What are multiplication and division?

[Ashish Shukla]

Hey,
I have often times wondered what is multiplication? Repeated addition is OK but for some reason it doesn't satisfy me. For example:
2*2cm is linear because it scales 2cm on the same dimension but 2cm*3cm is not scaling, it spans 2 dimensions. It seems as if the flow of operation takes a 90 degree turn when 2nd term i.e. 3cm is multiplied, whereas 2cm + 3cm keeps the operation confined to 1 dimension, if we again multiply 4cm to it i.e. 2cm*3cm*4cm, the flow takes another 90degree turn and now spans 3 dimensions. This makes me think there is much more to multiplication than just repeated addition.

Same with division: if I divide 2cm*3cm*4cm = 24cm^3 by 8 cm, then there are 2 distinct operations here, either divide the value i.e. 24 by 8 first, that would result in 3 cubes of 3cm^3 volume, OR you divide the dimensions first cm^3/cm which would result in an area cm^2 of value 24.

So I think * and / signify construction and deconstruction somehow which is way more than repeated addition and I don't even know what to say for division.

Please help...

 

[Mark44]

Hey,
I have often times wondered what is multiplication? Repeated addition is OK but for some reason it doesn't satisfy me.

In overly simplistic explanations, multiplication is sometimes defined as repeated additions. This is fine for integer values, but once you extend things to real numbers and beyond, it's too simplistic.

For example:
2*2cm is linear because it scales 2cm on the same dimension but 2cm*3cm is not scaling, it spans 2 dimensions.

Sure. When you multiply a length by a length, you get an area. If the lengths are in centimeters (cm) the area will be in square centimeters (the notation is $\text{cm}^2$).

It seems as if the flow of operation takes a 90 degree turn when 2nd term i.e. 3cm is multiplied, whereas 2cm + 3cm keeps the operation confined to 1 dimension, if we again multiply 4cm to it i.e. 2cm*3cm*4cm, the flow takes another 90degree turn and now spans 3 dimensions.

This represents the volume of a box with the measurements you gave. The units for volume here are cubic centimeters (in notation, $\text{cm}^3$).

This makes me think there is much more to multiplication than just repeated addition.

Same with division: if I divide 2cm*3cm*4cm = 24cm^3 by 8 cm, then there are 2 distinct operations here, either divide the value i.e. 24 by 8 first, that would result in 3 cubes of 3cm^3 volume, OR you divide the dimensions first cm^3/cm which would result in an area cm^2 of value 24.

No, you wouldn't get 3 cubes of 3cm^3 volume. $\frac{24}8\frac{\text{cm}^3}{\text{cm}} = 8 \text{cm}^2$. The division happens in two parts: 24 is divided by 8 to give 3, and cm³ is divided by cm to give cm² as the unit.
About the only examples I can think of that might do this would be if you were constructing ratios of the volume of something to the length of one side. So whether it makes sense or not, we can still talk about this ratio and give meaning to the units involved.

So I think * and / signify construction and deconstruction somehow which is way more than repeated addition and I don't even know what to say for division.

Please help...

At the simplest level (say about 4th grade of elementary school), we can note that 3 * 4 is the same as 3 + 3 + 3 + 3 or that it is also the same as 4 + 4 + 4. Multplying 2.5 by 3.6 is a little harder to explain as repeated addition, and $\sqrt 3 \cdot \sqrt[3] 5$ is even more difficult.
If we limit the conversation to whole numbers, division can be explained as repeated subtraction. In fact, the long division method does exactly that, but once you leave the realm of integers and rational numbers, things get difficult again.

 

[Delta2]

You are sort of right that multiplication creates extra dimensions and division removes or drop us to lower dimensions.

In Linear Algebra, the set of real numbers $\mathbb{R}$ is a vector space that has dimension 1. Addition is well defined in this vector space and give us as result something that lies within the vector space $\mathbb{R}$, so we remain in a vector space with dimension 1.

However multiplication , is not well defined in vector spaces. IF we view multiplication as a mapping(and it turns out we can define such a mapping $f:\mathbb{R}\times\mathbb{R}->\mathbb{R^2}$ with $f(x,y)=(x,y)$ ) from two elements x,y of the vector space $\mathbb{R}$, to one element (x,y) of the vector space $\mathbb{R^2}$ then multiplication is "sort of jumping" in a vector space with higher dimension (the dimension of vector space $\mathbb{R^2}$ is 2) (generally the dimension of vector space $\mathbb{R^n}$ is n, when we multiply n real numbers altogether we can view it as sort of jumping into the vector space $\mathbb{R^n}$).

Similarly division can be seen as a mapping from the Cartesian product of vector spaces $\mathbb{R^n}\times\mathbb{R^m}->\mathbb{R^{n-m}}$, that is it drops the dimension from n to n-m.

 

[FactChecker]

Whether you know it or not, you have asked a profound question. The subject of "abstract algebra" tries to answer questions like that in a rigorous way. The operations of "addition" and "multiplication" must be defined so that they work together in a coherent way. Properties of multiplicative association and distribution of multiplication over addition are required. A "ring" is a basic concept where that is done. See https://en.wikipedia.org/wiki/Ring_(mathematics). (Actually the "pseudo-ring" is more basic. It doesn't require a multiplicative identity, 1. The set of even integers is a pseudo-ring.)

As you indicate in your question, there are many different ways that the two operations of addition and multiplication can appear. The operations of addition and multiplication in your examples are operating on spaces of different dimensions.

 

[WWGD]

I always thought of the operation of division being done in the setting of fields or some structure where inversion is defined , with $$ a/b:= ab^{-1} $$ or if not, on a Euclidean domain, where you can say $$ a/b $$ can be worked through $$ a=bq+r ; r<b $$.

