Excellent work!

The world need more people like you , Cheers !

thank you so much, thats such amazing explanation ,

Sir, you are just expliaing the derivation like others but the logic is still left unexplained. what is the logic ???

There are unknown way to visualize subspace, or vector spaces. You can stretching the width of the x axis, for example, in the right line of a 3d stereo image, and also get depth, as shown below. L R |____| |______| TIP: To get the 3d depth, close one eye and focus on either left or right line, and then open it. This because the z axis uses x to get depth. Which means that you can get double depth to the image.... 4d depth??? :O It's possible to add 4d depth to a isometric cube, by moving the skew square side along the x axis of the right cube, individually, and leave the left cube unchanged. It's even possible to create an isometric/true perspective hybrid, by leaving the north west/south east line unchanged, while adding perspective to the north east/south west line. Would be awesome, if someone make a game (even KZbin video) that way! Nice video! :-) p.s You're good teacher!

underrated video 🔥

you help me a lot to understand this. Thank you... Wish you got millions subscriber in few days...

Can you make more videos of those please? You are amazing!! Thanks for all the tremendous effort!!

Very helpful explanations and animations.

pretty good. what about negative 3 in the bank?

Software?

This is the explanation that hammered down the nail for me. Thank you very much my great sir

same channel

Multiplication is repeated addition, WRONG! Multiplication is repeated addition OR repeated subtraction. The X in multiplication is the symbol for groups. First number tells you if you are adding or subtracting & how many times.

Basically the first number is your trajectory and the second number dictates if it continues that direction or flips the other way. Negative flips while positive keeps it going. Interesting

One of the best explanations I've seen of this topic. Well done!

Amazing... I didn't even know about the first axis type.I am realizing this channel is what I have been looking for, for a while. I love hearing and seeing the teaching in live experience, both teaching and demonstration.

Please read the book "Four-Dimensional Descriptive Geometry"'. The four 3D spaces are easy to see. kzbin.info/www/bejne/fqGthZKlmJiqaq9r

I'd explain -3*-4 as taking away negative 4 three times. If we're starting at zero, taking away negative four three times leaves us at positive 12, because taking away a negative is the same as adding.

Same as speaking in double negatives, which is how I just rationalized this explanation: by saying I am not NOT going to eat dinner, in effect I'm saying I'm going to eat dinner.

So you can take 2 negatives and equal a positive so 2x2=4 and -2x-2=4 lol so why even use the negative

This is the worst explanation I've ever heard of.

As a visual learner, this explained disjointed sets way better than my professor could. Very much appreciated

It's nice to see another axiom of mathematics!

I got it, then I didn't get it... The equation and/or demonstration at 11:55 only makes sense if you start with the number 1... But that number came right out of nowhere, and is not in the written statement. How, in reading (-3)x(-4), am I supposed to know that I need to start with the number 1, then complete the equation, versus starting with the number zero, or even starting with the number 17 or 39? conversly (just to make my questioning point very clear) when I multiply 3 x 4 to equal 12 I am starting with 3 then multiplying it by 4 to get 12.... I feel like in that example, I am starting with zero, then doing the work. I wouldn't even know how to start at 1 when multiplying 3 x 4... I mean, that could be 13 as 12+1, or it could be 3+1x4... oh... is that also 12? if we do the multiplication first? or would it be assumed that starting at 1 for 3 would equal 4 then times 4 to get 16? I Don't know, but still, you can see what my original question is getting at right?

I searched for so long to find someone who explains it with a visual representation and acknowledges that it creates a knot in the head. Many just present (-1)x(-1)=1 as a fact that needs to be accepted.

This still doesn't answer the question of why, which is the claim of the video. It's a good explanation of negative numbers and multiplication but not a conceptual answer to the why does a negative times a negative equal a positive.

omg most helpful god bless you thank you!!!

Soooooo Complex

Excellent explanation, thank you

thanks for this.

Please excuse all my thinking out loud here! :) And thanks for the video. I learned myself.

