When you accidentally multiply matrices the way, but nobody notices (Reddit r/mathmemes)

Рет қаралды 104,515

29 күн бұрын

Matrix multiplication is quite complicated but sometimes we could multiply them entry by entry. Of course, this does not work for all the matrices but only for the well-designed ones!
This meme is from Reddit r/mathmemes
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Пікірлер: 141

@johnchestnut5340 28 күн бұрын

You "broke" the pattern with a green pen. Black and Red alone were expected. We entered another dimension/world? Thank you for the video.

@orangeoranges-mw2sb 28 күн бұрын

theres been videos where hes used like 4 different pens like in his all in one calc question

@johnchestnut5340 28 күн бұрын

@@orangeoranges-mw2sb I never suggested that this is the only instance. But having not seen all of his videos, this did break what appeared to be a pattern. It's also implied in the name.

@iwanadai3065 28 күн бұрын

this feels like when they made i = sqrt(-1)

@stephenbeck7222 28 күн бұрын

@@johnchestnut5340 his name is already broken because we normally don’t refer to markers as pens. When he first started making videos he used a doc camera (or iPhone?) pointed at his paper notebook and used actual black and red pens on the paper. But the whiteboard videos are the best. He should get a bigger whiteboard - or make a huge chalkboard set up like his friend Michael Penn.

@johnchestnut5340 28 күн бұрын

@@stephenbeck7222 You can argue that pens are not markers and vice versa. I can argue that electric cells are not batteries unless configured together into battery. Neither is pertinent to my comment. But you have been heard.

@StephenMarkTurner 28 күн бұрын

Love that "oldie but a goodie" simplification of 16 / 64.

@billcook4768 28 күн бұрын

These are the math equivalent of Dad Jokes.

@white9763 28 күн бұрын

Bprp helped me so much by teaching me calculus and now he is teaching matrices? Huge W

@hmmm6200 28 күн бұрын

same. we started doing limits rn, theyre super easy for me thanks to this channel

@white9763 28 күн бұрын

@@hmmm6200 my high school became easy asf because of our master bprp, I've seen basically everything in this channel, he made me love math, I am forever Grateful to bprp

@matthewbauerle7153 28 күн бұрын

It works for diagonal matrices because of all the zeros. If it works for anything else it’s a happy accident.

@benjfr5723 24 күн бұрын

For some triangular matrices because you only have to handle the nondiagonal coefficients too

@Ny0s 28 күн бұрын

I love these "wrong way but not really" rules

@SatoruSnipes 27 күн бұрын

bro wrote a book called you can do calculus

@theslickestcrabaround5588 28 күн бұрын

In the beginning he goes "ok, so let's discuss what's going on" and i initially thought he was saying "ok, so that's disgusting" and really i agree

@galaxygamer2156 28 күн бұрын

I’m self teaching linear algebra and this video is super simple and clear to understand! I would love to see more linear algebra videos!

@molybd3num823 28 күн бұрын

I recommend watching 3Blue1Brown's "Essence of linear algebra" playlist

@lilyoy7942 28 күн бұрын

r/mathmemes mentioned 🔥🔥

@maximilianarold 28 күн бұрын

Are we stupid? And is Jessica welcome here?

@smallcube-zn2mm 21 күн бұрын

@@maximilianarold women not welcomed

@blackpenredpen 28 күн бұрын

New mechanic just dropped!! 196-182*0.5=7!! Most Redditors didn't get this! kzfaq.info/get/bejne/d7eZerOlx8DTYH0.html

@highviewbarbell 28 күн бұрын

double factorial

@Rando2101 28 күн бұрын

Holy hell

@Gears_AndGrinds 28 күн бұрын

If you're gonna use the double factorial like that then you might as well define the the double factorial of "dropped" and the factorial of "this" lol

@SatoruSnipes 27 күн бұрын

Dude. You got my best friend to write math textbook called you can do calculus. thank you.

@ANIKDAS1356 24 күн бұрын

Please solve integral of (x÷dx)

@Celestia1323 19 күн бұрын

When you accidentally write the title of the video wrong, but nobody notices

@cosmicvoidtree Күн бұрын

Well, I only just now noticed so I guess yeah

@mndtr0 28 күн бұрын

It's actually product...

