Proportionality Constant (Units)?

[Capiert]

I was always stunned
 by the complexity.

Proportionality constants exist (almost) everywhere
 (in Physics),
 & their complexity has always stunned me;
 but (peculiarly)
 I have NOT seen (a textbook) anywhere
 on how to make (=derive) them
 (as though (derivation (of their units) is) avoided);
 so I thought
 I would give it a whirl (=try)
 (on my own,
 as DIM
=Do it myself).

(It’s only common sense.
Dealt with intuitively, due to the(=my) confusion.)

Recap:

1 of the most basic (& yet easy (but tricky)) concepts
 (rarely found
 in text books)
 is deriving (=how to derive (them))
 proportionality_constant’s “units”.

---
Disclaimer1:

(I suspect)
E.g.
Maybe
 (perhaps (because))
 from the (misleeding?=confusing, conflicting)
 contraproductive (brainwashing) statement(s)
 (e.g. from Sears; & Alonso):
 that units are NOT
 (suppose to be)
 multiplied
 by the symbol’s number (value(s)),
 (but) when they (perhaps) really are (multiplied),
 instead;
 but few (people=physicists (have)) admit(ted) it.

(Even)
 although NOT perfectly
 ((multiplied) sometimes)
 due
 to singular versus plural (units, syntax).

Maybe some people
 get the basic idea right=correct.

Math
 is suppose
 to be an exact science (sometimes).

Units are (typically) NOT included
 into formulas
 until the number values are (present(ed)).

((Rule 1:
 (Simply)
 include the units;
Rule2:
 & then copy them (units)
 to the constant.

Rule3:
But how?))

Note:
I’( ha)ve used square_brackets for units, here.
---

E.g.
Hooke’s law
 (for)
 a spring.

The proportionality
 k=F/(-x)
 (looks like a quotient=ratio,
 to me, &)
 is the Force F
 as main actor (influencer, cause);
 wrt the negative displacement distance -x
 as secondary actor=slave (result, effect)
 e.g. which is in the opposite (=negative) direction.

(Simply)
 include
 the(ir) (multiply_)units;

  k*[N/m]=F*[N]/(-x*[m])

 then “copy” (them, units)
 to the constant,
 for equality.

Keeping in mind
 that a [Newton] is
 [N]=[kg*m/(s^2)]
 &
 Force is
 F*[?]=m*[kg]*a*[m/(s^2)]
 the unknown units for Force are (simply)
 [?]=[kg]*[m/(s^2)]=[N].

Applying the same method
 for Hooke's proportionality constant
 k*[N/m]=F*[kg*m/(s^2)]/(-x*[m]), cancel [m/m]=[1/1]=1
 k*[kg/(s^2)]=F*[kg/(s^2)]/(-x), swap sides.

The k constant’s (unique) units are
 [kg/(s^2)] kilogram per second(s)_squared,
 or
 [N/m] Newton(s) per meter.

Again, take the ratio (=quotient) including (multiplied) units
 k [N/m]=F*[kg*m/(s^2)]/(-x*[m]), swap sides
 F*[kg*m/(s^2)]/(-x*[m])=k*[N/m], *(-x*[m])
 F*[kg*m/(s^2)]=(-x*[m])*k*[N/m], rearrange
 F*[kg*m/(s^2)]=-k*[N/m]*(x*[m]), /[kg*m/(s^2)=N] remove(=divide_by) units
 F=-k*x, gives us the standard recognized formula (Hooke’s Spring Force).

E.g.
A (weight_)scale
 could displace its spring
 -1 [cm]=-0.01 [m] (down),
 per [kg].

That force (weight)
 Wt=m*g=1*[kg]*9.8*[m/(s^2)]=9.8*[kg*m/(s^2)]
 would be 9.8 [N]
 for 1 [kg] (mass).

Using Hooke’s spring law(=formula)
 F=-k*x, swap sides
 -k*x =F, /(-x)
 the spring’s proportionality constant (e.g. ratio) is
 k=F/(-x), let the Force F=Wt weight Wt=m*g
 k=m*g/(-x), let the mass m=1 [kg] & the gravitational_acceleration g=-9.8 [m/(s^2)]
 k=-9.8*[kg*m/(s^2)]/(-0.01 [m]), 1/0.01=100
 k=980*[kg*m/(s^2)]/[m]), cancel [m/m]=[1/1]=1
 k=980*[kg/(s^2).