I was motivated to think of the following thanks to your video; so, that's good. However, the video didn't explain the point. [Another similar video was completely unsatisfactory; yours is better and more productive] However, ultimately getting to (-1)(-1) = 1 is a tautology. Of course, it's mathematically the case, but it doesn't explain why it is the case in terms of multiplication as repeated addition and subtraction (as someone else commented). The root problem is that presenting multiplication as repeated addition or subtraction on a number line is a fail. Clearly, if I start at 4 and multiply by -2, then I subtract 4 twice [4-4 = 0, 0-4 = -4], and end up at -4, not -8. Whyyyyyyyyyyyyyyyyyyyyyyy? Below is my attempt at an explanation (inspired by the example of your video). However, I tried the explanation below to someone very bad at math, and it didn't seem to light up the understanding. So, maybe it's still a fail. Rather than thinking of multiplication as repeated addition and subtraction in terms of a number line (as the example above crashes on), let x(y) mean something like "put y blocks from your hand onto a table x times," then we see that the operation involves either moving blocks (in your hand) onto the table or (from the table) into your hand. So, if we first do 3(4), this moves 3 sets of 4 blocks (from my hand) to the table, giving 12 blocks on the table (as expected). Then given -3(4) or 3(-4), this will involve moving groups of blocks from the table to my hand, leaving zero blocks on the table (and thus convincing myself that I have somehow "reversed" what I did with 3(4) to begin with]. [Also, while -3(4) and 3(-4) are obviously mathematically interchangeable, it is not intuitive how "removing" -4 blocks 3 times or "removing" 4 blocks -3 times is actually the same thing, if it is. But set that aside for here.] However, if there were not already blocks on the table, then after -3(4) or -3(4), I will have borrowed or stolen or subtracted 12 blocks from the table to my hand, and there will be a "hole" or "deficit" of -12 blocks on the table (as expected). Hmm. No one told us that we could "steal" from an empty table in this way, but apparently we can! Also, no one told us that we apparently have a stock of blocks in our hands to be setting down on the table (in groups of 3) in the first place, but apparently we do! Where did these come from? From the fact that moving blocks (that already exist) to and from spaces is a better metaphor for presenting multiplication as repeated addition or subtraction. As such, the situation involves blocks moving from the table (to my hand) or from my hand (to the table). As such, if (-3)(-4) will involve some kind of "opposite" movement compared to -3(4) or 3(-4), then that movement is precisely borrowing or stealing or subtracting blocks from the HAND to the table, with a result that there will then be 12 blocks on the table (and a "hole" of -12 blocks in my hand). QED

the fundamental flaw with this argument is that 3 · 4 doesn't means adding the number 4 three times to itself but two times. Indeed here there are two additions 4 + 4 + 4 namely 4+4 is one addition and 8 + 4 is the other. there are no other addition. This make impossible to explain (-3)·(-4) in terms of repeated additions, but nevertheless it is still possible if one can find the true meaning of multiplication.

It's not elegant

Even more important is to realize that the flipping is actually a rotation left into a transverse plane 1st (we call it "times i"). And then rotate again ( hence i^2) to reach the final flipped state caused by x -1.

Multiplication is repeated addition OR repeated subtraction

great

(-1)= flips number to negative (-1)= flips number to positive And the pattern repeats This is an easy way for people to understand negative integers in multiplication and division. Interestingly, integers with addition and subtraction is a bit more challenging.

10:28 we can see that it HAS to be 1, it is proven to be 1. But how does (-1)x(-1) make +1? What is happening? What does (-1)x(-1) mean?

This is like my 8th video on the topic, still...don't....get it! Your basically creating an arbitrary rule then applying that rule, "Multiplying by -1 creates a flip around 0." But you didn't arrive at that rule by the same rules that you used to describe the previous equation so how valid is your rule? i.e. So, 3 x 4 = 3 sets of 4 which = 12. And 3 x -4 is 3 sets of -4 which = -12. Great. Makes sense. Ok, well, if 1 x 1 is one set of 1 which = 1....THEN...-1 x -1 = one negative set of -1 which = -1. That's consistent logic across all three equations. But a negative x a negative is NOT trying to be consistent with it's counterparts (like we would expect and what is intuitive) instead YOU (meaning mathematicians) are FORCING it to be consistent with something else, namely, the distributive property. But WHY must it be consistent with the distributive property when it is inconsistent with the previous intuitive rules that were set up???? No one answers this question.

Thank you so much!!! I have been away sick from school and watching your video caught me up in everything I missed today!! You’re the best

is this considered a rigorous proof?

it looks so great

Great video, thank you very much!

Why doesn't this channel have more views or subs? Like I mean, I learned a LOT from this. Thank you so much

0:11 chat-elegant-girls.online

Very well explained...👍👍 What is the degree of freedom when we only talk about choosing the angles of the triangle?

REALLY NICE . KEEP ON THE GOOD WORK !

I really like the use of colors and animations 😊👏 Handwriting on points as well 😉