@minratos6215 28 күн бұрын

hadamard product also known as element-wise product

@justsaadunoyeah1234 28 күн бұрын

Euler-Gauss-Euclid-Archimedes-Leibniz-Newton-Ptolemy-Mascheroni-Shanks-Noether product

@asheep7797 28 күн бұрын

obviously the EULER(-farey) product, with EULER written in 96 point, and farey optionally written in 1 point.

@codahighland 27 күн бұрын

You joke, but it is, and it has valid uses.

@samueldeandrade8535 27 күн бұрын

@@minratos6215 I see you are a man of good taste ... I prefer the Kronecker product.

@Horizon24129 28 күн бұрын

Tomorrow is my exam i was studying matrix and determinants this notification pop up

@shahaansahni 28 күн бұрын

Looks like a tini-mini 6th grader tried to attempt this logically 😂😂

@kristinborn8882 24 күн бұрын

Thank you for teaching me how to multiply matricies! I tried it with some random numbers, and it worked!

@NinjaBear1993 11 күн бұрын

Oh gawd, I never thought others might actually multiply matrices like that. Then I realized, others probably hate matrix math that they will try to get the answer with tricks.

@ReeBoot-h2e 8 күн бұрын

When it comes to deep learning this is correct way to do it, it's called pointwise multiplication.

@vocabpope 20 күн бұрын

7:56 "that's not a b it's a six" I appreciate so much that someone else experiences the deep personal horror of "whoops that letter/number/word writing sucked let me fix it" *draws literaly EXACTLY the same fucked up shape as before*

@davidramos4707 28 күн бұрын

I wish I didn’t need more of this, but I really do

@Appreciation-q9v 28 күн бұрын

Thank you

@Peter_1986 26 күн бұрын

Element-by-element multiplication honestly feels like the most intuitive type of multiplication; I always thought that it was a bit weird how you are supposed to do that awkward row-column multiplication.

@ShrekPNG 25 күн бұрын

That row-column multiplication is necessary though, because the resulting matrix has the same effect as the two matrices used to make it. Say you have a rotation matrix A and a scaling matrix B. AB would be rotation then scaling all in one matrix

@Tryss86 24 күн бұрын

The idea is that matrices represent linear maps, and multiplication of matrices is then the same thing as composition of the linear maps. if A and B are the matrices of two linears maps a and b and X and Y two column matrices representing two vectors x and y then Y = ABX is just another way to write y=a(b(x)). And this linear map matrix correspondance is the real reason why we bother with matrices at all

@Peter_1986 24 күн бұрын

Yeah, I realise that the row-column multiplication is the real way to multiply matrices; it just felt a bit strange to me when I first started doing it. That said, element-by-element multiplication is of course sometimes a desired form of multiplication as well, like maybe if you have various tables of data and want to perform several individual multiplications at the same time.

@ge97aa 24 күн бұрын

Given that the answer is correct, why would you assume the person who generated it used the wrong method?

@slava6105 27 күн бұрын

5:00 since regular multiplication is associative, can we add another 4 equations from multiplying matricies in reverse order (AB = BA = C, C is the wrong way)? Ok, some of them doesn't matter, but seeng AB=BA would be also invalid in general

@Leo-pd8ww 6 күн бұрын

Ah yes, the broken clock law.

@jrbaum4644 26 күн бұрын

Thank you for helping me on calculus 2

@giovannielias8153 13 күн бұрын

0:50 cursed fraction

@sharkysharkerson 15 күн бұрын

I didn’t get how this was wrong because I was multiplying the correct way and got the same answer. Then I saw the calculation you tried in the beginning.

@Tletna 28 күн бұрын

There's nothing wrong with multiplying the same elements in each matrix. You just get a different result than the dot product....

@bachvaroff 27 күн бұрын

That'd be the Hadamard's which is just fine…

@crtwrght 27 күн бұрын

When it said "multiply matrices the wrong way" I did thought they meant column times row it got the right answer. Never even occured to me to lazy-multiply the entries.

