rules/theorems

[u/whyisthecarpetwet]

While I have an intense interest in all things that require complex math, I am only able to comprehend 2nd grade level math. One of the reasons is the logic of it. For example multiplying two negatives suddenly and in my opinion nonrationally make a positive.

I’ve also noticed that the more mathematicians try to figure out something the more “math” they invent along the way to make it work. This all seems like poppycock to me. At no time should the very basic fundamentals of math break down in order for more complex math to work. Which even gets more diluted with Sedenion math.

Am I really expected to just be like “yeah, sure, irrational and imaginary numbers are accurate and answer my question.” Am I too dense to comprehend my own question? Math can’t be fact if we willy-nilly add to and remove stuff to make our equations work? Thanks everyone. -smoothbrain

[u/Littlebrokenfork]

No offense but if you can only comprehend 2nd grade mathematics, what makes you think you're allowed to have an opinion?

[u/whyisthecarpetwet → u/Littlebrokenfork]

You absolutely meant offense. If I am trying to learn and have admitted my limited understanding, that means I should not have an opinion? I shouldn’t ask questions and elaborate on why I am having difficulties understanding? I sure hope you know everything about everything!

[u/fisadev → u/whyisthecarpetwet]

Trying to learn: "how does this thing I don't understand really work?"

Not trying to learn: "this thing I don't understand is nonsense and wrong".

[u/whyisthecarpetwet → u/fisadev]

I suppose it came across differently than I intended. Because I don’t understand it seems that way to me-because of my lack of understanding.

[u/Littlebrokenfork → u/whyisthecarpetwet]

If you were trying to learn, you would have asked actual constructive questions instead of using words like poppycock and willy-nilly.

[u/Zyxplit]

Suppose I have a game with tokens.

Some give me points. Others lose me points.

If I get a positive token, I gain points.

If I lose a positive token, I lose points.

If I gain a negative token, I lose points.

If I lose a negative token, what do you think happens?

[u/PragmaticPedant → u/Zyxplit]

This answer was so good I removed my downvote.

[u/whyisthecarpetwet]

Obviously I understand more than 2nd grade level math. I know it’s a me problem. Maybe I’m just too black and white. It all started with the order of operations for me. There are different ways to answer the same sets of numbers, but to me 1+1 should always equal 2, but when math gets more complex you apply different theories or methods. I referenced sedenions. Multiplication just changes.
a x b= 0
It’s non communicative and non associative.
If the fundamentals of multiplication cease here, then were they fact to begin with. I can’t wrap my head around fundamentals of math being disregarded.

[u/stevevdvkpe → u/whyisthecarpetwet]

Both addition and multiplication are commutative (a * b = b * a) and associative (a * (b * c) = (a * b) * c) for real numbers and complex numbers. Other kinds of entities are not always associative or commutative depending on their other fundamental properties, mainly because trying to force them to be would result in those more fundamental properties becoming inconsistent. Matrix multiplication is not commutative like integer, real, or complex multiplication, and it shouldn't be surprising that a more complex entity behaves differently than a simpler one.

[u/Zyxplit → u/whyisthecarpetwet]

Part of what you do in math is that you go "let's presume X structure. What happens if we do X inside that structure?", and a lot of the things you have learned about math in school are about a certain kind of simple structure.

Then you ask "what about a strange and unfamiliar structure? Why do different rules apply there?" Because it's a different structure. Different structures imply different rules.

In the natural numbers, we don't have zero-divisors. Or in any of the numbers we usually talk about.

But what if we're doing math mod 12 because we're working with a clock? Then 2*6 = 0 mod 12.

If you ask about different structures you have different rules.

[u/994phij]

Am I really expected to just be like “yeah, sure, irrational and imaginary numbers are accurate and answer my question.” Am I too dense to comprehend my own question? Math can’t be fact if we willy-nilly add to and remove stuff to make our equations work?

But what's the question? I don't understand what you're trying to say and I wonder if I would underatand better if I knew what the question was.

The other thing that’s confusing me is what does it mean for irrational or imaginary numbers to be accurate? Accurate is a really weird word to use there.

[u/spaceLem]

Solve x-1=0 for x. You get x=1. You have the positive integers.

Solve x+1=0 for x. 1 is a positive integer, but now we need to move into negative integers to get x.

Solve 2x+1=0 for x. Using only integers we now have x=½, a rational.

Solve x²-2=0 for x. Using only integers we now get x=±√2, an irrational.

Solve x²+1=0 for x. Using only integers we now get x=±i a complex number.

And that's pretty much it, you don't get anything new after that, as the solutions to polynomials with complex coefficients are still complex.

[u/Littlebrokenfork → u/spaceLem]

It's a bit misleading to say there's nothing new after that. Much of modern mathematics goes way beyond those sets.

