, dividing, and (maybe) more exotic things like squares and square roots. You probably learned most of these basics already.’

Mmm. What does wikipedia say arithmetic is? From https://en.wikipedia.org/wiki/Arithmetic:

Arithmetic is an elementary branch of mathematics that deals with numerical operations like addition, subtraction, multiplication, and division. In a wider sense, it also includes exponentiation, extraction of roots, and taking logarithms.

Hmm. Idk. This feels to me as if they needed a term to denote all of “simple” mathematics. Then again I don’t really know what algebra is and how that definition would work. Idk, this just doesn’t feel well defined to me.

That toolbox is algebra. Algebra is the language of all advanced mathematics.

How all we talking? Actually all? If so, what about Algebra allows for that?

Algebra gives us tools to take our concepts from arithmetic and make them general, meaning that we can use the concepts not just for arithmetic problems, but for other sorts of problems, too.

Is this why? Instead of having 2 (3 + 5) = 2(3) + 2(5), we can instead have a (x + y) = ax + ay. Is it this variable(?) language that defines Algebra?

Hmm. So I don’t understand negatives and therefore 1 well. I do understand 2 and 3 I think. I do understand the fourth one too.

You’ll know all these things not because you’ve blindly applied some calculation, or because you’ve memorized some formula–instead, you’ll understand the mathematics behind all these expressions.

I remember on my first go through that not being the case for me.

Unfortunately, different mathematicians and different textbooks may use slightly different words for the same concept, in the same way that what an American calls a “truck” is called a “lorry” by people in Great Britain. So, before we go any farther, we want to make sure that we all agree on some of the words that we’re going to use.

An integer is a number without a fractional part

That includes 0. Its a number and it has no fractional part.

So an integer is a whole number/it has no fractional part. A positive number is a number greater than 0. A negative number is a number less than 0. A nonnegative number includes positives numbers and zero. A nonpositive number includes negative numbers and 0. A nonzero number is any number that is not zero or, in other words, a positive or negative number.

Oh yeah, some text will be screenshotted when copy and pasting it doesn’t go well.

Elliot · August 16, 2025, 3:17pm

It said you should understand that by the end of the chapter. Did you finish the chapter yet? Do you understand it yet? If you have trouble, share some of the book’s explanation about it and say what the problem is.

Eternity · August 22, 2025, 9:23pm

Chapter 1.2 Addition:

1.1) The order/organization of numbers doesn’t change the outcome of addition. 2 and then 3 or 3 then 2 doesn’t change the outcome. They’re the same.

Looks like I got it right.

Answer

Commutative property was introduced in the answer. a + b = b + a. Numbers can move around “commute” anywhere on one side of the equation (when it comes to addition).

1.2) I would answer it the same as my answer to 1.1.

Looks like I more or less got it right. I think my overall reasoning is fine? But I wasn’t going step by step counting the squares in a particular order like they were so idk.

Answer

Associative property was introduced here. a + (b + c) = (a + b) + c. numbers can be added in any group(?) first. they can associate however they want.

1.3)

a.) I would just say the same thing as 1.1. The order of adding stuff doesn’t matter. I guess another way to put it: let’s say you have a pile of things, it doesn’t matter how you split up that pile, it still adds up to one singular quantity.

b.) 472 + (219 + 28) = 472 + 28 + 219 = 500 + 219 = 719

Idk about a. I didn’t explain it in the way they did I feel, but I understand what they’re saying. b I got right.

Answer

1.4) (2+12+22+32) + (8+18+28+38) = 2+8+12+18+22+28+32+38 = 10+30+50+70 = 10+50+70+30= 60 + 100 = 160

Got it right but looks like there was an easier intended way. 40 + 40 + 40 + 40.

Answer

1.5)

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 + 17 + 18 + 19 + 20

1 + 19 + 2 + 18 + 3 + 17 + 4 + 16 + 5 + 15 + 6 + 14 + 7 + 13 + 8 + 12 + 11 + 9 + 20

20 + 20 + 20 + 20 + 20 + 20 + 20 + 20 + 20 + 20

200

oh I got this wrong. hmm. i’m missing a 10 in my summation. i think i got too caught up in making “perfect” little pairs of 20s that I just ended up ignoring the 10.

Answer

1.6) Zero adds nothing to a quantity.

I guess I got this right?

Answer

Exercises from chapter 1.2:

1.) Got it right.

My Work

Solution

2.) Got it right.

My Work

Solution

3.) Got it right.

My Work

Solution

4.) Got it right.

My Work

Solution

Eternity · August 22, 2025, 9:25pm

I have finished it before in the past. I’m going through it fairly fast and I’ll re-post that section soon.

I think my issue was that negative numbers never felt intuitive for me. Similar to how I understand subtraction as removing one quantity from another, but not adding a negative quantity to a positive quantity.

