# Physical meaning of multipication?

• MeChaState
In summary, Jeff says that multiplication has a physical meaning, but this meaning depends on the context in which it is applied.
MeChaState
Hi,

I think I can relate the mathematical operation (+) to a physical entity. Let's say we know for mass it makes sense to say (m1) + (m2) = (m1 + m2) and that has an physical meaning.

But what is the physics behind multiplication (x)? Let's say we accept a/b as a x 1/b and let's say we can measure (Dt) AND we can measure (Dx). Does v = Dx/Dt means something physical?
I mean it sound to me like it is just an mathematical object without any real meaning in physics.

I need help, I really need help.

Thanks

David

Is your question here about the physical meaning of multiplication or about the physical meaning of the derivative?

multipication, dx/dt is a division which is define by a/b as a x 1/b.
Thanks

If you want a physical significance to multiplication doesn't it make sense to you that if you have 2 m1s then the total mass is 2m1? or if you have m2 m1s, then the total mass is m1m2? If the addition property you showed seems intuitive, so should this.

I really don't understand what you're asking for because first you say you want to know about multiplication, but then you start talking about division unnecessarily and then your example is a derivative (not really division).

Nabeshin
In your example 2 has no dimension and it is an integer! But if you use L = V * t, it doesn't mean the same to me.
To division:
Isn't division the inversion of multipication? As far as I remember, in higher Algebra division a/b is defined as a x 1/b.
Please forget the derivative, I wanted just state a number divided by another one, how large or infinitesimal is not important.
Thanks

MeChaState said:
I think I can relate the mathematical operation (+) to a physical entity ... multiply.
Multiplication could be considered a shortcut to add up equal sized groups, for example, 3 crates of apples, with 4 apples per crate = 12 apples.

The inverse operations are normally defined with a third variable.

Subtraction: c = a - b is defined as the value c for which b + c = a
Division: c = a / b is defined as the value c for which b x c = a

Start with positive integers (and zero). Subtraction introduces negative numbers, and division introduces fractions.

For signed multiplication, think of water being poured (+) or drained (-) at 2 gallons per minute from a tank, with the current water level on the tank labeled as "zero gallons".

If water is being poured into the tank at 2 gallons / minute, then 3 minutes from now, the water level in the tank will show (+2 gallons / minute) x (+3 minutes) = +6 gallons.

If water is being poured into the tank at 2 gallons / minute, then 3 minutes before now , the water level in the tank would have shown (+2 gallons / minute) x (-3 minutes) = -6 gallons.

If water is being drained from the tank at 2 gallons / minute, then 3 minutes from now, the water level in the tank will show (-2 gallons / minute) x (+3 minutes) = -6 gallons.

If water is being drained from the tank at 2 gallons / minute, then 3 minutes before now, the water level in the tank would have shown (-2 gallons / minute) x (-3 minutes) = +6 gallons.

Lets take one of the cases,
lets take the case (+2 gallons / minute) x (+3 minutes) = +6 gallons
where we have the units (gallons / minute) x (minutes) resulting (gallons) which is fine and I can understand and is the "shortcut to add up equal sized groups".
There are physical dimensions like Force set as F = m x a. or I = E x D.
How could we interpret the multiplication? Can you still see it as a adding up group?

Thanks

Jeff let me please ask you in this way, if multiplication is just "shortcut to add up equal sized groups" it must be possible to express any equation with (x) into an equation with more or less (+)s.
How would you express F = m x a n terms of addition only, without any multiplications?
I know it sounds weird, sorry, I am really confused here!

Thanks again

The physical interpretation ultimately depends on the context in which it is applied. Going with Jeff's interpretation, yes Newton's second law can be interpreted in terms of group addition. For example,

It require 1N to accelerate one unit mass at 1m.s-2. Therefore is takes 3N to accelerate three unit masses (m=3) at 1m.s-2.

It is useful, and perhaps necessary, in *physics* that mathematical operations have a physical meaning. However, it is not necessary for a mathematical operation or operator to have a physical meaning for it to be defined.

Last edited:
MeChaState said:
multiplication ... Can you still see it as a adding up group?
The concept of multiplication goes beyond the simple concept of scalar times group of objects. I was just using a very simple example. In the water in tank example, a rate of flow was being multiplied by time to calculate a net change in mass versus time.

Concepts like acceleration involve subtraction and division:

average acceleration = (velocity_1 - velocity_0) / (time_1 - time_0)

mass could be considered as a group of atomic particles.

Force = mass x acceleration, results in unit definitions such as:

Newton = kilogram x (meter / second) / second.
pound = slug x (foot / second) / second.

Multiplication can also be generalized to apply to mathematical abstract objects, such as matrices or vectors.

