# Why is equal sign used in physics?

• goldust

#### goldust

Might seem a little silly, but I just can't seem to get my head around it. In math, when we write (a + b)^2 = a^2 + 2ab + b^2 or 2 = 1 + 1 , we mean the quantity on the left side of = and the quantity on the right side of = are the same. In physics, when we write F = ma or E = mc^2, what's on the left side of = cannot a priori exist unless what's on the right side of = is given. I don't get how in physics, as does in math, the part on the left side of = and the part on the right side of = can represent the same quantity. Further, I don't get writing the unit of acceleration, which is m / s / s as m / s^2, simply for unit cancellation in manipulation of equations in physics. m / s / s has real world physics meaning. m / s^2 does not, because s^2 does not have meaning in the real world like s does. Is it me, or is notation used in physics somewhat sloppy and need new notations or symbols just for physics? Any help or explanation is greatly appreciated!

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Uh ... what do you do with a case like x = 2y + 3 ? Do you say that x cannot exist unless y exists? It seems like that math statement is identical to the ones you don't like in physics.

I DO agree w/ you that there is a fundamentally different meaning between the two uses of the equals sign but I don't see why the "assignment" usage is a problem.

Equalities in physics are just exact relationships between different things. If you walk in a straight line using equal steps of length ##L##, the distance you will have walked with ##n## steps is ##d = n L##. It is a simple relationship between the distance walked and the number of steps.

In physics, when we write F = ma or E = mc^2, what's on the left side of = cannot a priori exist unless the what's on the right side of = is given.
In my example, what existed first, the distance walked or the number of steps?

Further, I don't get writing the unit of acceleration, which is m / s / s as m / s^2, simply for unit cancellation in manipulation of equations in physics. m / s / s has real world physics meaning. m / s^2 does not, because s^2 does not have meaning in the real world like s does.
When measuring something, units are just references to another real-world thing for comparisons. The measured quantity is a real-world thing, and you carry the units just for reference. When it becomes a mathematical object, i.e., when you use it in a equation, the unit behaves just like any other mathematical object. You say that s2 doesn't have meaning in the real world. What about m2? m3? m4?

is notation used in physics somewhat sloppy and need new notations or symbols just for physics?
Quite the contrary. The mathematical formulation of physics is what makes it an extremely well grounded representation of nature. It is when hands starts waving that things get sloppy...

"In my example, what existed first, the distance walked or the number of steps?"

Without steps, how can there be distance?

"You say that s2 doesn't have meaning in the real world. What about m2? m3? m4?"

m^2 is area or 2D volume. m^3 is 3D volume. m^4 is alleged 4D volume. s^2 I don't know what that is.

"Equalities in physics are just exact relationships between different things."

I think this is the correct way of looking at it. F = ma relates the 3 quantities from different realms together, as in = is a relation symbol and not an equal symbol as it is in math.

This one describes some of the reasons why the = sign can be confusion for students. http://www.intmath.com/blog/the-equal-sign-more-trouble-than-its-worth/4986

I suppose the best way to understand equations in physics is to think of the fact that the two numerical values on either side of = are the same, and not bother with manipulation of units.

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Without steps, how can there be distance?
But without distance, how can you have taken steps? OK, I admit this might not have been the best example, but my point is that the relation usually goes both ways. If you have one, you must have the other. In F=ma, you could argue that the force comes first, and then the acceleration, but if you have acceleration, there must be a force.

"You say that s2 doesn't have meaning in the real world. What about m2? m3? m4?"

s^2 I don't know what that is.
But that is very different from saying
s^2 does not have meaning in the real world
m4 also doesn't have meaning in the "real world", but you find it different because you can imagine its meaning.

F = ma relates the 3 quantities from different realms together, as in = is a relation symbol and not an equal symbol as it is in math.
The moment physical quantities become mathematical objects, I don't see the difference.

"m4 also doesn't have meaning in the "real world", but you find it different because you can imagine its meaning."