That spring (law) formula
 F=-k*x, swap sides
 -k*x=F, /(-k)
 can be manipulated
 to find the displacement
 -x=F/k
 for (e.g. calibrating
 to) the mass m=Wt/g
 -x=m*g/k, *(-1)
 x=-m*g/k, let g=-9.8 [m/(s^2)] & k=-980*[kg/(s^2)
 x=-m*(-9.8 [m/(s^2)])/(-980*[kg/(s^2)), 9.8/980=0.01 & [(s^2)]/[(s^2)]=1/1=1
 x=-m*0.01 [m/kg].

E.g. let mass m=1 [kg]
 x=-1 [kg]*0.01 [m/kg], [kg/kg]=1/1=1
 x=0.01 [m]=1 [cm].

All pretty obvious.

Disclaimer 2:

Such a constant k
 (as ratio),
 also looks like
 it could serve (well)
 as a variable, instead
 (e.g. if needed,
 when a (so_called) constant
 is NOT constant,
 at all).

If that'( i)s the way
 things are done(?);
 then it would be a help
(to me).

Edited October 26, 2023 by Capiert
Typo.

[swansont]

It’s usually covered the first couple of days of class.

You can always multiply by 1, i.e. equal quantities, or multiply both sides of the equation by the same factor. Identical units in the numerator and denominator cancel, just like in simplifying fractions.

These concepts are rooted in simple algebraic manipulation. It’s assumed you can do this if you are doing physics.

Units and the quantity are indeed separate. You need not know the numerical amount in order to manipulate the units.

Proportionality constants ensure both sides of an equality are indeed equal.

 

 

[studiot]

1 hour ago, Capiert said:

Proportionality constants exist (almost) everywhere
 (in Physics),
 & their complexity has always stunned me;

Indeed but Physics uses more than just simple (linear) proportionality.

A very simple example would be the statement

 

The Kinetic energy of a moving body is proportional to the square of its velocity.

 

There is also what is known as inverse proportionality.

Edited October 26, 2023 by studiot

[Capiert]

My (major) problem (obstacle)
 is intuitively (a) psychological (block(age))
 when using the 2 (thus) inconsistent ((conflicting(?) syntax) methods
 (at the same time?)
 while dealing
 with the units (derivation).

Please let me explain
 what I observe (mathematically).

I estimate
 I automatically
 get the algebra (answer) wrong
 with (a fake) ~40%.

That is
 I (repeatedly)
 make the same mistake;
 but recognize it
 (just in time),
 to correct it
 (so that
 (in the end)
it does NOT seem
 like I am making a mistake (at all);
 although I really do ERROR!).

I find that very peculiar,
 why I can NOT directly
 proceed to the correct answer;
 but instead
 I must always make the ERROR 1st;
 & then go thru the routine
 of correcting it 2nd.

That is a waste of time;
 & a risky proceedure
 (e.g. if severely distracted).

But, what is happening
 in my mind
 does NOT make SENSE
 unless there is a reason for all that (NONSENSE, detour).

I must conclude
 that the k=F/(-x) construct
 is EITHER a ruff forced_fit
 which does NOT (naturally) belong (together);
 or else there must be another explanation
 (which I (might) have missed)?

What I see is,
 units are typically
 NOT included
 for the symbols
 F & -x
 (but) until the number( value)s
 are stated (=declared).

However, k does NOT have any (units,
 at all before that);
 thus those units are NOT known;
 but still (NO_units means they) must be found
 ..from somewhere!

Strictly speaking,
 for me,
 I have 2 alternatives:
 e.g. either to borrow them (units)
 from F & -x
 by cross_multiplying
 them to k;
 but then they are (wrongly) inverted;
 & so I correct that inversion;
 & notice that
 I have only copied those units
 from F/(-x)
 in(to) the same positions.