Jeff Reid said:
The concept of multiplication goes beyond the simple concept of scalar times group of objects. I was just using a very simple example. In the water in tank example, a rate of flow was being multiplied by time to calculate a net change in mass versus time.

Concepts like acceleration involve subtraction and division:

average acceleration = (velocity_1 - velocity_0) / (time_1 - time_0)

mass could be considered as a group of atomic particles.

Force = mass x acceleration, results in unit definitions such as:

Newton = kilogram x (meter / second) / second.
pound = slug x (foot / second) / second.

Multiplication can also be generalized to apply to mathematical abstract objects, such as matrices or vectors.

Yes,Newton and pound and everything,what is "per something" is easy to imagine and explain.But what about this?:R=specific electrical resistance x length of the conductor/cross-sectional area.Where is "per something"?I understand this formula,but it could be for example:R=specific electrical resistance + length of the conductor - cross sectional area?

thedy said:
Yes,Newton and pound and everything,what is "per something" is easy to imagine and explain.But what about this?:R=specific electrical resistance x length of the conductor/cross-sectional area.Where is "per something"?I understand this formula,but it could be for example:R=specific electrical resistance + length of the conductor - cross sectional area?
You could see addition as a specific case of multiplication wherein the mulitplier is 1.

i.e:
R=specific electrical resistance + length of the conductor - cross sectional area where length = 1.

If you keep up this line of reasoning, you'll never make it through higher math than arithmetic. It is probably best to just drop it as being pointless.

My 'physical' point of view about multiplication is that it represents the relations between quantities. This point of view it is not formal but it is how it was presented to me when I was first introduced to science.Consider some physical quantity q that increases or decreases when another physical quantity p.We can write this as $$q\propto p$$.To get an expression that has an equal sign so we will include a 'correction constant C'(which a can be considered a physical quantity in itself).We now have $$q=Cp$$. Let's say that there is another quantity n that causes q to decrease when it increases but does not effect p in any way.This means $$C\propto \frac{1}{n}$$ by including another constant k we get $$q=k\frac{p}{n}$$.This process can continue if we find new quantities and relations between them and q. I do not know if this helps you but it helped me understand how some equations are determined empirically, especially in E&M.

MeChaState said:
multipication, dx/dt is a division which is define by a/b as a x 1/b.
Thanks
This is an absolutely wrong understanding of what a derivative is.I suggest you really concentrate on learning calculus

MeChaState, maybe I'm on the wrong track but having studied primary teaching (don't look at me that way; at least I studied something) maybe you're just after a more basic answer. If not then just ignore; but just in case:

If the issue is: what does it mean to multiply by a fraction...

If ':' represents 1 whole apple then the following represents 4 x 5 apples:

:::::
:::::
:::::
::::: = 4 x 5 apples = 20 whole apples in total

If '.' represents half an apple then the following represents 4 x 1/2 apples:

.
.
.
. = 4 x 1/2 = ... = 4 half apples

If we stick the apples back together

... = :: = 2 whole apples in total so: 4 x 1/2 = 2

That explains why multiplication of fractions works.
However, in physics division is quite often representative of the rate of something per something; rather than as an array of equal rows of one thing as I have shown.

What you are looking at are: rate = frequency / quantity
It could be for every 6 vehicles that you pass 2 are green cars.
This would be:
Rate of green cars per vehicle = 2 green cars / 6 vehicles.

This is written as: Rate = number of green cars / vehicles or R = g / v.

As you've said this can be written as R = g x 1/v.

You can't make rates fit nicely into arrays like I did for the apples.
I showed the arrays simply to show that multiplication by fractions works.
But you would know that g x 1/v = g/1 x 1/v = (g x 1)/(1 x v) = g/v.

But you can see physically that our above example gives:
2g/6v which is equivalent to 1g/3v
This can further be taken to 1 x g x 1/3 x 1/v
= 1 x 1/3 x g x 1/v
= 1/3 g/v or 1/3 green cars for every vehicle

Obviously for every vehicle that passes 1/3 of it is not going to be green car and the rest of the vehicle something else but it just gives us a 'rate' that it occurs at. We can then use that rate in other calculations.

Most often rates use time as the quantity ie: rate = how many / how often
For example speed = distance / time
You can measure you distance traveled over 2hrs to get a speed of 200km/2hrs,
but this is the same as 100km every hour.
ie speed = 200km/2hrs = 200km x 1/(2hrs) = 200 x km x 1/2 x 1/hrs
= 200 x 1/2 x km x 1/hrs = 100 x km/hr = 100km/hr

I hope this is what you needed.

... it sound to me like it is just an mathematical object without any real meaning in physics.

It is a genuine miracle that our physical world so often matches human kinds mathematical theory...But much of math may have no significant physical meaning; that's why physics, for example, uses experimentation to confirm one of perhaps many models...Only the one(s) that matches physical observation apparently has physical meaning...in our universe.