Isn't m^4 the volume of a hypercube in 4D space having side length 1 m? How would s^2 be interpreted? Methinks it's safe to write it as m / s / s or more specifically as (m / s) / s, a combo of speed and time.

I think, say in physics when we write F = ma, there are three quantities from 3 realms, force, mass, acceleration, and the numerical value on the left side of = and the numerical value on the right side of = must be equal, this is why the = sign is appropriate, since the two numbers are equal in quantity.

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This one describes some of the reasons why the = sign can be confusion for students. http://www.intmath.com/blog/the-equal-sign-more-trouble-than-its-worth/4986
I don't think this is the place for a rant, so I'll keep quiet this.

I suppose the best way to understand equations in physics is to think of the fact that the two numerical values on either side of = are the same, and not bother with manipulation of units.
You will have trouble going far in physics with such a mindset. You will quickly get into manipulating functions describing physical systems (think quantum mechanics), where there are lots of equal signs but few numerical values. And without units, errors are eay to make. Also, units inform you a lot on the nature of the physical quantity with those units.

Isn't m^4 the volume of a hypercube in 4D space having side length 1 m?
But space has only three dimensions.

How would s^2 be interpreted?
It's nothing more than time times time.

Look, I don't want to carry on much longer in this discussion. If you're having some trouble understanding some concepts in physics because of the use of equal signs or units that you can't picture in your head, I hope you will find the proper help.

"You will have trouble going far in physics with such a mindset. You will quickly get into manipulating functions describing physical systems (think quantum mechanics), where there are lots of equal signs but few numerical values. And without units, errors are eay to make. Also, units inform you a lot on the nature of the physical quantity with those units."

Haven't ventured into quantum mechanics myself, but wouldn't variables be used in place of numerical values? I agree unit checking can be useful for quick verification. However, I contend that writing (m/s)/s as m/s^2 does more confusion than good, especially for novice students in physics.

The way I understand it, for any object at any given time, it has a force F applied to it, a mass m, and an acceleration a, and F = ma holds true because the two numerical values on either side of the = sign are equal. For novice students, I think it could be confusing if they think of other factors like energy, momentum and so on. Methinks the best way to see past this problem is to think of it as F = ma when the values of all other factors are fixed.

In my view, F = ma should not be interpreted as "if given mass m and acceleration a, what force F should be required". This would imply that force does not yet exist when mass and acceleration already exist. All three factors should be assumed to already exist for the object in question. F = ma would then mean that the numerical quantity F and the product of the numerical quantities m and a have equal numerical values.

Thus, in my view, for novice students in physics, it should be taught that numerical values of variables and physical units should be separate rather than mixing the two together. This implies, when looking at equations in physics, only numerical values should be considered, just as with equations in math.

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"You will have trouble going far in physics with such a mindset. You will quickly get into manipulating functions describing physical systems (think quantum mechanics), where there are lots of equal signs but few numerical values. And without units, errors are eay to make. Also, units inform you a lot on the nature of the physical quantity with those units."
The Quote button is useful for quoting what someone else says, and sets that text apart more clearly than just a pair of " characters.
Haven't ventured into quantum mechanics myself, but wouldn't variables be used in place of numerical values? I agree unit checking can be useful for quick verification. However, I contend that writing (m/s)/s as m/s^2 does more confusion than good, especially for novice students in physics.
m/s2 is just notation that has been developed for convenience. The notation is consistent with arithmetic that let's us write (1/2)/3 as 1/(2 * 3). Novice physics students have a lot of new stuff to learn, and IMO this is small potatoes.
The way I understand it, for any object at any given time, it has a force F applied to it, a mass m, and an acceleration a, and F = ma holds true because the two numerical values on either side of the = sign are equal.
Yes, and the units are equal as well.
For novice students, I think it could be confusing if they think of other factors like energy, momentum and so on. Methinks the best way to see past this problem is to think of it as F = ma when the values of all other factors are fixed.

