Multiplication, how this shit makes any sense to you?

I haven't cringed this hard in weeks...
months?

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If you multiply any number by 1, you get the number, simple concept right? (3)(1) = 3, (200)(1) = 200, So if you multiply any number by less than one, your answer will be smaller than the original number. Jesus Christ this thread is autistic

Go back to grade school. You have a candy bar. You give half to your best friend and keep half. You then eat half of what you kept saving the rest for later. How much candy bar are you saving to eat later as a fraction of the whole candy bar you had at the beginning?

>dotting two scalars together
what did he mean by this?

>10cm x 10cm = 100cm^2
>0.1m x 0.1m = 0.01m^2 = 1cm^2
I remember freaking out about it in middle school

number field is trivial inner product space over itself, where dot product is field multiplication

1.1*2=2.2 explain that

Half of one half is pretty easy to grasp. Is this bait?

Count to 10 half way
You get 5

Count to a half (0.5) half way (0.25)

>counting all the numbers between 0 and 0.5

rekt

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count only natural numbers obviously

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Yes, multiply any number by anything lower than 1 and the result will be lower.

3x3 means you give 3 people 3 oranges each, therefore (3+3+3)

1/2x1/2 means you give 1/2 people 1/2 oranges each, therefore (1/2+1/2) which is 1 QED

A dot B is equal to the magnitude of A times the magnitude of B times the cosine of the angle between them. Since all scalars reside on the number line the angle between them is 0, cos(0)=1, and the magnitude of a scalar is the scalar itself. Just think for a little bit and it'll make sense to you. There's a reason elementary multiplication can be written as A*B or AxB

You're a fucking retard, it's confirmed now.

You have stumbled upon something that most people just ignore.

We don't teach kids arithmetic correctly and as a result a lot of people don't actually grasp what something simple such as multiplication actually is. Anybody who tries to explain multiplication using groups of apples or w/e makes my head bleed, and yet that is how we explain it to children. To fully understand this, we have to go back to the beginning of mathematics.

Ok let's begin, first you start with addition on the numbers 1,2,3,... (which is called the natural numbers, denoted N). Everybody understands this correctly, it is numbers that actually exist, with an operation called addition which is easily understood by children (counting your sheep or w/e). But you are missing two things. An identity and an inverse.

An identity is an object (number if you will since we are calling this a set of numbers) with the special property that with respect to our operation, it leaves things unchanged. ie 3+x=3. This object is 0 (the additive identity), and itself has a fascinating history. It is very non-obvious and almost all ancient civilizations never discovered it. If you actually think about 0 you realize it doesn't actually exist in the real world, it is the very first abstract object in mathematics and the most important.

The other thing we are missing are inverses. Additive inverses are numbers that undo or reverse addition. ie 3+x=0. These numbers are the negative integers, one for each natural number, so -1, -2, -3,... ect. Likewise if 1 is supposed to represent ( you have 1 sheep) then the concept of -1 is unnatural, you can't have -1 sheep, that is an abstract concept.

both of these abstract objects are hugely important if you want to advance mathematics so lets instead consider our number system as (..., -3, -2, -1, 0, 1, 2, 3,... ). We call this the integers or Z. (notice that subtraction isn't actually a thing, all you need is addition. Subtraction is just a fancy word for addition with these new fancy inverse numbers we added).

Ok good, now we are ready to move to the next level and introduce multiplication. In fact we already have multiplication. Multiplication is just iterated addition. If you want to add a lot of numbers together, and you are lazy, it is easier to multiply them instead (notation wise). So we define another operation on the integers. a*b= a+a+...+a (where their are b + signs). Notice that also a*b=b*a (this is not obvious). Now there is a problem here, and this is where the (5 groups of 3 bananas) horseshit falls down. This definition only makes sense for natural numbers currently (what the fuck does 0 or -3 "+" signs mean? these numbers are unnatural) We fix this by "choosing" what these expressions mean in such a way that everything is consistent and makes sense.

Notice that:

1) 0*a = 0+0+ .. + 0 = 0 (so if we want consistency we need a*0 = 0, likewise we set 0*0 = 0)

2) -(a*b) = -(a+a+ ... +a) = -a+ -a + ... + -a = -a*b (and so we set a*-b = -b*a = -(b*a) = -(a*b) )

Notice that from 2) we get the result that -1*-1 = -(-1*1) = -(-1) = 1, which often confuses kids when they first see it.