If you have a formula like F = ma -- this means, F is directly proportional to m AND F is directly proportional to a, independently.

In Newton's gravitational formula, the mass of one body is multiplied by the mass of another. This means -- there is a gravitational attraction between each part of one body and each part of the other.

In other words, the gravitational force is proportional both to the mass of one body AND to the mass of the other.

Hope that helps...

gonegahgah said:
That explains why multiplication of fractions works.
However, in physics division is quite often representative of the rate of something per something; rather than as an array of equal rows of one thing as I have shown.

Most often rates use time as the quantity ie: rate = how many / how often
For example speed = distance / time
You can measure you distance traveled over 2hrs to get a speed of 200km/2hrs,
but this is the same as 100km every hour.
ie speed = 200km/2hrs = 200km x 1/(2hrs) = 200 x km x 1/2 x 1/hrs
= 200 x 1/2 x km x 1/hrs = 100 x km/hr = 100km/hr

That is what I m saying about."Per something".But as I say,could you try explain R=specific electrical resistance x length of the conductor/cross-sectional area?I don t see connection between your apple example and this one.I know that addition in this case is error,but if formula needs just have orientational meaning and not real meaning,like this:4 apples plus 3 apples equals 7 apples,then doesn t matter,what I use.Addition or multipication/division.We can see just decreasing or increasing of variable.Not real value.If I divide cross-section area,I don t know,how many electrons don t pass over conductor.We know just,that R will be decreasing.That s all what we know,dont we?

Hello MeChaState-
Biological cells learned to multiply by dividing. Does that make sense? Exponential growth is equivalent to multiplying by a constant factor at regular intervals.
Bob S

DaveC426913 said:
You could see addition as a specific case of multiplication wherein the mulitplier is 1.

i.e:
R=specific electrical resistance + length of the conductor - cross sectional area where length = 1.

How do you mean" where length = 1 ".Could you give me other example?I don t understand it.thanks

thedy said:
But what about this?:R=specific electrical resistance x length of the conductor/cross-sectional area.Where is "per something"?I understand this formula,but it could be for example:R=specific electrical resistance + length of the conductor - cross sectional area?
DaveC426913 said:
You could see addition as a specific case of multiplication wherein the mulitplier is 1.

i.e:
R=specific electrical resistance + length of the conductor - cross sectional area where length = 1.
thedy said:
How do you mean" where length = 1 ".Could you give me other example?I don t understand it.thanks
Nevermind, I think I see what you're getting at in your above post. I'll requote:
R=specific electrical resistance x length of the conductor/cross-sectional area.
Where is "per something"?
The "per something" you are asking about is per unit area of the cross section.

If the cross section is 1in^2 then R is simply SER x length.
If the cross section is 0.5in^2 then r is SER x length / 2.

Thanks,so it really seems,that every formula is per something.Of course if includes division.

## 1. What is the physical interpretation of multiplication?

The physical interpretation of multiplication is the process of combining or grouping equal sets or quantities to determine the total amount. In other words, it represents repeated addition. For example, 2 x 3 can be interpreted as 2 groups of 3 or 3 + 3, which gives a total of 6. This can be seen in many real-world scenarios, such as counting objects or calculating area and volume.

## 2. How is multiplication related to scaling and proportion?

Multiplication is closely related to scaling and proportion as it involves the concept of increasing or decreasing a quantity by a certain factor. When multiplying two numbers, the result is a scaled version of the original numbers. For instance, multiplying a length by a width gives the area, which represents a scaled version of the original dimensions. Similarly, proportions can be solved using multiplication, where the two ratios are multiplied to find the missing value.

## 3. Can multiplication be used to represent division?

Yes, multiplication can be used to represent division in the form of inverse operations. Just as addition and subtraction are inverse operations, multiplication and division are also inverse operations. For example, 20 ÷ 4 can be represented as 20 x 1/4, where 1/4 is the inverse or reciprocal of 4. This concept is important in understanding fractions and solving equations involving fractions.

## 4. How does multiplication relate to the concept of dimensions?

In mathematics, dimensions refer to the measurement of space, such as length, width, and height. Multiplication is used to calculate the total area or volume of an object, which is an essential aspect of understanding dimensions. For instance, multiplying the length and width of a rectangle gives its area, which is a two-dimensional measurement. Similarly, multiplying length, width, and height gives the volume, which is a three-dimensional measurement.

## 5. Can multiplication be applied to non-numeric quantities?

Yes, multiplication can be applied to non-numeric quantities, such as vectors, matrices, and complex numbers. In these cases, multiplication represents a combination or transformation of these quantities, rather than repeated addition. For example, multiplying two vectors results in a new vector that represents a combined effect of the original vectors. This concept has applications in physics, engineering, and other fields of science.

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