I DON'T (think I) have to say
 how little mathematical SENSE
 that makes
 (to me),
 with a 1_sided abracadabra copy
 (of units),
 from F/(-x)
 to the (empty=NO_units) k side.

It is simply a (functional) fix=repair
 just to make things work
 without explaining
 why (things fail,
 or why they should work,
 but do NOT,
 without that trick (of the trade).)

Disclaimer:

EVERYBODY KNOWS
 how to make
 the units (derivation,
 work correctly;
 by repairing=correcting it);
 BUT NOBODY tells (me) why
 (that trick is NEEDED (at all);
 other than that it is NEEDED).

My mind deals with that problem
 like a paradox (conflict);
 & WANTS to shut (it) off
 like a clap TRAP.

It is like asking
 (for) a yes or NO answer
 but the priority (dominates, &)
 immediately locks up
 (=latches)
 into the NO priority,
 as stolen!

It's VERY sticky
 gumming up my thinking process(es).

ONLY a "corrected"
 wrong_answer
 will function correctly.

 

Edited October 31, 2023 by Capiert
typo

[Bufofrog]

48 minutes ago, Capiert said:

My (major) problem (obstacle)
 is intuitively (a) psychological (block(age))
 when using the 2 (thus) inconsistent ((conflicting(?) syntax) methods
 (at the same time?)
 while dealing
 with the units (derivation).

Do you have some sort of parentheses fetish? 

[swansont]

F=-kx

This is an equality

What is on the left side is equal to what is on the right side. This applies (separately) to the magnitude, the direction, and the units

If force is in Newtons (N) and x is in meters (m), the units for k will be N/m, because N/m * m gives you N.
m, being in both the numerator and denominator, cancels, similar to what would happen if you were simplifying a fraction (e.g. 3/3 =1)

This is not considered to be mysterious.

[Capiert]

19 hours ago, Bufofrog said:

Do you have some sort of parentheses fetish? 

Absolute(ly).

18 hours ago, swansont said:

F=-kx

This is an equality

What is on the left side is equal to what is on the right side. This applies (separately) to the magnitude, the direction, and the units

If force is in Newtons (N) and x is in meters (m), the units for k will be N/m, because N/m * m gives you N.
m, being in both the numerator and denominator, cancels, similar to what would happen if you were simplifying a fraction (e.g. 3/3 =1)

This is not considered to be mysterious.

The cancelation
 (of units)
 is NOT my problem.

My (automatic) inversion
 of the units
 is my (internal) problem.

Edited November 1, 2023 by Capiert
typo

[swansont]

17 minutes ago, Capiert said:

Absolute(ly).

The cancelation
 (of units)
 is NOT my problem.

My (automatic) inversion
 of the units
 is my (internal) problem.

Inversion?

Let’s say you have the equation 2x = 5

You can divide both sides by 2, and get x = 5/2

If you have F=-kx, you can divide both sides by x and get k = -F/x, or you can divide by k and get x = -F/k

These are math rules that you seem to be having trouble with. 

 

 

[Capiert]

On 11/1/2023 at 5:01 PM, swansont said:

Inversion?

Let’s say you have the equation 2x = 5

You can divide both sides by 2, and get x = 5/2

If you have F=-kx, you can divide both sides by x and get k = -F/x, or you can divide by k and get x = -F/k

These are math rules that you seem to be having trouble with. 

Thanks for your attempt
 to try & understand
 "my" problem.
(It helps me try to look at it closer.)

But I think you missed it.

My problem
 is the clash
 between inconsistent syntax.

I.e. 2 different syntaxes=formats.
(They (DON'T) mix like oil & water.)
(It's a MASTER & SLAVE relation.)

(BEFORE derivation:)
Both F & x have units;
 BUT k does NOT.

Thus k's units
 "must" be "derived"
 from them.

That is a 2 step process
 instead of ONLY 1 step.
(I recover=repair it.)

k's units do NOT exist then
 (before that)
 until they are derived.

(& it is a tricky 1_way street,
 until done!)

Attempting to (algebraically) manipulate
 F's & x's "units"
 before that (derivation, for k's_units (obviously?))
 will fail.

That's the catch,
 the whole problem.

