# Meaning of Multiplication and Division in Physics

Mathematics is all around sciences.It is also considered a language.In language everything should have meaning.

I know addition and subtraction meaning.Same things[quantities ,etc.] can add or subtract . It cannot be done on different things.

What is the meaning of multiplication and division in sciences[Physics,.. etc].I know what it means in real life.Multiplication is higher form of addition.Division is higher form of subtraction.

What does F=ma means. m is in kg. a in ms-2.How can both multiply.They cannot add or subtract.

Why any quantities can multiply or divide,contrary to adding and subtracting.

• starprince

## Answers and Replies

Staff Emeritus
I know addition and subtraction meaning.Same things[quantities ,etc.] can add or subtract . It cannot be done on different things.

What do you mean "different things"?

What is the meaning of multiplication and division in sciences[Physics,.. etc].I know what it means in real life.Multiplication is higher form of addition.Division is higher form of subtraction.

It is the same as it is in math.

What does F=ma means. m is in kg. a in ms-2.How can both multiply.They cannot add or subtract.

I could easily add or subtract or add M and A. I just wouldn't be using the F=ma equation. That is only useful in certain contexts. The letters themselves are just placeholders for numbers, and the equation itself is only useful for real life applications when the numbers you use come from the force applied, the mass of the object, and the acceleration. Mathematically F=ma is identical to X=cd if you plug the same numbers in.

On the other hand, the reason WHY m times a is equal to the force applied in real life is a mystery. That's just the way it works. In some other universe it may be the case that F=2ma, or F=ma2. But not in our own universe.

Why any quantities can multiply or divide,contrary to adding and subtracting.

Because otherwise it wouldn't be useful to us. I can take the numbers I plug into F=ma and do a million different things with them and it would all be mathematically correct. They just wouldn't mean anything in real life unless I use them in the appropriate equation.

dipole
You've completely misunderstood his question, Drakkith. He's not asking why any specific physical law has a specific form, he's asking what is the physical meaning of multiplying or dividing physical quantities.

What do you mean "different things"?

He means different physical quantities which have different units.

It is the same as it is in math.

No I don't think so. In math, the statement $ab = a+a+a+... b$ number of times is valid and makes perfect sense when dealing with pure numbers, but in physics the statement $ma = m+m+m+m...$ is nonsense, because $a$ is not a number, but a physical quantity with a magnitude which is represented by a number. In physics when we write down a multiplication, we're really talking about a different kind of multiplication than the kind we use in normal arithmetic. However, we represent certain aspects of the world by real numbers, to which we can apply the normal rules of arithmetic.

I could easily add or subtract or add M and A. I just wouldn't be using the F=ma equation. That is only useful in certain contexts. The letters themselves are just placeholders for numbers...

No, you can't. As I said above it makes no sense to write down an equation like $F = m + a$ because $m$ and $a$ have different units.

So, as far as the OP's original question goes... I think it really depends on the situation. What it means in a physical way to multiply something like mass and acceleration may be different than what it means to multiply something like force and distance. However, just because we can represent many quantities by numbers, especially in one dimension, doesn't mean they are numbers.

I have no good general response to the question, and I think it's a rather deep one and one I've thought about myself before. I hope some other more enlightened persons can weigh in.

• Astronuc, valenumr, reetinder singh and 2 others
Staff Emeritus
You've completely misunderstood his question, Drakkith. He's not asking why any specific physical law has a specific form, he's asking what is the physical meaning of multiplying or dividing physical quantities.

Huh. Well then ignore everything I said.

Mentor
we create new physical quantities from multiplication and division operations:

length * width = area with units of measure of meters^2

or

distance/time = average velocity with units of measure of meters/second

or

F=ma --> we've defined a new quantity called force with units of measure newtons (kg*meters/second^2)

We add and subtract the same kinds of quantities but create new ones when we multiply or divide.

Differentiating or integrating operations are also used to create new quantities that aid us in understanding a physical system.

The key is that the equations must be internally consistent in units of measure.

• Astronuc
What it means in a physical way to multiply something like mass and acceleration may be different than what it means to multiply something like force and distance.
And multiplying force and distance itself can have two different meanings (work vs torque).

• sophiecentaur
technician
Huh. Well then ignore everything I said.
What do you mean by this ?

DiracPool
we create new physical quantities from multiplication and division operations

Yeah, I like that explanation. We are creating new relationships between otherwise isolated phenomena in order to label and quantify some emergent property of a system wherein those hitherto isolated phenomena/physical quantities interact. This is the utility of the multiplication and division operators in physics. As noted above you can't do that with the addition and subtraction operators because they must act on like units.