Elliot · August 23, 2025, 3:34pm

So what does the book say about negative numbers and what are your thoughts about what it says?

Elliot · August 23, 2025, 3:45pm

I don’t think this is clear enough and I didn’t think the book’s answer is clear enough either. One way to approach it better IMO is to relate addition to counting. Changing the grouping or location of units doesn’t affecting counting, which is only affected by the number of units. (Addition basically means count, then count again without starting back at 0.)

If you know how to count 5 pennies, you should see how moving them around on a table, in groups or just random positions, won’t affect the count. Counting puts units/objects in 1 to 1 correspondence with numbers and the count only changes when you add more units or take some away.

Does this make sense to you?

Your 20 wasn’t in a pair either.

Eternity · September 2, 2025, 4:51pm

I don’t think the books explanations for many things is too clear either. I think they offer difficult problems and I assume they do do a good job of creating math olympiads and what not, but I don’t think they do the best job of making you really understand the math like they claim they do.

Hmm. That makes sense. You have 5 apples on a table and then someone puts 3 more on a table. You can add them up by counting all the apples on table. It doesn’t matter where on the table the apples are. You still can count them up to 8.

Elliot · September 2, 2025, 5:32pm

yeah. it also doesn’t matter which apple you count first or second. order doesn’t matter. you just have to count all the apples and count each apple only once. you have to get one to one correspondence (of objects to counting numbers) right.

Eternity · September 2, 2025, 7:00pm

Chapter 1.4 Negation:

So -2 + 2 can be thought of as starting at -2 on a number line and then moving right two spaces (+2).

The key concept behind negation is numbers summing up to 0? I wonder if they’re claiming something along the lines of negative numbers being discovered because people were looking for stuff that adds up to 0. I remember hearing that this was why “imaginary” numbers were discovered. People looked for them because Euler proved something related to polynomials (afaik).

Ok.

1.16)

a) To the one above it? Each number is one larger than the one above it.

b.) 0

c.) 1

d.) 2

Answer

Oh. I didn’t read it carefully enough “are on the number line”. -1 is related to -2 on the number line by being one unit to the right of -2.

a) each number is three less than the number above it

b) -3

Answer

Ok. I get that I guess. There proof is: if we do x + -x first we have - - x. Which is x. If we do -x + - - x we’re left with x. so x = - - x.

Ok.

Ok. This one feels kinda pointless to me. -x, in my head, is kinda defined as negative 1 times the number its attached to.

1.18)

(-2) x 3 = -1 x 2 x 3 = -1 x 6 = -6 = -1(2 x 3)

Ok.

Problems w/ Answers:

So my issue with this chapter on negation is that I felt like it didn’t explain negation to me.

Why can I go left of the number line? What is a real world example of a negative number? What are we actually doing when we multiply a negative number? Idk. I think I have more questions but its hard to say them.

I guess my main confusions are:

1.) Why are we able to go left of the number line here?

2.) What is a real life example of a negative number? I think this is more so what I’m stuck on. Something I’ve commonly heard in school was that subtracting isn’t a thing, just adding negative numbers. But how? I see taking away of two apples from five apples. I don’t see the person “adding” negative two apples.

Elliot · September 3, 2025, 12:13am

That is a valid concept but certainly not the single key concept. You can also connect negation to counting backwards or to rotating points on the number line by 180 degrees with 0 as the pivot.

Have you tried to brainstorm any real life examples of negative numbers?

Suppose the number line represents distance from home. So you drive to the grocery store and go right on the number line. Then you drive home and go left on the number line.

Why wouldn’t you be able to go left on a number line? You can go left or right along a line you draw on the ground.

Eternity · September 4, 2025, 1:54am

Kinda? Nothing methodical. One example that always come to mind that I probably should have mentioned is debt.

The issue I have with debt is that it kinda feels fake in my head to consider that as a real life example of a negative number. I think I’m wrong on my thinking on this but I haven’t really resolved it: personal finance/economics and with that debt feel like something people made up. Since we “made it up” it feels wrong to consider debt a real example of negative numbers.

Brainstorming for about 5 minutes though:

debt
i’ve heard of negative charges in chemistry and stuff are those related to negative numbers?
same with negative poles in physics with magnets. are those related to negative numbers?
this is just something I came up with on the spot, but like holes or something? like if you’re filling in a hole I can see that as being represented as a negative number. like if a hole is 5m^3 then you’re adding 5m^3 to the hole to fill it back up. i can see why that volume could be represented as a negative number,

uhh thats about it

Well you’re going right and then going left. You’re moving along values that, in my head, actually exist. Going left from 0 is where the confusion is. Hmm. Though I say that negative numbers on a coordinate plane kinda deal make sense to me. There just directional things.