Maybe my thread's title
 should have been named
 "(Proportionality_Constant's) Units ((perplexing), derivation)?"
 instead.

Or something like that?


 

 

 

 

Edited November 2, 2023 by Capiert
typo

[studiot]

Proportion, proportions and proportionality do not necessarily refer to equations in Maths or Science.

 

Consider the following question.

 

If there are 10 grammes of sugar in a 225 millilitre cup of drink

How many grammes of sugar are there in a 175 ml cup of the same drink ?

 

Please indicate by what method you would solve this question to try to find your difficulty.

Edited November 2, 2023 by studiot

[swansont]

22 minutes ago, Capiert said:

BEFORE derivation:)
Both F & x have units;
 BUT k does NOT.

Thus k's units
 "must" be "derived"
 from them.

k has units. you just don’t know what they are. That’s not the same thing.

Like when you see 2x = 5, you don’t know what x is until you solve the equation. It doesn’t magically become 5/2 only when you solve it. There’s no time dependence here. 

 

 

[Capiert]

1 hour ago, studiot said:

Proportion, proportions and proportionality do not necessarily refer to equations in Maths or Science.

Consider the following question.

If there are 10 grammes of sugar in a 225 millilitre cup of drink

Then I have a Density
D=m/vol=10 [g]/0.225 [L]=1 [g]/0.0225 [L] (wrt a simple mass of 1 [g]); or else)
D=44.444.. [g/L].

That is where the Density's units (suddenly) pop up.
Only by using a specific (numbers) example
 can we (suddenly) see the units.
It (=That pop_up, inclusion)
 goes (=happens)
 effortlessly.
(It's (easy (&)) automatic.)

But I typically do NOT use numbers
 algebraically,
 because irrational_numbers are very messy.
I usually deal with (general) variables
 (for (very loosely speaking:) "any" number),
 & (=but)
 typically only use (exact) values (rarely)
 when I need
 some form of extra orientation.

The disturbing part
 for me
 here in Physics
 is the inconsistent syntax (& or method?);
 which causes me to runamuck.

I would prefer
 all symbols
 had (also) their units
 (multiplied)
 with them
 to be (obvious &) consistent;
 but you (Physicists) DON'T deliver that (consistency)
 because it looks messy.
((&) It's (also) NO fun!)

& then peculiar problems (occasionally) start to happen
 that I can NOT (always) explain
 or else I have difficulty explaining
 (because they are so rare, & foreign).

1 hour ago, studiot said:

How many grammes of sugar are there in a 175 ml cup of the same drink ?

If my Density is
 D=m/vol, swap sides
 m/vol=D, *vol
 m=D*vol.

My new mass (would be)
 m=D*vol=44.444.. [g/L]*0.175 [L]=0.777.. [g]

1 hour ago, studiot said:

Please indicate by what method you would solve this question to try to find your difficulty.

How can you find (=derive) the (proportionality_constant's) units
 without number( value)s?

Answer: NOT known.
My problem is here at D=m/vol;
 there is NO Problem after the 2nd "="
 at
 "10 [g]/0.225 [L]".

There is something missing for me
 before the 2nd "=".

Maybe I also need
 a (fake, (temporary) dummy, placeholder)
 unit's_variable symbol too?
 (for (premature) consistency)
 e.g.
 Density, is
 D*[units]=m*[g]/(vol*[L].
?

But it clashes
 in my head brutally.

Thus Failure.

 

Edited November 2, 2023 by Capiert

[studiot]

Please don't take this as a personal cirticism but you are definitely overthinking this by a very long way.

Small wonder you are having trouble.

 

As I said, strictly proportionality is not about equations, although of course proportionality can lead to an equation.

 

The way to calculat the answer to my question by proportion is as follows.

 

225 millilitres of drink contains 10 grammes of sugar

So

1 millilitre of drink will contain 1/225 as much sugar.  That is 10/225 grammes.

So

175 millilitres of drink will contain 175 times as much sugar or (10/225) * 175 grammes or about 7.8 grammes.

 

Check: 175 ml is about three quarters of 225ml so I would expect it to contain about three quarters as much sugar or about 7.5 grammes.

