# B Understanding units

#### Matt Smith

I am an student in physics department and I will graduate very soon.In my past four year I have done a good work and get a nice gpa.But as I get more and more familiar with advanced content such as quantum field theory,I become very comfusing about what I learn in my high school. Because I think I don't really "understand" the knowledge in high school,even I am very familiar with how to calculate them.

For example,how does the numbers "add" and"multipy".It won't make me very confusing if the numbers are just real numbers,mathematically.But no one tell me a percise theory about how the numbers operate between too numbers with unit,since the physical quantity can be written as "R + unit”.

It sounds very reasonable that 10cm is equal to 0.1 meter.But It doesn't mean “10cm+1m”is the same as "0.1m+1m".I just mean,the addtion principle should be defined carefully.If not,maybe somthing weird will happen.But I don't remmenber my high school teacher having taught that.I just know,1kg/1 dm^3=1kg.cm^(-3).And 1kg/0.00001m^3=1*10^6kg.m^(-3)=1kg.cm^(-3).

But the point is I have totally forgotten the reason.And I think the operation between real numbers is the not same as the what we use in physics.Because in physics we are talking about numbers with unit,not pure number.

I think I really need some book about it to read.But all lecture notes in college never mention it.Even the classical mechanics course for freshman will talk about V=dr/dt,E=1/2mv^2.But it almost drive me crazy what heck is m times v^2.

Secondly,I am much confusing about the charge.Why the positive charge will have a positive value and the negetive charge will have a negtive value.We never think about it when we are measuring mass,length.As I have said before,I don't really remember what I learn in my high school.I always think since the two kind of charge is different,we use + and- to make a difference.Can't we use other symbol to show the difference?

For example,we can say positive charge is 3 C,and the negative one is 3 c*.If we accept we can use negative value to represent a negative charge.Why won't we define the mass as -4 nkg or something else.Maybe I think to much but I just can't stop myself.Please help me .

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#### mjc123

10 cm = 0.1 m. But 1 kg/1 dm3 =is not 1 kg cm-3. It is 1 kg/1000 cm3 = 0.001 kg cm-3. And 1 kg/0.00001 m3 = 105 kg m-3. I think you left a 0 out. You will constantly go wrong if you're not careful about attention to detail.
Strictly, you can't add things that are in different units. You have to convert them to the same units, e.g. 10 cm + 1 m = 0.1 m + 1 m = 1.1 m.

But you can multiply. The units are the product of the units of the multiplicands. Thus for E = 1/2 mv2, if m is in kg and v in m/s, E is in kg m2 s-2. We have a name for this unit, we call it the joule.

As for charge, we find by experiment that there are two kinds of charge. Like charges repel and unlike charges attract, with a force that is proportional to the product of the magnitudes of the charges, and opposite in sign for the like and unlike cases. And equal amounts of unlike charges can neutralise each other to give zero net charge. So it makes sense mathematically to give one kind of charge positive values and the other negative values - which is which is a matter of historical convention.

We only know one kind of mass, which always attracts other matter, so mass is positive - there's no reason to give any mass a negative sign.

#### BvU

Homework Helper
Hello Matt,

Your post shows an inquisitve mindset and you don't just accept anything you hear, see or read: good characteristics for a scientist. Seems to me the PF is a good place for you... you get people to answer your questions and put you right if you make a mistake. One example:
since the physical quantity can be written as "R + unit”.
It is important to reconsider this: a physical quantity is a real number times a unit. 4.5 kg indicates a mass that is 4.5 times as heavy as a standard mass. Multiplying (or dividing) quantities means multiplying values AND multiplying (or dividing) units.

And a math example: addition is only defined for quantities within a group (butlanguage has widened that). Your 10 cm + 1 m requires a conversion to either cm or m before the addition of the values can be performed. Conversion in the metric system is relatively easy with powers of 10 and with centi = 1/100 so 1 cm = 0.01 m (and therefore 1 = 0.01 m/cm), your 10 cm + 1 m becomes 10 cm * 0.01 cm/m + 1 m = 0.1 m + 1 m = 1.1 m.

Your confusion about charge: do you have the same problem wit your bank account ? A shortage is considered negative so that we can add a negative charge to an equal amount of positive charge to end up with something neutral. Extension of the set of positive numbers plus zero to all numbers allows subtraction for all possible pairs of values.

#### Dale

Mentor
But no one tell me a percise theory about how the numbers operate between too numbers with unit,
They work precisely like any other algebraic quantity. If 100 y = 1 x then:

10 y + 1 x = 0.1 x + 1 x = 1.1 x

It doesn’t really matter if x is a meter or a pure number.

Why the positive charge will have a positive value and the negetive charge will have a negtive value.
Because the force between two charges can be either attractive or repulsive.

Why won't we define the mass as -4 nkg or something else.
Because the force between two masses is always attractive.

#### A.T.

Can't we use other symbol to show the difference?
You mean like and ?