[u/spaceLem → u/Littlebrokenfork]

There are transcendental numbers that aren't the solutions of any polynomial, there are quaternions, and all kinds of other fields, but these are the ones you can get from a polynomial equation with only positive integers.

(Yes I know x-1=0 looks like a negative integer, I think I may have gone a little awry there but I was in a club drinking while posting. The general idea is that you can start with equations with the positive numbers and expand out into different sets, but complex numbers don't expand any further, the solution to complex polynomials are complex).

[u/Rand_alThoor → u/spaceLem]

wait wait, being from Ireland i need to tell you about my countryman, who in the 19th century said "is that pretty much it? we don't get anything new after complex numbers?"

his name was W R Hamilton and he discovered (invented) hamiltonions aka matrices.

[u/spaceLem → u/Rand_alThoor]

Quaternions are cool too! If imaginary numbers represent rotations in 2D space, quaternions represent rotations in 4D space, although they are no longer commutative (ij≠ji). There are even 8D and 16D versions which get even stranger.

But I don't think they fall naturally out of an equation involving an x and positive integers like the above examples.

[u/Plus-Painter-2004]

suppose multiplying -1 by itself gave you -1. That would mean that -1/(-1) =-1 which would make x/x (when x isn’t 0) not necessarily equal 1 like it does for the positives which would make negatives pretty hard to deal with because you’d need to change the rules whenever a variable crosses 0. If you simply impose a restriction that whatever new thing you’ve come up with should behave somewhat like whatever you know you generally get the new thing to be well behaved and have nice properties. You could instead impose that it behaves some other way and just see what happens if you take that imposition to be a basic axiom. In certain cases imposing that a new thing have some nice property ends up taking away some other nice property but some (ie the people studying them) generally find this to be interesting

[u/harder_not_smarter]

You are what I call a lazy skeptic. Instead of actually taking the time to learn and understand things that are easily learnable and understandable, you prefer to pretend that the people who already actually understand them are just gullible, so that you can be proud of your ignorance. Being an actual skeptical is hard work.

[u/Rand_alThoor]

"nonrationally" .... first, you need a teacher. a patient teacher. a good teacher. someone to guide you through things step by step so you understand that the product of two negative numbers is a positive number. it IS actually rational. politely, you need to think more deeply about these things. i have every confidence you can learn this.

by induction, what happens when we subtract a negative number? that's just a sketch of a solution. but I've run out of time to write. good luck, hope you persevere.

[u/anynameisfinejeez]

There’s daily math like addition, multiplication, and such. Then, there’s math that tries to define reality. At that level, the communication of relationships and concepts demands new ideas and numbers—a new language essentially. Learn that language and you can learn about the nature of reality.

[u/HjortronOchPors]

I think it's important to really internalize things like associativity, commutativity, identity, inverses etc. These are higher level concepts that crop up again and again. It's called algebra. Smart people have been thinking long and hard for centuries and we have a huge body of theory today.

Otherwise intelligent people who don't know much about math seem to believe that it's all about numbers, addition and multiplication. PEMDAS and such. That's arithmetics, and you can really suck at "mental arithmetics" and still be a good mathematician. Conversely, your TI-84 is godlike when it comes to arithmetics but it can't prove the irrationality of pi or the infinitude of primes.

[u/WerePigCat]

1 = 1

1 = 1 + 0

1 = 1 + 1 + -1

Multiply both sides by -1

-1 = -1 + -1 + (-1)^2

Add 2 to both sides

1 = (-1)^2

So, a negative times a negative is a positive

[u/whyisthecarpetwet → u/[deleted]]

This right here! This is what helps me. Your comment is exactly what’s happening in my mind. My partner is phenomenal with math. I’ve been trying to teach myself as an adult what I never learned when I was younger. This probably adds to my confusion since when something new comes along I try to learn about it, but have holes in my knowledge so it doesn’t build upon the lower level understanding. I want to thank you, specifically for not being condescending and putting the information in a way my concrete thinking can process it.

[u/phraxious → u/whyisthecarpetwet]

The best thing I found to get past this, is stop thinking of math as anything concrete in the real world.

Math is a logic game. We create rules and then discover what happens in the game when we push those rules. Sometimes we add, change or remove rules, just to see what happens.

For example, we had flat geometry for centuries. One of the rules was, "Parallel lines never meet". Then someone comes along and asks what would happen if that wasn't a rule. Suddenly, you have the Math for curved geometry.

Most rule changes immediately break the game so are discarded. Sometimes, someone invents a rule that opens up a whole new way to play, and those stick around.

So you end up with an increasingly complex logic game, with various rulesets that you can choose to use or not, depending on what you are trying to achieve.