 

Read this a few times I have laid it out in great detail.

 

[swansont]

2 hours ago, Capiert said:

Then I have a Density
D=m/vol=10 [g]/0.225 [L]=1 [g]/0.0225 [L] (wrt a simple mass of 1 [g]); or else)
D=44.444.. [g/L].

That is where the Density's units (suddenly) pop up.
Only by using a specific (numbers) example
 can we (suddenly) see the units.

Who is “we”?

Density is m/V

if mass has units of kg, and volume is expressed in m^3, the units are kg/m^3. Or you could use grams/cm^3

No numbers. They aren’t necessary.

2 hours ago, Capiert said:

I would prefer
 all symbols
 had (also) their units
 (multiplied)
 with them
 to be (obvious &) consistent;
 but you (Physicists) DON'T deliver that (consistency)
 because it looks messy.
((&) It's (also) NO fun!)

This is a “you” problem. Stop blaming physicists.

[Capiert]

On 11/2/2023 at 8:39 PM, studiot said:

Please don't take this as a personal criticism but you are definitely overthinking this by a very long way.

Thank you for the help.

On 11/2/2023 at 8:39 PM, studiot said:

Small wonder you are having trouble.

Good that you noticed.
But my wrong answer was a typo.

On 11/2/2023 at 8:39 PM, studiot said:

As I said, strictly proportionality is not about equations, although of course proportionality can lead to an equation.

Then (it=proportionality is about), estimating?

On 11/2/2023 at 8:39 PM, studiot said:

The way to calculate the answer to my question by proportion is as follows.

 

225 millilitres of drink contains 10 grammes of sugar

So

1 millilitre of drink will contain 1/225 as much sugar.

That math has always turned my head (crazy).

On 11/2/2023 at 8:39 PM, studiot said:

That is 10/225 grammes.

Agreeable.

On 11/2/2023 at 8:39 PM, studiot said:

So

175 millilitres of drink will contain 175 times as much sugar or (10/225) * 175 grammes or about 7.8 grammes.

Also fitting.

On 11/2/2023 at 8:39 PM, studiot said:

Check: 175 ml is about three quarters of 225ml so I would expect it to contain about three quarters as much sugar or about 7.5 grammes.

It all seems to make sense.

On 11/2/2023 at 8:39 PM, studiot said:

Read this a few times I have laid it out in great detail.

Studiot, I am sorry (disappointed) I typed the wrong answer for you.
 I had tried,
 but I made a (stupid) decimal ERROR.
I had it right in Excel
 but I copied it wrong.
(Too many problems with my eyes.)

On 11/2/2023 at 7:35 PM, Capiert said:

Then I have a Density
D=m/vol=10 [g]/0.225 [L]=1 [g]/0.0225 [L] (wrt a simple mass of 1 [g]); or else)
D=44.444.. [g/L].

If my Density is
 D=m/vol, swap sides
 m/vol=D, *vol
 m=D*vol.

My new mass (would be)

 m=D*vol=44.444.. [g/L]*0.175 [L]=7.777.. [g]

[NOT 0.777].

Considering, that mistake had NOT happened,
 why (then) should I do otherwise
 (& estimate)
 when it is so simple (& exact)?

I mean,
 I assume,
 I still would have (probably) made
 the same (or similar) mistake.
(Blurry vision, +.. . Reading glasses
 are NOT going to help that.
Nor had the optometrist's cortisone brought permanent success.)

On 11/2/2023 at 10:07 PM, swansont said:

Who is “we”?

We is we naive (NON_physicists)
 who have NOT a clue
 (what is happening).
Abracadabra.

Sometimes it is
 & then sometimes it is NOT.

On 11/2/2023 at 10:07 PM, swansont said:

Density is m/V

if

There you go.

"Now" you begin constructing.

On 11/2/2023 at 10:07 PM, swansont said:

mass has units of kg, and volume is expressed in m^3, the units are kg/m^3.

Although they are NOT visible
 before (hand).

On 11/2/2023 at 10:07 PM, swansont said:

Or you could use grams/cm^3

That is all fine Swantsont.

