I found your blog randomly and saw your post about struggling with math. I hope this isn’t out of line coming from a total stranger, but have you been tested for the learning disability called dyscalculia? I have it myself and getting tested for it answered a LOT of questions about why math was so difficult for me. If you have it, you can get accommodations for it in school! Sorry if this was inappropriate, but I didn’t want to leave without dropping that info just in case it may be helpful.
aw hi there! not inappropriate at all, thanks for caring! 🙏 Yes i actually have, when i was about 15 i was evaluated for dyscalculia. I was apparently super close to getting the diagnosis but was a little too quick with some assignments to actually get it 🙃 I got an autism diagnosis later on though which made sense with my uneven cognitive profile. I did get some accommodations for my autism during this math course, at least. I hope youre doing okay stranger 🌸
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I'm trying to make sense of My Strawberry Film and the colors.
Ryo is a Brown Boy and Hikaru is a Blue Boy.
See the brown? Yes!
And see the blue? Yes!
But the red/pink ball of 💕love💕 has to mean something, right?
So we have the boys' colors, but what about the lady friends?
I think Chika (CHICA!) is a Green Gal. (I hate the Japanese filter because is it yellow? Is it green? Is it bad eyesight?)
Which means Minami is our strawberry-colored ball of 💕love💕 since Hikaru is in love with her.
But why these outfits on the double date?
Chika had a green dress withthat red/pink cardigan, and Minami wore the green colorful sweater with the pink scarf and coral bag while the boys wore their colors.
And Chika has that green and pink scarf.
So what is going on with these girls?!
Make it make sense, Mary!
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You have to know that I LOVE talking math and I am used to people stopping listening after three words so - don't worry if this isn't anything for you after all but also please ask any and all questions <3
Without further ado: Addition and multiplication explained structurally as I understand them in as simple terms as possible
Do you know what a function is? - starting with the hardest part right off the bat but that's math for you 😅 So a function is like a recipe of what you do to things in one pot, named 'set' to return things from another set (so another pot). The sets could be anything, meaning there could be anything in either pot, even other pots. The things inside the set are called 'elements'. So a function takes an element from the first set and then returns an element from the other.
You might know functions as like f(x) =x^2 or something but they can look very diverse.
Addition and multiplication are such functions that follow certain rules. We call them 'operations'. (I think we do that because they take elements from the same (not really but kinda) set that they return elements from but I might be mistaken. Never been good with remembering names.)
So if we perform these operations on a set X, they will take elements from the set X×X and return an element from X. What does that mean? Well, in easy words it means they are functions that independently choose two (always 2!) elements from X and return one element from X as their sum/product.
And then the following rules apply for both addition and multiplication: (I am going to use the symbol * for either operation now)
They are associative, that is a*(b*c) =(a*b) *c (this literally means that within pure addition/multiplication the order of performing the operation is completely random - well as long as we don't change the order of the letters themselves. So by this rule alone, we can't start with the operation a*c because there is a 'b' in between but the choice between starting with ab vs b*c is free)
There is a neutral element. That is, there exists an element e in X such that e*a=a*e=a for all a in X. For addition this is the 0. No matter whether I add 0 from the left or the right to any number, I still get that same number. For multiplication this is the 1.)
There is an inverse. That is, for any element a in X there exists an inverse element b such that a*b=b*a=e, meaning if we perform the operation on a and B we get the neutral element. For addition, this is b=-a because a+(-a) =0 and for multiplication this is b=1/a because a1/a=1 (note the different neutral elements here). (also note that for multiplication in the real numbers we have to exclude the 0 here as it is the only number without an multiplicative inverse)
The operation is commutative, that means a*b=b*a. And THIS is the rule that basically tells us it is COMPLETELY random in which order we perform the operation AS LONG AS we have pure addition or pure multiplication.