#### Matt Smith

10 cm = 0.1 m. But 1 kg/1 dm3 =is not 1 kg cm-3. It is 1 kg/1000 cm3 = 0.001 kg cm-3. And 1 kg/0.00001 m3 = 105 kg m-3. I think you left a 0 out. You will constantly go wrong if you're not careful about attention to detail.
Strictly, you can't add things that are in different units. You have to convert them to the same units, e.g. 10 cm + 1 m = 0.1 m + 1 m = 1.1 m.

But you can multiply. The units are the product of the units of the multiplicands. Thus for E = 1/2 mv2, if m is in kg and v in m/s, E is in kg m2 s-2. We have a name for this unit, we call it the joule.

As for charge, we find by experiment that there are two kinds of charge. Like charges repel and unlike charges attract, with a force that is proportional to the product of the magnitudes of the charges, and opposite in sign for the like and unlike cases. And equal amounts of unlike charges can neutralise each other to give zero net charge. So it makes sense mathematically to give one kind of charge positive values and the other negative values - which is which is a matter of historical convention.

We only know one kind of mass, which always attracts other matter, so mass is positive - there's no reason to give any mass a negative sign.
It seems reasonable to think like that.But it is too intuitive.There are many points to be not so strict.For example,why numbers with different unit can not be added.My question is ,is there any mathematical language which can define the addition and mulplication percisely?For example,"if we have a set Q{all p/q,p∈Z,q∈Z}.For any p/q∈Q and c/d∈Q,p/q+c/d=(pd+cq)/qd.Don't you think your definition is too physical?Or maybe we never want to have a percise theory.If we forget it,we still have more problem.In physics,everything is abstract.E=1/2mv^2.du=Tds-pdv+udn.▽×E=-partialB/partial t
If we say E,what is the unit ?Joule,eV?If we say V,what is the unit?m/s,km/s?One thing is sure,we never care about which unit is used!There must be a mathematical theory about it,which make the physical law "unit independent".I really can't accept your explanation ,bacause I am not a high school student,but a college student.

#### Matt Smith

They work precisely like any other algebraic quantity. If 100 y = 1 x then:

10 y + 1 x = 0.1 x + 1 x = 1.1 x

It doesn’t really matter if x is a meter or a pure number.

Because the force between two charges can be either attractive or repulsive.

Because the force between two masses is always attractive.
No no no.Your explanation almost make me think I understand it.But in fact ,not.You can't say 10 y + 1 x = 0.1 x + 1 before you define the +.For example,we all know 2=2/1.But you can't say 2+7/8=2/1+7/8 before you define the +between rational number.Although in the history of math,we have to adjust our addition law to make the two equal.But firstly ,we should define it.
Even it sounds right when you just use your intuition.But I am not a kid,and I don't need to calculate" 1m+1cm".We always deal with E=1/2mv^2.du=Tds-pdv+udn.▽×E=-partialB/partial t.We never say which unit the quantity use.One thing is sure,we never care about which unit is used!There must be a mathematical theory about it,which make the physical law "unit independent".When we add two abstract quantity such as E1,E2,it seems weird to talk about the unit they use.What I really want is a abstract ,mathematical explanation,such as "if we have a set Q{all p/q,p∈Z,q∈Z}.For any p/q∈Q and c/d∈Q,p/q+c/d=(pd+cq)/qd".

#### DrClaude

Mentor
No no no.Your explanation almost make me think I understand it.But in fact ,not.You can't say 10 y + 1 x = 0.1 x + 1 before you define the +.
You have to accept that the normal rules of algebra apply.

Treating units as algebraic quantities, everything works out. You can add quantities that have different units, but you get nowhere
$$10 y + 1 x = 10 y + 1 x$$
if there is no relation between $y$ and $x$, as is the case if $y = \mathrm{m}$ and $x = \mathrm{s}$. But if $y = \mathrm{m}$ and $x = \mathrm{cm}$, then you can use the relation $\mathrm{m} = 10^2 \mathrm{cm}$ to simplify the equation to a single numerical quantity with a unit
$$10 y + 1 x = 10 (10^2 x) + 1 x = 101 x$$

The problem with your question is that you are seeking a mathematical answer for what is a physics problem. Mathematically, there is no problem with adding quantities with different dimensions, but you obtain a result that you can't map to the physical world. Physically, it doesn't make sense to add numbers with different dimensions.

#### Matt Smith

10 cm = 0.1 m. But 1 kg/1 dm3 =is not 1 kg cm-3. It is 1 kg/1000 cm3 = 0.001 kg cm-3. And 1 kg/0.00001 m3 = 105 kg m-3. I think you left a 0 out. You will constantly go wrong if you're not careful about attention to detail.
Strictly, you can't add things that are in different units. You have to convert them to the same units, e.g. 10 cm + 1 m = 0.1 m + 1 m = 1.1 m.

But you can multiply. The units are the product of the units of the multiplicands. Thus for E = 1/2 mv2, if m is in kg and v in m/s, E is in kg m2 s-2. We have a name for this unit, we call it the joule.

As for charge, we find by experiment that there are two kinds of charge. Like charges repel and unlike charges attract, with a force that is proportional to the product of the magnitudes of the charges, and opposite in sign for the like and unlike cases. And equal amounts of unlike charges can neutralise each other to give zero net charge. So it makes sense mathematically to give one kind of charge positive values and the other negative values - which is which is a matter of historical convention.

