What is Multiplication?

Every decade or so a vehement debate/discussion flares up searching/demanding for a definitive, concrete, absolute answer to the question: What *is* multiplication? (It is often linked to the hope of finding a concrete, real-world manifestation of negative times negative being positive.) I am entering the debate--a number of years late to the latest round--to share my thoughts on the matter, such as they are.

[JackNimble-gftq]

This is the best multiplication video ever. I come back to this video every few years.

[matthewderen]

James, Your videos are the finest most important math videos I have ever seen for helping Highschool Math Teachers know how it should feel for students to see a lesson. I watch your videos over and over and my students benefit from your influence. Thank you, Thank you! for posting all your work!!

[nandakumarmunaganti]

Loved your way of explaining things, Thank you so much Sir.

[devrimturker]

" Negative Math: How Mathematical Rules Can Be Positively Bent ",
Book by Alberto A. Martinez

[SteveHeller]

Great fun! Perhaps you answered the question "What ARE multiplication?"

[mootown]

Came for “what is multiplication” - stayed for watching you sneak teach people group theory ...

[THOPE]

Omhay this is the exact video I'm seeking for. Most videos about the meaning of multiplication are just basic and elementary level videos. I became curious about this concept because I'm baffled specifically in kinematics why for example they multiply mass by acceleration to have a Force. My intuition just can't handle it

[mohammadhaseeb]

U teach in minutes what I haven't learnt in years altogether.
Thank you

[blingpup]

Greatest Math Video I have ever watched.

[jgy]

Exactly, this is exactly why I personally love maths! Identifying this patterns and structures and just running with them and seeing where they lead. If you continue with this line of thought just a bit further, you get structures like groups and rings, with are quite interesting and can help you to study all different types of things, from the very abstract, like how "shapes" relate to each other (Algebraic topology), to the very concrete, like the geometry of molecules and the physics of the smallest things. By the way, I wonder if it could be appropriate to present operations in the negative numbers to kids (in my country we learn them in school during 7th grade, I don't know the situation in the US) in a similar fashion, in a sense of defining them with the properties of natural number operations, and them following the logic consequences of that? I wonder if one could make that work, I feel like it should, but I'm not a teacher, so what do I know. (I do plan on eventually becoming one, but right now I'm just a humble high school student)

[alonzoarcher]

These types of explanations should be taught right around (preferably before) the time students tend to start demonstrating confusion with the seeming inconsistency in what is considered multiplication. In general, seeking to clarify common difficulties for students in a way that can scale as they advance is a gift that never stops giving. Excellent job!

[oksolets]

Brilliant, you are doing great great things

[lindaeakin]

Awesome tour of the number system. I also recommend Richard Feynman's Lecture on Algebra, especially if you're interested in moving on to include exponentials and logarithms as operations.

[Abc-bnzh]

Thank you for this beautiful and easily comprehensible lecture, kind sir. You have cleared my doubts about the meaning of multiplication. Now I know what channel to watch when I wanna study more about maths. Thank you, sincerely. Have a great day, sir.
/subscribed

[kimrimbey]

Thank you!!!! I just posed this question to my class last week. We've not yet derived an answer. Thank you for such an intriguing way of introducing the notion that sometimes we must rely on systems and patterns. My simple method for showing why a negative times a negative is a positive is to use a pattern approach with a simple example (sort of like your equation example approach). If 2x(-3) = -6 and 1x(-3) = -3 and 0x(-3) = 0, then continuing the pattern would result in (-1)x(-3) = 3 and (-2)x(-3) = 6, etc. This, of course, assumes the premise of the properties you outlined in the video.

[MarkHuberDataScience]

I feel I have to jump in to defend scaling, because if done right it gives the correct answers for multiplication without the need to first presuppose an axiomatic system. By "done right" I mean that you have to include both scaling and rotation.

Suppose I can transform a vector by scaling (stretching it) and rotating it. Call that a scale+rotate transform. Then if I transform a vector twice, call the first scaling+rotate a, and the second scaling+rotate b. Then there is a final vector that is the result of using both these operations called scaling+rotate c. Then multiplication is just saying that c is b times a.

For example, scaling+rotate of (-3) means triple the length of the vector, and flip it over (rotate) so it is moving in the opposite direction. Scaling+rotate of (-2) means double the length of the vector, and flip it over (rotate) so it moving in the opposite direction. If I first transform by (-3) and then by (-2), then the original vector is pointing in the original direction, and 6 times as long as before. Hence 6 = (-2)(-3).

This definition of multiplication has the added benefit of incorporating matrix multiplication, which as you know does not always follow your axiom of ab = ba.

[memersid]

I did enjoy the video but it wasn't that helpful tbh but i really loved the video and you've planted a seed of curiosity and new believe system in my brain ....thanks for that ...im subbing just for that❤

[lindaeakin]

Please note the unit m^2 is a "square metre". Admittedly (like most people?) I think "metres squared" when I write it down, but using this leads to ambiguity ... viz. two metres squared = 2m x 2m = 4 m^2

[arcomarco]

Marvelous video! But
I would still say that multiplication can be understood.