@FocalLens-hk6tj 19 күн бұрын

I feel like you are the Ethan Hunt of MATHEMATICS

@darcash1738 27 күн бұрын

Yeah this post was cool. If we equate the formulas we get b1*c2 = 0 and c1*b2 = 0. Then our other equations inform our other required zeros. For b1 = 0: c1 = 0: (a1 or b2 = 0) & (d1 or c2 = 0) b2 = 0: c1[a2 - c2] + c2d1 = 0 For c2 = 0: c1 = 0: b1[d2 - b2] + b2a1 = 0 b2 = 0: (c1 or a2 = 0) & (b1 or d2 = 0) We were given that c1, c2 = 0. So 6(2-4) + 4*3 = 0, which is true, which is why the post fulfills the reqs. Similarly for b1, b2 = 0: c1[a2 - c2] + c2d1 = 0 Choose c2, c1, and a2 to be whatever; but then you have to calculate d1. Say c2 = 1, c1 = 2, a2 = 4. d1 = -6 X. 0. 4. 0. 2. -6. 1. Y. Can choose X and Y freely.

@user-iq1zw1tf3f 28 күн бұрын

sir rubberd the whole 7 just to cross it

@TheFastProgrammer 27 күн бұрын

doesnt matter which video, lambert w functions sure to show up

@pattmahiney 22 күн бұрын

So if this and if that, then this. Got it. Thank you for sharing.

@Damic_Damic 22 күн бұрын

Looked at the thumbnail and multiplied it the correct way and did not understand what's wrong with it... Did not even occurred to me to do it that way 😂

@emanuellandeholm5657 28 күн бұрын

One could also try to find examples with the dot product in the wrong order, ie. col dot row.

@vladimirilyushko5614 21 күн бұрын

Can you get all solutions for a in terms of n in a=n*sqrt(n+sqrt(a))?

@jamescollier3 28 күн бұрын

Cool... 😊 That endin lol

@johnbyrne1022 28 күн бұрын

It's easy to remember to dot the first row with each column on the second matrix. Then you get a number for each of those dot products. The hard part to remember is, do these numbers form a *row* or a *column* of the new matrix? I've never been able to come up with a good way to remember that, other than just memorizing it.

@Greenicegod 26 күн бұрын

I just noticed the resulting number goes in the intersection of the row and column that made it

@johnbyrne1022 26 күн бұрын

@@Greenicegod Nice, thanks!

@alyme_r 27 күн бұрын

just a reminder to watch the vampire matrix stand up maths video

@Archimedes_Notes 25 күн бұрын

😂😂😂😂😂. The gamma function should work here. Upper triangular matrices but very surprizing🎉

@popularmisconception1 28 күн бұрын

I always hated how matrix multiplication is taught. Rows times columns errrggh.. what? Where have I been, did I make a mistake? I always get confused. So I have devised a better method. I imagine the matrices are rotated 45 degrees to right and the result matrix is like a lego brick down between them. That determines its size. Then I imagine the numbers are like balls in a tivoli machine, just waiting to fall down to their place, just block by some imaginary obstacle. Then I release both of the obstacles at once and from both sides numbers start falling in their respecive line. And the numbers that hit each other are the ones that multiply and all the numbers that hit in the same place are added and the result is the number in that place. And then I just rotate the resulting matrix 45 degrees left and put it where math teacher wants it. I don't know if I explained it right so you can imagine it, but this is what I do in my head to prevent myself from making mistakes so I don't have to think in terms of confusing rows and columns. There is also a way to do the same thing without the 45 degree rotation, if you just imagine the right matrix is just above the left matrix but just to the right, so the resulting matrix is just right of the first and just below the second and the multiplied vectors are basically just intersections..., but the gravity thing kinda helps me... I guess I better make a video. :D Maybe one day

@drachefly 28 күн бұрын

My linear algebra teacher did the same thing but without rotating. You just raise the second one up above and the resulting matrix fits in between them. Later, when I set up matrix multiplication for people who hadn't taken this particular course, I found out that this was not standard.

@samueldeandrade8535 27 күн бұрын

If you really like blocks like that, you would love Kronecker product of matrices. Haha.

@jasnesciemnienie9107 22 күн бұрын

The best way to think about it is that you sell apples. In the first matrix rows correspond to different days of selling and columns are the numbers of different kind of apples that were sold. Column in another matrix tells us the prices of those different kind of apples and the result of multiplication is the amount of money you get for every single day of selling. And if we have multiple columns in the second matrix, then it's just hypothetical scenarios where you set different prices for your apples ;)

@sukantasarkar7519 22 күн бұрын

can anyone recommend a similar channel like this for other subjects too..