We only know one kind of mass, which always attracts other matter, so mass is positive - there's no reason to give any mass a negative sign.
I just can't accept any addition principle.
You have to accept that the normal rules of algebra apply.

Treating units as algebraic quantities, everything works out. You can add quantities that have different units, but you get nowhere
$$10 y + 1 x = 10 y + 1 x$$
if there is no relation between $y$ and $x$, as is the case if $y = \mathrm{m}$ and $x = \mathrm{s}$. But if $y = \mathrm{m}$ and $x = \mathrm{cm}$, then you can use the relation $\mathrm{m} = 10^2 \mathrm{cm}$ to simplify the equation to a single numerical quantity with a unit
$$10 y + 1 x = 10 (10^2 x) + 1 x = 101 x$$

The problem with your question is that you are seeking a mathematical answer for what is a physics problem. Mathematically, there is no problem with adding quantities with different dimensions, but you obtain a result that you can't map to the physical world. Physically, it doesn't make sense to add numbers with different dimensions.
I don't know what had happened to me.I am very sure about what you have said before I learn linear algebra and abstract algebra.But now,it is difficult to take this problem as a pure physics problem.But if we ignore the math problem,how can we use differentiation,intergal?We need a+b=b+a, (-a),(a+b)+c=a+(b+c).Mathematically,calculus is more difficult.We should have metric space,upper bound,completeness...... The point is,if I accept a physical,intuitive explanation,I am not sure the calculus can work .

#### Matt Smith

I just can't accept any addition principle.

I don't know what had happened to me.I am very sure about what you have said before I learn linear algebra and abstract algebra.But now,it is difficult to take this problem as a pure physics problem.But if we ignore the math problem,how can we use differentiation,intergal?We need a+b=b+a, (-a),(a+b)+c=a+(b+c).Mathematically,calculus is more difficult.We should have metric space,upper bound,completeness...... The point is,if I accept a physical,intuitive explanation,I am not sure the calculus can work .
We always define "+"by set theory.But in physics,it vanished,dispeared.In fact I think I should go to the math department when I am searching a master degree.

#### Dale

Mentor
But in fact ,not.You can't say 10 y + 1 x = 0.1 x + 1 before you define the +
It is defined the same as it is for any other algebraic quantity.

There must be a mathematical theory about it
Yes, algebra

There must be a mathematical theory about it,which make the physical law "unit independent"
Physical laws are not unit independent in general. See the difference between Maxwell’s equations in SI vs cgs units.

What I really want is a abstract ,mathematical explanation,
Units are algebraic quantities.

I just can't accept any addition principle.
That is not a problem of definitions or math. That is a personal choice. If you personally choose to be confused then we really cannot prevent it.

Units are treated just like any other algebraic quantity. The definition is clear and complete. The confusion is your choice, not a failure of definition.

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#### stevendaryl

Staff Emeritus
In lots of cases, why something works is sort of by definition. If you ask why we can add and subtract units, it's because we don't measure something in units unless addition and subtraction works for them. So length has units, but not shape. You can say something is round, but it doesn't make sense to talk about roundness units, since we don't know what it would mean for one object to be twice as round as another object, nor do we know what it would mean for an object to be as round as two other objects added together.

#### Matt Smith

It is defined the same as it is for any other algebraic quantity.

Yes, algebra

Physical laws are not unit independent in general. See the difference between Maxwell’s equations in SI vs cgs units.

Units are algebraic quantities.

That is not a problem of definitions or math. That is a personal choice. If you personally choose to be confused then we really cannot prevent it.

Units are treated just like any other algebraic quantity. The definition is clear and complete. The confusion is your choice, not a failure of definition.
In lots of cases, why something works is sort of by definition. If you ask why we can add and subtract units, it's because we don't measure something in units unless addition and subtraction works for them. So length has units, but not shape. You can say something is round, but it doesn't make sense to talk about roundness units, since we don't know what it would mean for one object to be twice as round as another object, nor do we know what it would mean for an object to be as round as two other objects added together.
It is a great idea,honestly speaking.Like what happened in quantum mechanics,we use spin angular momentum and magnetic moment to describe a electron,even it can't really spin and it don't have a real magnetic moment(a generalized one percisely).But ,anyway,I won't study physics anymore,maybe I have more talent in math.

#### Matt Smith

It is defined the same as it is for any other algebraic quantity.

Yes, algebra

Physical laws are not unit independent in general. See the difference between Maxwell’s equations in SI vs cgs units.

Units are algebraic quantities.

That is not a problem of definitions or math. That is a personal choice. If you personally choose to be confused then we really cannot prevent it.

Units are treated just like any other algebraic quantity. The definition is clear and complete. The confusion is your choice, not a failure of definition.
Thank you.If we accept 100cm=1m,we can get a set which include all length.So maybe I can define for any x and y belong to the set,x+y=the total length.I could accept it in this way.I just think your definition is not so straightfoward

"Understanding units"

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