Force=kg*meters/second^2. I always thought the balance of units of all of these physical terms (such as force, work, etc.) was downright goofy and confusing. But I think I see their utility more now. Interesting.

• sophiecentaur and AlexS
Thanks guyss .......... I got the concept .......

technician
On the other hand, the reason WHY m times a is equal to the force applied in real life is a mystery. That's just the way it works. In some other universe it may be the case that F=2ma, or F=ma2. But not in our own universe.

I would disagree with the statement F = 2ma indicates a 'different universe'
The relationship between physical quantities is best given as F ~ ma (proportional)
The numerical factor indicates the units used in measurement.
If you had written F = 2.2ma it would be a fairly good indication of Newton's law in this universe if m was measured in lbs, and a in m/s2
F = ma2 could be another universe but that is science fiction (as far as I know)

AlexS
Interesting conversation, I think that the criterion with unit is very important.I actually think that after you have this fundamental defining formulas such as average speed=total distance/total time, you can obtain other formulas such as Galileo's formula(v square=v0 square+2ad) just doing the math,without any trouble. But,in my view, the problem is still with this fundamental equations. For instance, why the speed is distance per time and not distance multiplied with time? I see the proportionality between time and distance (if speed is constant), but I still cannot perfectly understand this problem.

willem2
The distributive law seems to be logically required for F, m and a.
Suppose you accelerate a mass, that's composed of two smaller masses m1 and m2, since the masses are attached to each other a, is the same for both masses and for the total mas, so you have mtot * a = Ftot = F1 + F2 = m1 a + m2 a. Also if a = 0, then F=0, another multiplication axiom m*0 = 0.

Gold Member
You've completely misunderstood his question, Drakkith
That's a bit harsh, ain't it? Any given mathematical operation can be fitted to a whole set of different physical situations with different quantities. There are Logical Rules that limit which operations can be applied to which relationships between physical quantities; some operations are commutative or distributive and some aren't. How far do you want to take this discussion and how deep into Maths Analysis do you want to go?
When we learn Maths we start with discrete quantities and slide gently into applying the same rules to continuous variables. When we derive any Physics equation from basics, we start with discrete steps and extend into a continuum. Calculus is a great example of this when the results always involve "limit as x approaches zero" statements.

I must say though, it gives me a weird feeling that Maths seems to have sprung up, out of nowhere, in the heads of humans who have constantly been trying to describe and predict the world around them. Just why should it be common to everything we experience?

• Philip Koeck
OldYat47
The basic issue is dimensional analysis and the definitions of various physical quantities. For example, F=ma could be in English or SI units as long as they agree. Force is defined as mass times acceleration, so both sides of the equation have the same dimensions. The meaning of, for example, F = (20 kg) X (2 m/s^2), F = [(40)(kgXm/s^2)], is that the numbers give the magnitude, the dimensions give the system of measurements used. You could not, for example, say F = (20 slugs) X (2 m/s^2) because the systems of measurement are not the same. Force is not defined as (slugs X m/s^2) or (kg X ft/s^2). You have to be consistent. The dimensions are handled algebraically.

And it pays to be careful. You may see lbm (pounds mass) instead of slugs. lbf (pounds force) or just lb (pounds) as weight, and kg (kilograms) is commonly used as a measure of weight (actually 1/9.81 kg mass measured by a scale on the surface of earth, which should be Newtons to be accurate).

• KY_MU
pixel
I must say though, it gives me a weird feeling that Maths seems to have sprung up, out of nowhere, in the heads of humans who have constantly been trying to describe and predict the world around them. Just why should it be common to everything we experience?

There's a paper by Wigner titled "The unreasonable effectiveness of mathematics in the natural sciences" that may provide some answers: http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html

OldYat47
pixel has referenced an excellent article. On thing that the author discussed in an interview I heard was that mathematics (or at least the way we understand mathematics) doesn't always work (weather prediction was an example he used). It's an interesting question whether it's our mathematical inability to explain certain phenomena because we haven't figured out the math yet, or if the phenomena are truly outside the realm of mathematics. I tend to believe the former, just because.

Homework Helper
One mathematical model for dealing with different units is to consider the area under a curve (or between two curves) where x and y axis have different units. The area can be considered to be the limit of a bunch of squares with sides Δx and Δy as the size of the squares approaches zero and the number of squares approaches infinity. For simple cases with constant values, then the area is just the product of the sides of a rectangle x by y, with x and y having different units, such x = mass and y = acceleration, where the area would be force.