Little capybara gif in between to take a breather:
[ID: gif of a happy capybara swimming with oranges. end I'd]
So what do we have until now:
There is no subtraction or division. That's a lie they tell you in school. Well, I mean, there is but those operations are covered in case 3 by the inverse. Like, really, structurally division and subtraction ARE NO DEFINED OPERATIONS. They are simply what we call when one of the elements we pick is the inverse of an element that would feel more 'natural' to pick. (hah, pun intended because the ones that feel natural to pick tend to be from the natural numbers) What do I mean by that? So if I asked you to give me an example for an addition, you would probably come up with something like 3 + 1 or 8+5. You would most probably not pick 1 + (-9), even though -9 is a real number, even an integer and something we should learn in school how to handle. At least if we get a good teacher. But 1 + - 9 is an addition on the reals! Same for the multiplication (1/6)*2, also a rather simple multiplication on the reals. (You may have noticed that I used the 'inverse' in the second place in the first case and the first place in the second case so obviously that is random, too. I could also do (-1)+(-3), bam : addition!) (btw I am putting inverse in '' on purpose here, because the inverse of the inverse is the original element again. So while we can say -3 is the inverse of 3 it is as true that 3 is the inverse of -3. Like subtracting 2 by - 3 would be calculating 2 +3, you know? Confusing? Yeah, that's why we like to ignore the existence of subtraction/division :))
With pure addition or multiplication - read with addition/subtraction OR multiplication/division there is NO SET ORDER TO THE OPERATIONS WHATSOEVER. This is because of point 1 and 4: point 1 says that for any expression with only one operation performed several times I can start with whichever one and continue until all have been performed. Left to right, right to left, from the middle outwards, jumping around - does not make any different. Point 4 says that in addition, I can randomly exchange the order of elements in addition, so really I can solve the expression HOWEVER I WANT (within the rules, of course, but the order is entirely random). Like that's what these rules are FOR.
Okay, but what if we mix addition and multiplication now? Then we get one (1!) extra rule:
5. For any elements a, b, c in X we have: (here now * is only multiplication and + is addition)
a*(b+c) = a*b + a*c (distributive law)
So this is the only thing that tells us how to handle a mixture of addition and multiplication. That's why we have the 'multiplication before addition' rule - there just is no structure for doing it the other way round! You know?
Like, with the distributive law we know what to do when multiplying something with a sum. But there is no rule for what to do when adding to a product. So we first have to solve the product to get an element in the set of real numbers, because for that we know how to perform the operation. That's where the hierarchy comes from (I believe).
Funfact: in Germany this (multiplication before addition) is the only rule taught in school (or at least it was when I went to school in Germany). They don't have pemdas or whatever but 'Punkt vor Strich' which directly translates to 'Point before stroke' bc multiplication is one little point and division two little points. And addition is a Swiss cross and subtraction is a horizontal line. I have to admit it messed me up a little seeing as I very quickly used fractions / is also a stroke rather than a point.
And then the last casual and very obvious rule is 'brackets help us understand things so they are stronger than anything else'.
In conclusion:
there is no hierarchy between multiplication/division and addition/subtraction respectively because they are literally the same thing
order within the same kind of operation is completely arbitrary
you can always only perform an operation on two elements of your set OR perform the distributive law and this leads to the 'point before stroke' rule
Brackets are your friends and thus you prioritise them <3
I hope this makes sense, halfway?
If you want to look this up in more mathematical terms/clean definitions, then I have given you the rules for a 'field' and 'abelian groups' here. Also you might want to familiarise yourself with functions if you have trouble understanding this. I feel like getting what a function really is is THE entry step into understanding math. (but this is a personal opinion and I bet there are mathematicians who would fight me over that)
Because these are actually more general algebraic constructs and not just what happens in the reals when we add/multiply. They ARE the most simple examples of what you encounter in algebra imo but that might be because we are used to dealing with the reals/rational number from child age. Anyway, there are loads of different objects that work differently but with similar rules. Vector spaces and rings are lovely for example, if you want to have a look there. Or modules or algebras if you'd like it to be a little more difficult ;)
Have an image of some butterflies as a treat because you read until here <3
[ID: image of four butterflies on a shrub. end ID]
(I hope you like butterflies. I love them :))
(also, I really hope I didn't make any mistakes up there 😅 but more or less this is what Lilly meant by 'equal weight' of operarions)
holy fucking shit you are smart
i feel like this all makes sense (???) but i have one question: what tf is pure/not pure multiplication and addition? like, i understood everything but that just like messed with my brain 😭
i feel like this is such a good description of mathing that my brain is in shock lol
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