 

[FactChecker]

In a ring (with a multiplicative identity element, 1) the question is whether an element, x, has a (left, right or both) multiplicative inverse, x^-1 where (for a right inverse) x⋅x^-1 = 1. In that case, division by x is defined as multiplication (on the right for a right inverse) by x^-1.

 

[Mark44]

All: keep in mind that this is a 'B' thread.

 

[symbolipoint]

How complicated can people make this!

Take a number, any number, like 6.
Add a few 6 and figure how much you have. That is multiplication.

Take a number, preferably something big that for example, was built by summing some 6. Take 42.
Start with 42, and take 6 away from it until no more can be taken away. That is division.

 

[WWGD]

How complicated can people make this!

Take a number, any number, like 6.
Add a few 6 and figure how much you have. That is multiplication.

Take a number, preferably something big that for example, was built by summing some 6. Take 42.
Start with 42, and take 6 away from it until no more can be taken away. That is division.

...In Integers. OP wanted , as I understood it, a more general definition, for different settings.

 

[FactChecker]

All: keep in mind that this is a 'B' thread.

I may be guilty, but I thought that the OP showed some unusually abstract curiosity. I liked that.

 

[WWGD]

All: keep in mind that this is a 'B' thread.

I sort of agree with FactChecker. My goal is to whet the appetite without putting pressure nor expectations that the
OP fully understand things. I think it is a good policy to give people a head start on the more difficult part so they can
deal with it without much pressure.

 

[Mark44]

Some quotes from this 'B' thread. The idea of these tags is that replies should roughly match the indicated level.

IF we view multiplication as a mapping(and it turns out we can define such a mapping $f:\mathbb{R}\times\mathbb{R}->\mathbb{R^2}$ with $f(x,y)=(x,y)$ ) from two elements x,y of the vector space $\mathbb{R}$, to one element (x,y) of the vector space $\mathbb{R^2}$ then multiplication is "sort of jumping" in a vector space with higher dimension (the dimension of vector space $\mathbb{R^2}$ is 2) (generally the dimension of vector space $\mathbb{R^n}$ is n, when we multiply n real numbers altogether we can view it as sort of jumping into the vector space $\mathbb{R^n}$).

A "ring" is a basic concept where that is done. See https://en.wikipedia.org/wiki/Ring_(mathematics). (Actually the "pseudo-ring" is more basic. It doesn't require a multiplicative identity, 1. The set of even integers is a pseudo-ring.)

In a ring (with a multiplicative identity element, 1) the question is whether an element, x, has a (left, right or both) multiplicative inverse, x^-1 where (for a right inverse) x⋅x^-1 = 1. In that case, division by x is defined as multiplication (on the right for a right inverse) by x^-1.

Note that the multiplication and division asked about in the OP was more to do with how the units work, than about multiplication/division in the reals, vector spaces, rings, or domains.

I think it is a good policy to give people a head start on the more difficult part so they can
deal with it without much pressure.

Is it a good policy if the "head start" goes completely over the victim's head? The concept of "drinking from a firehose" comes to mind.

 

[FactChecker]

Some quotes from this 'B' thread. The idea of these tags is that replies should roughly match the indicated level.

I got here without ever seeing a 'B' level indication. Now I see it in an intermediate screen if I get here a certain way, but that was not the way I have been getting here.
CORRECTION: I DO see the 'B' on this page. I also see it in the main page alerts. I just never noticed it for some reason. My bad.

 

[Mark44]

I got here without ever seeing a 'B' level indication.

I come in the same way all the time (via a browser on a desktop), so I see the level tag on all threads here. How do you get here and not see the level tag - by a smart phone?

 

[FactChecker]

I come in the same way all the time (via a browser on a desktop), so I see the level tag on all threads here. How do you get here and not see the level tag - by a smart phone?

Sorry. I probably came here initially when it was the "latest" on the main page and after that, from alerts. Chrome browser on desktop PC. (I probably would have been guilty even it I had seen it. I may have gotten carried away.)

PS. I don't see it on this page. Am I supposed to?
CORRECTION: I DO see the 'B' on this page. I also see it in the main page alerts. I just never noticed it for some reason. My bad.

 

[Kumar8434]

What multiplication basically achieves in these situations counting the units. Draw a 1cm x 1cm square and call its size 1 unit area. You can't measure how big the unit is. You can only take a look at it and see how big it is. However, you can measure the size of other surfaces in terms of the unit. In a 15cm x 3 cm rectangle, 45 of your unit surfaces can fit into it. So, its area is 45 units. In a 1cm x 1.5cm rectangle, one of your unit surface can fit into it. But wait, there's still room left for half your unit surface. So, its total area is 1unit + 0.5 unit=1.5 unit. In a circle of radius 1 cm, 3 of your unit surfaces can fully fit into it. But there's still some room left, so you try filling the rest of it with fractions of the unit surface. It turns out that you can keep filling it with smaller and smaller fractions of your unit surface, but the circle will never be full. Its area in terms of your selected unit is a never ending number.

It gets even weirder in Physics. You were driving at 5m/s for 100s. How much distance you moved? 5m/s added to itself 100s times? No. Call the distance you'd move in 1s at 1m/s speed to be the unit distance. Then you just multiply 5x100 to measure how many unit distances you moved.

 

[coolul007]

Multiplication defines a two dimensional area, 2x4, Pi x e, etc. Two values to a rectangle. Division transforms this two-dimensional area (rectangle) to a single dimensional value (line). I restrict it to two dimensions as we only know how to do operation on 2 numbers at a time.