@nyandyn 28 күн бұрын

Shift the right matrix up so that the resulting N*M matrix fits in the empty space. Then do dot product between the corresponding row and column vectors for each element in the result. Can't get it wrong accidentally that way.

@NobodyYouKnow01 26 күн бұрын

"these aren't the same, but here's the system of equations where they are"

@breadmusic1 21 күн бұрын

It looks like you misspelled numbers with hose underlines

@rishavgamerz7460 26 күн бұрын

Please explain the integral solved by RON GORDAN I am a huge fan and you explain miraculously 😊

@Beeblebrox6868 26 күн бұрын

I keep thinking I'm finally going to understand matrices but... nope.

@annekekramer3835 23 күн бұрын

Just curious, why does he write the matrix multiplication like that? I'm used to writing them down in a square, where the first matrix (X) is written in the bottom left, the second matrix (Y) in the top right and the answer matrix (Z) gets into the bottom right. That way it's visually much easier to see what to multiply. Like this: |Y X|Z

@SimonBuchanNz 23 күн бұрын

(forgive me if this covers stuff you already know) Because multiplication is symbolically represented in algebra by simple adjacency: eg "ax = y" is multiplying a and x. When you use literal numbers, eg "3•4 = 12" you need the dot (or ×) to disambiguate, but otherwise no operator is needed. For matrices, equivalently, the symbols are traditionally uppercase letters, often starting with T for the varying values, eg "AT = U" has three matrices, but while the literal form is the square brackets around the grid of values, it's still as a whole a single value in an algebraic expression. So the only thing left is: why don't you need a different operator for multiplying matrices than numbers? The confusing answer is that matrices actually are numbers! Just like whole numbers, integers, real numbers and complex numbers are all different structurally but can be treated as numbers by adding and multiplying, following sensible rules you learned in early math classes like "(x + y) + z = x + (y + z)" and "0x = 0", so do matrices (and vectors, quaternions, polynomials, etc. to some degree) The sets of these rules and what structures and operations follow them is group theory.

@PhrontDoor 22 күн бұрын

This one was fun..

@always_be_ur_frd 27 күн бұрын

l want to see bprp Solving 2017IMO p3

@nicholasscott3287 26 күн бұрын

So, are there an infinite number of matrix pairs where this sort of naive matrix multiplication works, then?

@runenorderhaug7646 27 күн бұрын

If i am honest I thought that this was gonna be a joke on that because there is two square matrices if the same length he flipped which matrices was in front

@alcar32sharif 26 күн бұрын

Scalar Multiplication vs Matrix Multiplication.

@Kavukamari 27 күн бұрын

doesnt this method of "multiplication" actually have a name and usage as well? i thought i remembered something really specific you can use this for

@khansahab1974 26 күн бұрын

Can you pls do sin(x^2)=2sinx?

@skeptica 28 күн бұрын

what is the cross product?

@carultch 28 күн бұрын

The cross product is a product of two vectors, that returns a third vector that is mutually perpendicular to both of the vectors. The magnitude of the cross product vector, tells you the product of magnitudes of the two given vectors as well as another factor that measures how "crossed" (i.e. perpendicular) the two source vectors are. That factor, is the sine of the angle between them. The cross product is not commutative like normal multiplication, and produces the direction of its output according to the right hand rule. An application of the cross product is torque. The radius vector from the pivot point, crossed with the force, tells you the torque that the force applies; i.e. the rotational equivalent concept to a force. By convention, we assign torque in the same direction as a standard right-handed bolt would move parallel to its axis, if you spin it in the direction you apply the torque.

@skeptica 28 күн бұрын

@@carultch Thanks!

@duckyoutube6318 28 күн бұрын

This stuff really reminds me of ancient greeks, chinese, and babylonian mathematicians. Its all very simple logic. But from that simplicity, we get some amazingly complex problems and solutions. Its really sucks that so much has been lost to history. Look at where we are today with just what we have saved, created, and built upon. I hope everyone here knows how special it is that we are able to gather here, and talk, and learn, and grow together.

@apokalypthoapokalypsys9573 28 күн бұрын

Nothing in mathematics is lost to history. Artworks like writings and statues can be lost, because they are unique, but mathematics is objective. We can discover the exact same principles that have been "lost", if any, and given that we have supercomputers now, it's safe to say that there is nothing in mathematics they knew and we don't. In fact, we've solved problems Euler and other geniuses couldn't, because of technical limitations. We've advanced fields of theories, like knot theory, that would have been impossible to progress without computers. So please, spare us from your profound sanctimonies.