In my view, F = ma should not be interpreted as "if given mass m and acceleration a, what force F should be required".
Why? F and ma are identical quantities. This formula has been around since Newton came up with his three laws. The equation gives a relationship between force, mass, and acceleration. If you know any two of these, you can solve algebraically for the one you don't know. Would you have people memorize these two other variants?
• m = F/a
• a = F/m
This would imply that force does not yet exist when mass and acceleration already exist.
No it doesn't. It merely states that F and ma have the same values. There is no sense of anyone of them existing before any of the others. You are reading much more into this equation than is there.
All three factors should be assumed to already exist for the object in question. F = ma would then mean that the numerical quantity F and the product of the numerical quantities m and a have equal numerical values.
That's exactly what it means.
Thus, in my view, for novice students in physics, it should be taught that numerical values of variables and physical units should be separate rather than mixing the two together. This implies, when looking at equations in physics, only numerical values should be considered, just as with equations in math.

BTW, I took a look at the blog whose link you posted. About the only point that I thought was reasonable was that the symbol used in an identity ought to be different from the symbol used in regular (i.e., conditional) equations.

For example, many textbooks write 2(x + 3) = 2x + 6 and 2(x + 3) = 10 using the same connecting symbol. This is probably confusing, because the first equation is an identity, an equation that is true for all value of the variable. The second equation is true for only one value of the variable; namely, x = 2. Some textbooks (and the blog post) use the symbol ##\equiv## as the connecting symbol in identities.

Most of the other points seemed silly to me, such as 1/2 being equivalent to 3/6 (not sure of the exact fractions that were used, but this was the idea}. 1/2 and 3/6 both represent exactly the same number, so the = symbol between them is appropriate.

Another example they gave was 5 + 9 = 7 * 2 as somehow being confusing because different operations are being used. This is confusing only if you don't understand that the two expressions represent the same number.

For example

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...

This one describes some of the reasons why the = sign can be confusion for students. http://www.intmath.com/blog/the-equal-sign-more-trouble-than-its-worth/4986

...

Who ever wrote that blog is not a mathematician nor does he even know much about math. Perhaps before writing something like that you should make and effort to learn the subject.

Physics uses math to model the real world, the math used in physics is the same math used by mathematicians. So the equal sign is the same.

Who ever wrote that blog is not a mathematician nor does he even know much about math. Perhaps before writing something like that you should make and effort to learn the subject.
My thoughts as well.

sometimes its not intuitive , sometimes only the values make sense
like how E=MC^2 is even possible ? you tell me that the energy of a particle is equal to its mass multiplied by light speed squared ? how can you intuitively even imagine something multiplied by light ?
i assume this is your question , and you probably think this is only true numerically , i sometimes tend to think so
but this is not true , its true numerically and physically
i wouldn't go much into MC^2 and how mass and energy are equivalent physically not just numerically because to be honest i haven't gotten much into that yet

but i am going to go again into the example of F=MA
you say they are from different realms ? how ? can you see force ? no
you only SENSE the force through its effect on other objects , and what exactly is this effect ?
you see the force push an object , what does pushing actually means ?
it means accelerating the object from V=0 to whatever velocity it gets
and is the force the same when someone pushes a small rock and when someone pushes a huge boulder ?
no , you say the later is stronger , he applies more force , how do you know ? he pushed more mass
so Force is just a manifestation of how something applies a certain acceleration to a certain mass
if you rearrange the equation where M=F/A
you say how could mass equal something i don't see divided by acceleration ?
again you might think of mass in this case , as the relation between F and acceleration
this might seem a little bit un intuitive , but it gets more and more intuitive as you understand the meaning of relation . the sign = does not mean what's on the right is what's on the left
you and your brother for instance can be = in weight
that does not mean you and your brother are the same copy , it just means you weigh as much as your brother

same thing goes to M=F/A
this means that Mass is equivalent to force over acceleration
again , what is mass ? can you see Mass ? no you can't
what you see is light coming from the object giving it its shape , you can only sense mass
how do you sense mass ? well , you try to lift the object , if it is easy to lift * accelerates with little force * then the body is not massive , you push it and you can't then you know this is a massive object because it took huge force to generate small acceleration