Other examples of multiplication in physics: torque = force x radius, power = force x speed, ... .

David Lewis
What does F=ma means? m is in kg. a in ms-2. How can both multiply? They cannot add or subtract.

When mass and acceleration are multiplied, it just means if you double the mass then you double the force. If you triple the mass, you triple the force, etc. And, if you double the mass and triple the acceleration, you will get 6 times the force.

Multiplication started out as shorthand for this abstract generalized relationship between numbers. The original application was counting. It was subsequently discovered some disparate physical quantities obey the same relationship.

• KY_MU
arul_k
What does F=ma means. m is in kg. a in ms-2.How can both multiply.They cannot add or subtract.

We also need to look at how we define certain quantities in science. Force is defined in terms of the accleration produced on a certain mass.

So the equation F=ma indicates that if you double the force you would get twice the accleration, mass remaining constant and experiment proves this to be correct.

• KY_MU
We are introduced to addition as a method of counting objects in groups. We are introduced to multiplication as a method of adding things that are grouped in equal quantities as a dozen of eggs in a carton packed 20 to a box. So we can find the total quantity of eggs by multiplying 12 by 20 instead of having to remove the eggs and count them. Addition expedites counting and multiplication expedites addition. Addition requires that we add only similar objects that is when you count you count something particular, cows, bulls, black sheep or herbivores, They can be different in actuality but must have a common characteristic to which the addition is to apply and the counting is appropriate answering the question "How many of this type do we have?"
Now in the use of multiplication to "non counting situations" as in mass x acceleration one can see that this is still in fact a counting or addition situation. The total force for example on a body is the sum of the forces on a multitude of elemental masses. There is the force acting on each real elemental mass but this force is the sum of the basic specification unit of the force i.e, Newton. Each elemental mass will have a delimited number of Newtons associated with it. So we have an array of force units associated with a number of masses. analogous to the total number eggs in a box containing a number of cartons associated with a dozen eggs count in each carton.

We are led to notice that units can be treated as mathematical quantities e.g., meter/meter =1 or meter⋅meters = meters2 which leads to dimensional analysis and of course the adoption of names for certain combinations of these quantities. • Astronuc
OldYat47
Think about it this way: F = m*a is algebraically the same on both sides of the equal sign. Force is defined as m*a so the equation reads (expanding force into its components) m*a = m*a. That's dimensional analysis, making sure the various quantities make sense algebraically. The magnitude of the equation is in the numbers. So for example, 10*m*a = 5*m*2*a. This makes perfect sense even if the definition of force doesn't exist. Similarly we could work equations with (550 ft-lbf/sec) without the "abbreviation" of horsepower.

All the quantities and defined units must follow the rules of algebra and calculus, so you can't subtract or add mass from or to acceleration.

Even though multiplication is often introduced as a form of addition for simplification, in reality it is not.

• meekerdb and (deleted member)
ogg
I may have missed it, but in case it hasn't been made explicit. "Addition" (or "multiplication" or "subtraction" or "division" or "integration" or "differentiation" or "taking the limit" or....) does NOT have a single unique meaning in Science/Physics. Adding groups, adding vectors, adding areas, adding lines, and adding scalars are DIFFERENT types of addition. Please look up the history of mathematics: counting (accounting) was considered almost completely different than geometry (areas of surfaces, volumes,...). The rules we use in the math we apply DEPEND on the need (the situation, the context, the problem). Sometimes a+b = b+a but sex and a dinner is a different date than dinner and then sex. (seriously, the "normal" properties of arithmetic (commutative property, distributive property, etc.) are NOT always correct in given contexts. I haven't looked into the philosophy, but I suspect there is no "bright line" distinguishing when it's meaningful to attach units to ratios and when its not. Since much of physics is ratios, I expect that interpretation (dimensional analysis) at some point requires a pragmatic (it works, so it's correct) approach.

• Lnewqban
David Lewis
Please look up the history of mathematics: counting (accounting) was considered almost completely different than geometry (areas of surfaces, volumes,...).

Geometry was originally used to count the area units of Egyptian farmland so the pharaoh knew how much property tax was due.

I can't seem to think of a quantity in physics than cannot be counted, included, accumulated, added, or summed in the simple strategy of arithmetic

OldYat47
gleem, that may be true working with individual units, but not when working with combined defined units. Mass and acceleration, for example, can be multiplied to give a resulting force, but they cannot be added or subtracted from one another. So algebra always enters early on.