I gave some thought to it when I was asked to explain why we multiply
converting units. The question was: Why when we convert miles to km we multiply
miles times 1.6 when miles are bigger?

In the beginning, I was confused: I don't know either, wait! It is illogical.
After a while (and a lot of frustration) I notice that as you said I started
doing it intuitively without thinking about logic.

I was incepted with the idea by the child who asked the question that
multiplying 3 x 2 is increasing the number three times two.

I went back to the first principles. when we multiply 3 x 2 we are increasing 2
times 3, not the other way around (which over time is forgotten).



I'm planning to write an article about it in the future.

Now I will provide a shortened version of it.



First, we must view math as a language. Every equation is a sentence and like
every sentence, semantics boils down to the context.

Before we start. Please note that I'm an idiot trying to understand the nature
of things.




What is multiplication?

Repeated add, area, scaling, unit conversion?

In my first principle thinking, it is a group of.
When translated to English it says 2 times, 3 elements

2 x 3 easy and was explained

area 2 m x 3 m = 6m^2 it is still grouping only in two dimension

We are not multiplying 2m times 3m but (2 times 1) times (3 times 1)

So we have 2 groups of 1 group that have 3 groups of 1 (think of it as a
2-dimensional matrix).




Scaling? 3x4m is
still 3 groups of 4 meters.

Unit conversion?

1 mile is a group consisting o 1.6km (more about it later).



3 x (-2) like it was said 3 groups of -2(debt) if I borrow from you $2 3
times I borrowed 3 groups of 2 dollars. I have 3 groups of elements that I owe you. And those elements consist of something I owe you -2 dollars Or
in other words,

-2 x 3? Is the same only we are looking instead of the elements themselves,
at the group as a whole that we own. I owe you 3 groups of 2 dollars.

As an example, we can use chocolate bars. They can be sold individually
or in groups.



If I take a carton that contains 10 bars.

I can look at this in different ways, either how much I must pay (positive)
assuming that one bar costs 1 dollar.

For one carton I need to pay 10 groups of 1(1 dollar) which gives me $10

Or how much I lose(negative). 10 groups of -1(1 dollar) or -10 groups of
1 (dollar). It boils down to how I look at it. As individual elements being negative or on an entire group.



It is like saying: I throw a ball. Some people will focus on Me others on
throwing the ball both lead to the same result with a different focus.



What about (-2) x (-3)?

I just put back two cartons of 3 bars that I owe (each worth 1$)


But let's go to the initial question that started all of this.

The units conversion

1 mile = 1.6 km

1 mile is a group o 1.6 kilometers.

what is the meaning of 1 mile?

1 x mile, mile here is a name of a group.

So we have 1 group of 1 mile which is a group itself consisting of 1.6km.
In the case of 3 miles, we have 3 groups of 1 group of 1.6km.



We also sometimes see the conversion factor written as 1 mile/1.6km



Great... now we have a division that is opposite to multiplication but why?

What division really means?

What If I say to you that division is actually two logics connected with each
other?

Let's look at examples.



We have 12 fish and 3 tanks.

we can divide them into 3 equal groups in this case of 4.

But what if we look at this scenario as 12 / 4 = 3

are we dividing them into 4 groups of 3?

Both these operations 12/3=4 and 12/4 = 3 can answer the same question of how
many fish goes to each tank.

What is going on?

It depends on our perspective.

We may look at the fish as families

If we want to write it in English sentence it means:

Divide 12 fish into x groups of 4 elements

Or not!

Divide 12 fish into 3 groups of x elements.



Most of the time logic is forced upon us depending on what we want to
calculate.



If we want to calculate the price of an individual element.



Like, I bought 10 bars and paid $20 for all, how much does one bar cost?

The equation would be 20 / 10 = 2 in this case one bar costs $2

But how we would write this in English?

Divide 20 dollars into 10 groups of x = 2

.

Now, I know that I have $20 and know that one bar costs $2, and want to know how much I'll get?

20/2 = 10

Divide 20 dollars into x groups of 2 = 10.

Notice something interesting

If you divide something by itself you are looking at how many groups it will be.

$20/$2

12 fish / 4 fish

If not like I paid $1000
for 250 products you are looking for elements of the groups

1000 dollars divided
into 250 groups of x elements = 4

For every product, you paid 4 dollars which means that 1 product is equal to a group of 4 dollars.

Suddenly 1 mile/1.6km starts making sense.

For better visualization
let’s take 10miles / 16km

Divide 10 miles into
x groups of 16 = 10

But hold on, I said
earlier: If you divide something by itself you are looking at how many groups
it will be.

We are indeed diving
something by itself both miles and km are units of length

Like with money:

1 Euro / 1.1 Dollars
in both cases, we are dealing with money.



For the same reason, the division is inverse multiplication.



12 fish divided into x groups of 4 = 3

Is opposite to the 3 groups of 4

And 12 fish divided into 3 groups of x = 4

Is also opposite 3 groups of 4

[motthebug]

i find it interesting that you find explanations of negative times negative using the abstraction of time "too convoluted, " but hey, completely symbolic ap9plication of distributive property isn't. Maybe it speaks to the elegance of logic ;)