Ok great, now we have defined multiplication on all of the integers. But remember, given an arithmetic operation, we want two things, an identity and an inverse. We have one but not the other. Notice that 1*b= 1+1+..+1 = b = b*1. That is multiplication by 1 doesn't do anything, it's a multiplicative identity (just like 0 is the additive one). But we do not have multiplicative inverses. So like before, let's throw in an abstract number for every integer and call it it's multiplicative inverse. These are the numbers (..., -1/3, -1/2,1/2, 1/3, ... ect) defined by for example 3*1/3 = 1.

Uh oh, what about 1/0? well there is a small problem. If we define 1/0 * 0 = 1 we are going to run into a big consistency issue, so we are going to exclude it. I should mention that there is a way to add in the expression 1/0 in a way that makes sense, but this expression is NOT the multiplicative inverse of 0, and adding it skrews around with our goal of defining arithmetic for ALL our numbers (namely 1/0*0 doesn't make sense, but that's not such a bad thing to do, you get a HUGE amount of mathematical progress by doing it, but that's a story for another day)

Ok great, but now we have some problems, one of which is your issue (what is 1/2*1/2 = ??) and also unlike before we have TWO arithmetic operations now, and we need to define expressions such as (1/3 + 1/5 = ?? ). Like before, we choose what these are in a way that makes sense and is consistent with everything we have done so far.

Notice that 1/a*b = 1/a+1/a + ... + 1/a = ??

call this expression b/a. That is 1/a*b=b*1/a=b/a. Notice that there is something subtle happening here. 4/2 is an expression that is already in our numbers (we defined 1/2*2=1, so it must follow that 1/2*4 = 1+1=2) but 3/2 is something new.

We need to impose a rule on our new set of numbers (expressions a/b were a and b are integers (b not zero), we call this the set of rational numbers, or Q) called an equivalence relation.

That is a/b = c/d if a*d= c*b (so 3/2 = -6/-4 because 3*-4 = -6*2 = -12. notice if we allowed 0/0 then it would be equal to everything, a big no-no ). We also see that any integer is in our rationals by setting a=a/1.

Now that we have a new way of looking at our numbers, we need everything that we have developed so far (addition and multiplication ) to carry over and be consistent.

Define addition on our rationals like this

a/b+c/d = (a*d+c*b)/(b*d)

and multiplication

a/b*c/d = (a*c)/(b*d)

Notice this is consistent with our arithmetic on the integers because a+b=a/1+b/1=(a+b)/1=a+b, and also a*b=a/1*b/1=(a*b)/(1*1) = a*b. Using the equivalence relation, our new arithmetic then extends these operations to all our new numbers.

So finally 1/2*1/2=1/(2*2)=1/4. Or (the inverse of 2)*(the inverse of 2)=(the inverse of 2*2). There are no cakes or chocolate biscuits here.

Now you have implicitly put a ordering on your natural numbers, you have defined 2>1 and so forth, which is great, so our numbers have geometry now (there is a notion of distance). You have also observed the property that for natural numbers, a*b>= a and b. But recall that 1

wtf have i done, the kid asked why 1/2*1/2=1/4 and i sperged out and wrote a fucking novel

I think it's simpler to grasp if you just say that multiplication is equivalent to multiple (hue) additions of the same number so if you have a*b you add a b times to zero (or vice versa). So if you have 1/2*1/2, how many times do you add 1/2 to zero? Half a time and what's half of 1/2? 1/4.

Imagine a square whose sides are length 1.
The area is ,1*1=1.
Now, imagine a square with side length ,1/2. The area of this square is 1/4 the area if the big square.

>half of 1/2=1/4

you are using the knowledge of 1/2*1/2=1/4 to show that 1/2*1/2=1/4

See

you're still doing it

So basically you want a proof of division and multiplication

Yo are you stupid? R is a vector space over itself.

That doesn't explain why -1*1 = 1 though.

The only serious answer.

1 cm^2 doesn't equal .01m^2

>1 cm^2
try 100