Homework Helper
I can't seem to think of a quantity in physics than cannot be counted, included, accumulated, added, or summed in the simple strategy of arithmetic
Adding the Celsius temperatures of all the students in a freshman physics class is of questionable value, even though they all have the same units.

I can't seem to think of a quantity in physics than cannot be counted, included, accumulated, added, or summed in the simple strategy of arithmetic

I guess I should added where permitted or sensible for adding temperatures is usually of dubious value as is adding densities and you know even adding masses together can be of dubious value unless there is a relationship or unifying connection between them, the same for example, resistance or battery voltages.

The OP questioned the meaning of multiplication in scientific applications. My post was to point out that I see multiplication basically has the same meaning as counting equal size groups where the groups size is precounted (summed) and usually interpreted as a rate or ratio. An note if the group size varies then this counting processes can be appreciated as a primitive form of integration.

Multiplication begets division as addition begets subtraction. I think when you look at a ratio as an indicated division and not stemming from a sum of things in a group you what is happening. A moving object for example put the number of miles traveled in a group with defined characteristic e.g.. one hour intervals. I guess I just been concentrating too much on my odometer counting of the miles while my watch is counting off the hours.

OldYat47
Again, I respectfully disagree. Multiplication and addition are fundamentally different as applied in common physics problems. And again, we can define certain physical properties and fundamental concepts with algebra where we could not do the same with arithmetic. And indeed a ratio is a division operation not stemming from a sum of things in a group.

Gold Member
It's easy to forget that Maths is only just another model of real life. You write down an equation or formula to represent what happens and you immediately make assumptions about the accuracy of that formula. Solving a simple quadratic equation for a computation of the motion of an object assumes that the variables are all continuous. Quantum Mechanics applies everywhere, so that implies that the energy changes are actually not continuous but discrete. Fact is that it makes not difference in most cases but you have to break away from that simple formula when the number of energy states becomes limited. The Maths model breaks down - no problem, unless you let it bother you. The 'faith' involved in using Maths should be just based on experience that multiplication, in most cases, produces results that can be verified (within the limits of experimental error).
3 (discrete) boxes of 12 (discrete) cakes gives you (correctly) 3X12 = 36 cakes. But you can only have complete boxes and complete cakes. 10s of acceleration of a car at 2m/s2 will only give an approximate answer of 10X2 = 20ms2, whatever accuracy you may claim to have measured the time and the acceleration.
If you want to introduce the concept of Chaos into the argument, the correspondence between Maths and Measurement becomes even more dodgy. The results for infinitesimal differences in input conditions for a chaotic model can be vastly different. The Butterfly can be absolutely anywhere around the flower but the Maths prediction assumes a discrete input of position - so many decimal places. Increase the number of decimal places by one and the possible outcomes can involve flat calm or a hurricane around your house.

valenumr
We also need to look at how we define certain quantities in science. Force is defined in terms of the accleration produced on a certain mass.

So the equation F=ma indicates that if you double the force you would get twice the accleration, mass remaining constant and experiment proves this to be correct.
Actually, the formal definition of force in Newtonian mechanics is change of momentum with respect to time. Varying mass isn't a problem.

• bhobba and PeroK
sysprog
Adding the Celsius temperatures of all the students in a freshman physics class is of questionable value, even though they all have the same units.
You can of course divide the sum by the number of students and get the arithmetic mean.

ergospherical
Actually, the formal definition of force in Newtonian mechanics is change of momentum with respect to time. Varying mass isn't a problem.
To the contrary, variable mass systems are way more subtle than you think. The expression ##\boldsymbol{F} = \dot{\boldsymbol{p}}## in fact only holds for systems of constant mass.

I won't go into details, but Spivak's Physics for Mathematicians: Mechanics I has some nice discussion on this issue, for example.

And a footnote, completely inappropriate for this thread so start a new thread for any followup:
In general treatments, the derivative of the momentum ##\boldsymbol{p} = \int_{\mathcal{D}(t)} \rho \boldsymbol{v} dV## of the medium within some domain ##\mathcal{D}(t) = g^t \mathcal{D}_0## must be evaluated using the Reynold's transport theorem of vector calculus, and the result involves a term ##-\int_{\partial \mathcal{D}(t)} \rho\boldsymbol{v}(\boldsymbol{v}-\boldsymbol{w}) \cdot d\boldsymbol{S}## in addition to the total force on the body. [Here ##\boldsymbol{w}## is the velocity of the boundary and ##\boldsymbol{v}## the velocity of the medium.]

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