Elliot · September 5, 2025, 3:29pm

Yeah debt and hole are good examples but there are others. The science stuff is more complicated so if you don’t already understand it then it may not help.

Suppose you own 10 cats and you want to make sure they all come in at night. You have a cat door. All day long, you count whenever a cat enters or exits your home. You count up one when a cat leaves and down one a cat comes in.

Night falls and your count is at 3. What does that mean?

Night falls and your count is at -2. What does that mean?

In both cases, consider a meaning other than a counting error.

Suppose you’re trying to make a scale balance. It’s the old type with two platforms. So on the left platform you have some object you’re trying to weigh. On the right you add weights. When it balances you can see how many weights of what types were needed.

Say the right is 3g too light. You’re short 3g of weights on the scale. Guessing, you add a 5g weight. Now how short are you? Now, how much more weight do you need to add to balance it?

Eternity · September 12, 2025, 2:24am

1.3 Multiplication

I skipped the beginning problems. I don’t see much value in trying them except for the first one, since it involves me trying to explain multiplication:

Using the blocks, and using my understanding that multiplication is just repeated addition, 2 x 3 is columns of 2 blocks added together 3 times. 3 x 2 is columns of 3 blocks added together two times.

A 2 x 3 is the same as a 3 x 2 as long as you flip it around.

I mean I think what I said is pretty close. Their explanation just feels lacking as to what multiplication is. This is, afaik, all we got for what multiplication is.

My understanding of multiplication is just repeated addition. I know it has some other uses like making a number line bigger and stuff related to scaling, But, idk how to put it, I have trouble conceptualizing those as part of my understanding.

I did the end of chapter problems. Nothing interesting I think. I can share the work if wanted. Below are both beginning chapter questions (like the one above) and end of chapter practice questions.

1.3 Problems

Eternity · September 12, 2025, 2:37am

Do we start at 10 or 0? It doesn’t say but I think it makes sense to start at 0 since it doesn’t specify and its like we’re counting the total number of cats. Just counting up or down.

cat leaves = +1

cat comes in = -1

well if night falls and your count is at 3 then three cats have left.

if night falls and your count is at -2 then two cats have come in. hmm but for them to come in, they’d first have to leave. if two cats left +2 and when they come back -2. so it should always end up at 0. the only thing i can think of at the moment is that you had two additional cats that aren’t yours come in.

left = object we’re trying to weigh

right = weights

if you add a 5g weight you’re now over 2. you need to add -2 to balance it?

Elliot · September 12, 2025, 3:45pm

When you start a day with all cats home, the count should start at 0.

Yeah. The count suggests there are extra cats in your house.

yeah. so do these examples make sense to you for negative numbers? a scale can have too much or too little weight on it, so you need positive and negative numbers to be able to represent both. you can be missing cats or have extra cats, so negative numbers help represent that.

Elliot · September 12, 2025, 3:49pm

Besides repeated addition, multiplying is scaling or counting with groups or dealing with 2-dimensional area instead of a 1-dimensional number line.

Multiplying 2x3 means figure out how many total things are in 2 groups of 3 (or 3 groups of 2 – either way works).

Eternity · September 14, 2025, 9:44pm

1.5 Subtraction Numbers

So subtraction here is defined as the adding of a negative number. Hmm. Ok.

One thing I’ve realized thats causing my confusion: I think I find it cool, weird, surprising, confusing, idk what that the same rules can apply to things that seem so different.

Multiplying can be seen as repeated addition and the multiplication I learn works like that. Multiplication works as scaling. Scaling here is referring to how things are X times bigger than other things. Like I have a line that measures 1 inch. I can have another line that measures 3 inches. The second line is three times the size. Correct?

Subtraction here can be seen as both taking away something and, in another way, be seen as adding negative numbers to something.

I think I’m having trouble accepting? understanding? that these things can represent, what feels to me, as many different processes.

Problems

Eternity · September 14, 2025, 9:48pm

They do. I think I like the scale example. That feels to me like a more “real” example of a negative number.

Eternity · September 17, 2025, 1:24am

1.6 Reciprocals

Just the way it’s worded bothers me. Why is that the question we really want to ask? Screenshot of the context:

Context

The reciprocal of a number X is a number that returns 1 when multiplied by X. That number is also called the multiplicative inverse of x. So a number times its inverse returns 1.

a) 0

b) Well, just using part (a): 0 doesn’t have a reciprocal because a reciprocal, in this case, would be a number we multiply against 0 to get 1. However, any number times 0 is 0. So there can’t be a number that we can multiply against 0 to get 1.

I skipped the rest of beginning of chapter problems, though they are provided below.

I did complete the end of chapter problems, nothing I think worth sharing though. Provided at the end together with the beginning chapter problems.

Problems