@duckyoutube6318 28 күн бұрын

@apokalypthoapokalypsys9573 Im sorry but i dont care what nasty things you have to say. So go be miserable alone. There is nothing to gain from talking with you.

@duckyoutube6318 28 күн бұрын

@apokalypthoapokalypsys9573 Just curious. Why do you gotta bring me down like that? Im just trying to be positive. Why be so mean? Sorry i have nothing interesting to say. Im sorry for being me. If i could change i would. You people make me feel like complete trash. And all i do is try to be positive. I dont understand. Im sorry for speaking. Im sorry for expressing anything. You guys win. I give up. I thought we were lucky to be here together. But now i know nobody feels that way about me. Im just annoying and useless. Im sorry. And it wont happen again. You win.

@OrdinarySonicfanMmKay 28 күн бұрын

Hello there

@henrygreen2096 27 күн бұрын

If I saw this video in undergrad I would have said “man everyone knows this” But now as a professor we needed AS MANY PEOPLE pointing this out possible. I could make a video EVERYDAY for the rest of my life on how to properly multiply matriciels and I will still have students multiply across on the final… you read that correctly… THE FINAL. As in after months of explaining, office hours, homework, quizzes, and tests… they still mess it up. There’s always one or two. 😢. Makes me want to quit, man.

@abacaabaca8131 28 күн бұрын

When you do matrix multiplication, you need to also "add" another term so that to assemble the real equations. Matrix in math is just a system to solve a simultaneous equations. In other words matrix is just an isolated system made from a bunch of simultaneous equations. Consider the circle equation. y(t)=r * sin (t) x(t)=r* cos (t) But, this is only true for the first quadrant i.e when x and y is always positive. Later they find out the equation is x(t)= x * cos(t) + y * sin(t) y(t)= x * -sin(t) + y * cos(t) This is the formula for a clockwise rotation in which the rotation is in the form of a circle. From here you can isolate the equations in a matrix form. [ cos(t) sin(t) ] [ x ] [ -sin(t) cos(t) ] [ y ] So, if you try to assemble the original equation from a matrix form, you must do "addition" or you won't get the original equations.

@zero-sl3bn 27 күн бұрын

you sure about that ?

@user-zg8ny5tp4g 28 күн бұрын

Integral: tan^2x ÷ (1+sec^4x ) dx ، how can you solve or just give me a hint

@zero-sl3bn 27 күн бұрын

you might've made some typo or there is something wrong in question if you are trying to find real solution but if you want complex solution just change above tan to sec and add and subtract sec^4 above and separate -1 and then in denominator just separate like a^2 + b^2 in complex numbers then it's just simple process ✌

@user-zg8ny5tp4g 27 күн бұрын

@@zero-sl3bn denominator would be (a^2 - b^2i^2)(a^2+b^2i^2)??? Did you mean this expression??

@user-zg8ny5tp4g 27 күн бұрын

@@zero-sl3bn then try with partial fraction??

@zero-sl3bn 27 күн бұрын

@@user-zg8ny5tp4g (sec^2 + i)(sec^2-i) and yup partial fraction by keeping sec^2 common in numerator

@user-zg8ny5tp4g 27 күн бұрын

@zero-sl3bn but how you can apply partial fraction to secx function, because we sec^2x in the denominator..so it would be (AX+C)÷ (sec^2x+i) + (BX+D)÷ (sec^2x -i)... but don't get it how to apply on secx function.

@macarioinmenzo3365 28 күн бұрын

wow im early

@arekkrolak6320 9 күн бұрын

What is this guy doing? This is not how uou multiply matrices

@geekonomist 28 күн бұрын

Why do you call it a matrix “multiplication” when in fact it is an arbitrary rigmarole? Why don’t you just arbitrarily divide all the numbers while you are at it? 2x3=6 is a multiplication. What you are doing with those two matrices is not at all a “multiplication”.

@benkelly2024 28 күн бұрын

It's not arbitrary at all. When you understand what's going on, it's very obviously the most natural way for matrix multiplication to work.

