Why Are Derived Units Formed by Multiplying and Dividing Basic Units?

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Discussion Overview

The discussion centers around the formation of derived units in physics, specifically why they are created through multiplication and division of basic units rather than addition or subtraction. Participants explore the implications of unit operations and the nature of mathematical relationships in the context of physical dimensions.

Discussion Character

  • Debate/contested
  • Conceptual clarification
  • Mathematical reasoning

Main Points Raised

  • Some participants argue that different units cannot be added or subtracted, as it leads to nonsensical physical interpretations.
  • Others propose that while addition and subtraction of different units is invalid, multiplication can be viewed as repeated addition, though this perspective is challenged.
  • A participant introduces an analogy involving toys to illustrate the potential for addition, but this is countered by others who assert that such combinations do not yield valid physical units.
  • Some participants emphasize that units represent distinct dimensions and cannot be interchanged or equated without losing meaning.
  • There is a suggestion that derived units arise from the equations that define physical quantities, such as area and speed, which inherently involve multiplication and division of base units.
  • One participant questions whether mathematics is merely a convention or if it reflects deeper truths about the universe, prompting further debate on the nature of mathematical constructs.

Areas of Agreement / Disagreement

Participants express disagreement on the validity of adding dissimilar units, with some firmly asserting that it is not permissible while others explore the concept more freely. The discussion remains unresolved regarding the philosophical implications of mathematics and its relationship to physical reality.

Contextual Notes

Limitations include varying interpretations of mathematical operations and the definitions of units, which are not universally agreed upon in the discussion. The relationship between mathematical constructs and physical reality is also a point of contention.

Deepak K Kapur
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Aa...ha... I can log in now...
So, without wasting much time (one never knows how much time one has...)
My next question...

I saw the list of derived units on wikipedia. All of them had the basic units either multiplied or divided.

Why aren't basic units added or substracted to get derived units?
 
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Well, you can't subtract different units from each other. I mean, what would "1 kg - 1m" mean physically?
In they end, units are dimensions. You can multiply dimensions (e.g. volume of a box is X*Y*Z), but you can't add/subtract them.
 
$175 - 327 hippopotamuses = ??
 
Chestermiller said:
$175 - 327 hippopotamuses = ??
Ha...ha
Well,

$175 - 327 hippopotamuses = $75 ( because 327 hippos cost $100)
I mean to say a relation can be found in addition or substraction also...
 
Deepak K Kapur said:
$175 - 327 hippopotamuses = $75 ( because 327 hippos cost $100)
I mean to say a relation can be found in addition or substraction also...
That is $175 - 327 hippo * (100/327 $/hippo) = $175 - $100 = $75

You only add or subtract same units. Never different units. Ever.
 
DaleSpam said:
That is $175 - 327 hippo * (100/327 $/hippo) = $175 - $100 = $75

You only add or subtract same units. Never different units. Ever.
So, $-hippo or $ + hippo is not allowed
but...
$ x hippo is allowed (if such were the case in reality)
Why so...multiplication is repeated addition only...
 
Deepak K Kapur said:
Why so...
What application would that have?
 
Deepak K Kapur said:
Why so...multiplication is repeated addition only...
Multiplication of integers can be represented as repeated addition. Not multiplication of reals.
 
DaleSpam said:
Multiplication of integers can be represented as repeated addition. Not multiplication of reals.
What is the limitation in doing so with reals?

Now...

10 dolls + 15 balls = 25 toys ( a new unit, I suppose)
and...
10 dolls x 15 balls = 150 toys

So we have distributed 'reals' in the second case..
 
  • #10
Deepak K Kapur said:
10 dolls + 15 balls = 25 toys ( a new unit, I suppose)
and...
10 dolls x 15 balls = 150 toys
These aren't physical units. In terms of physics all those counts are dimensionless numbers.
 