@geekonomist 28 күн бұрын

@@benkelly2024 ⁠your comment is arbitrary because, since it has no explanation, is neither true nor false. Why is it called a multiplication when in fact there are steps other than a multiplication involved? Why is the rigmarole needed? What facts of reality make all this necessary? If there are indeed reasons why this is necessary, why is calling this a multiplication not arbitrary, when it is quite clear that 2x3 is?

@carultch 28 күн бұрын

Because it has a lot of properties in common with multiplication. Consider how you'd use matrices to solve the following system of equations: 3*x + 2*y = 14 4*x + 5*y = 28 You'll construct a square matrix of the coefficients, which we call A. You'll construct a column matrix of x & y, which we'll call matrix X. You'll then construct a column matrix of the right hand side constants, which we'll call B. This produces the following matrix equation: A*X = B You can see that each entry of matrix A, is multiplied by one of the entries of matrix X, somewhere within this process. And just as you'd solve 3*x = 9 by multiplying by the reciprocal of 3, there's a similar method of solving this matrix equation by multiplying by the "reciprocal" matrix of A. A*X = B A^(-1) * A * X = A^(-1) * B A^(-1) * A = the identity matrix I, by definition. This is analogous to the idea of 1, where it is something you can multiply by anything, and return that same anything. I * X = A^(-1)*B An identity matrix multiplied by a column matrix, returns that column matrix. Thus: X = A^(-1) * B The hard part is finding the matrix A^(-1). For the 2x2 case where A = [a, b][c, d], the general solution is: [d, -b] [-c, a], all divided by (a*d - b*c) So this example: A^(-1) = [5/7, -2/7] [-4/7, 3/7] Multiply by the column matrix of [14][35] and get: X = [2][4] Which implies x=2 and y=4.

@benkelly2024 28 күн бұрын

@geekonomist I have pointed out that your confusion is simply the result of ignorance, but I have no interest in educating you. If, now that you are aware of it, you wish to remedy your ignorance, that is up to you. The subject you need to study is called Linear Algebra, and it is usually one of the first subjects taught in a mathematics degree.

@geekonomist 28 күн бұрын

@@benkelly2024 I asked ChatGPT. I gave it (along with you) shit for three rounds because it kept things absolutely ridiculously abstract (ie : Matrix A represents a Rotation (!?!) and Matrix B a Scaling (!?!). Then after repeated badgering, it finally yielded a real world example, called sales volume of apples and oranges at different prices in different stores. Funny how just the title alone answers all questions I have, and how NOBODY thinks in terms of apples and oranges when teaching and pontificating in comments. And no, it is not a "multiplication" when you combine the values of a spreadsheet of your prices and volumes of apples and oranges. "Multiplication" is an arbitrarily misnamed - rigmarole.

@styleisaweapon 26 күн бұрын

It most definitely is how we multiply matrices. Its called the Hadamard product and its a universal basis for how modern computer architectures compute all the other matrix products hyper-efficiently. Very disingenuous video.

@m3morizes 27 күн бұрын

Let X=[(a b), (c d)] & Y=[(e f), (g h)]. Then XY=[(ae bf), (cg dh)] if any of the following cases hold: Case 1: X=0 Case 2: Y=0 Case 3: a=b=c=0 ∧ g=0 Case 4: b=c=d=0 ∧ f=0 Case 5: b=0 ∧ e=f=g=0 Case 6: c=0 ∧ f=g=h=0 Case 7: X∈Diag_2(R) ∧ Y∈Diag_2(R) Case 8: X,Y∈LT_2(R) ∧ cg=ce+dg Case 9: X,Y∈UT_2(R) ∧ bf=af+bh Cases 1 & 2 are obvious. Case 7 reflects the fact that multiplying diagonal matrices is easy, that it's componentwise for the diagonal entries. Cases 3, 4, 5, & 6 are not as obvious, but straightforward to verify. Cases 8 & 9 are interesting for only requiring two entries to be 0, such that both X & Y are upper triangular (or lower triangular) matrices, along with requiring the componentwise multiplication to hold for the entry diagonally opposite the 0 entry.

@ClementinesmWTF 28 күн бұрын

This wasn’t even the way I thought they multiplied them wrongly. I thought they had transposed the multiplication and it does indeed work that way as well. I have a feeling (but not a proof) that any 2x2 matrix multiplication that abides by your proof will also have this quality of multiplying the same way in all three ways