  • #11
Your toys aren't a valid unit because each unit is not the same. A meter is a meter, and two different meters are not distinguishable. The same is not true of your toys unit. So it isn't a valid unit.

Your question has been answered. This forum is for education and not for debate.

If you wish to further push your idea of adding dissimilar units then you must provide a professional scientific reference supporting the practice. Failure to do so is continued personal speculation after being taught the correct physics.
 
  • #12
DaleSpam said:
Your toys aren't a valid unit because each unit is not the same. A meter is a meter, and two different meters are not distinguishable. The same is not true of your toys unit. So it isn't a valid unit.

Your question has been answered. This forum is for education and not for debate.

If you wish to further push your idea of adding dissimilar units then you must provide a professional scientific reference supporting the practice. Failure to do so is continued personal speculation after being taught the correct physics.

Ok. I got your point.

I DARE NOT post more...
 
  • #13
To extend that example:
Deepak K Kapur said:
10 dolls + 15 balls = 25 toys
15 dolls + 10 balls = 25 toys as well.
Therefore,
10 dolls + 15 balls = 15 dolls + 10 balls
Which reduces to
5 balls = 5 dolls
Divide by 5:
1 ball(s) = 1 doll(s)
you can also derive that every toy is a ball:
1 toy(s) = 1 ball(s)
which is certainly not what the calculation was supposed to mean. You can assume it to be true in terms of mathematics, but then there is no point in giving the same unit (toys) three different names.
 
  • #14
To coin a phrase, "you can't add apples and oranges."

Chet
 
  • #15
DaleSpam said:
Your question has been answered. This forum is for education and not for debate.

If you wish to further push your idea of adding dissimilar units then you must provide a professional scientific reference supporting the practice. Failure to do so is continued personal speculation after being taught the correct physics.
Quick add:

Math isn't really debateable and not necessarily even explainable in this way. It is a human invention, agreed upon by convention, and has the form it does, simply because it works. It doesn't need any other reason.

It is similar to other languages in this regard. There may be certain things about the English language you don't like, but you aren't entitled to change them unless you rise to a position of being a prominent expert who can drive the discussion to change the consensus (a discussion we won't entertain here).

So all this attempt to argue with the reality of how math works is pointless.
 
  • #16
russ_watters said:
Quick add:

Math isn't really debateable and not necessarily even explainable in this way. It is a human invention, agreed upon by convention, and has the form it does, simply because it works. It doesn't need any other reason.

It is similar to other languages in this regard. There may be certain things about the English language you don't like, but you aren't entitled to change them unless you rise to a position of being a prominent expert who can drive the discussion to change the consensus (a discussion we won't entertain here).

So all this attempt to argue with the reality of how math works is pointless.
Having gained strength, I say...

Does it mean, then, that math is just a brute fact...

In other words, would it suffice to say that...
THIS IS THE WAY UNIVERSE WORKS and THAT'S ALL!
 
  • #17
I disagree with that it's just an idiosyncrasy of math.
As I said before, units should be viewed as "markers" for a dimension. That dimension can be meter, or it can be "toy". So, while it may look tempting to consider a multiplication of units as an adding of them, that part really only applies to the scalars (e.g. the 5 in 5m) that we attach to those dimensions. The dimensions stand unchanged. That is, while a "5*2" gets mapped back to a single scalar, the "kg*m" stays a "kg*m".
 
  • #18
I would say that the derived units come out of the equations that compute those quantities:

area = length x width if length and width are in meters and since area = length x width then the units of area are meters x meters or m^2.

speed = distance / time so if distance is in meters and time in seconds then units for speed are meters/seconds.
 
  • #19
Deepak K Kapur said:
In other words, would it suffice to say that...
THIS IS THE WAY UNIVERSE WORKS and THAT'S ALL!
No, this isn't about how the universe works, it is just how math works. There is no implication that any particular equation reflects how the universe works and the structure of math doesn't necessarily say anything at all about the universe.
 
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