But a fundamental step is missing
 (which you obviously miss(ed)).

On 11/2/2023 at 10:07 PM, swansont said:

No numbers. They aren’t necessary.

This is a “you” problem.

Quite right.
I told you in advance.

On 11/2/2023 at 10:07 PM, swansont said:

Stop blaming physicists.

As my problem,
 I consider then
 that I should solve it,
 with constructs.

Edited November 22, 2023 by Capiert

[swansont]

2 hours ago, Capiert said:

That is all fine Swantsont

Who is Swantsont?

 

[Capiert]

---

1st
Please let me
 try a different perspective.

I have seen
 Physic’s tables
 where the units
 are stated as “per”.

Meaning the concept
 e.g. mass m
 would be
 ONLY a NUMBER;
 (thus)
 making it

 (e.g. the mass m as “ONLY number”;
 instead of a "number*unit" hybrid mix(ture))

 convenient
 for multiplying
 & dividing
 ONLY as NUMBERs
 in e.g. an Excel table (sheet).

NOW,
 to reverse (=swap)
 syntax
 for my convenience (only),
 please let (me make)
 the(=my, large symbol)
 Mass (concept)
 M=m*[unit(s)]
 (as construct)
 be made
 of (ONLY) a pure number
 (small character) m,
 & multiplied
 by its units
 in (square brackets)
 [kg]=[kilogram].

Then the “number” (of) mass
 m=M/[units]
 is the whole Mass_concept M
 but divided
 by its units [kg].

That is VERY IMPORTANT,
 because
 it has separated
 the (composite=hybrid=composite)
 Mass_concept
 into its basic (2) parts
 (of (ONLY the))
 number m
 versus units [kg].

In that form(at)
 (of (physic’s_)concept per (its) unit),
 we can strip
 ANY (physics)
 concept
 down
 into ONLY its NUMBER value;
 which is independent
 of ANY (awkward)
 (NON_Unifying=NON_mathematical;
 alphabetic, (instead of numerical),
 word (e.g. unit);
 thus dealing
 ONLY with MATH!
 (e.g. with NO other hassels!).

Motivation:

Why do I say all that blah blah blah?
Because it (=the number_variable, without units)
 is genial
 to be so unique.
I.e. It has (a lot
 of) math advantages.
E.g. We are NOT restricted
 to dealing
 with ONLY complicated (hybrid, mixture) relations.
Instead we have LESS
 to do,
 which CAN
 increase efficiency
 thus make things go faster
 & be (or at least seem)
 simpler
 & LESS complicated.

So, where are we NOW?

NOWHERE my friend.

But we can use
 what already exists.

& the results
 are astounding!

For instead
 of the units being multiplied
 by the number values;
 they are instead
 “divided” by the number( value)s!

& Thus as Sears
 & Alonso said=stated
 “NOT” multiplied!

But NOBODY
 could tell me why;
 because
 (if=when viewed only from that ("per unit") perspective)
 everybody (else)
 has been doing
 the math WRONG! (=Same method.)
(Otherwise NOT?)

Th(os)e (Physic’s) answers
 are also suppose
 to be a NUMBER
 but “per” UNIT!;
 instead of just beside the NUMBER (value).

Strange that Sears & Alonso
 could NOT have said more
 about those (per) units
 to speed up
 the (discovery) recogition
 process.

Disclaimer:

I personally
 did NOT expect
 that I would ever
 have gotten
 a solution (=reasonable, logical (explanation=) answer)
 to that problem;

 & I had (then, thus) thought
 I must brainwash
 myself;
 & ONLY memorize
 the method (technique),
 (always) with the fear (=concern)
 of forgetting
 how to do it=((the math) things)
 correctly,
 (if I had forgot)
 (eventually)
 mixing things up (again).

That puzzle (=paradox)
 is NOW solved.

(So let's give it a whirl.)

 

Please let
 Hooke’s (compressed_spring force)
 law
 F=k*(-x)
 be rewritten
 in Capital letters

 F=K*(-X).

& Retry:
 (using (Capital letter) constructs
 for: ((small letter) “number”_)variables;
 & “units”).

Please let
 the Hooke’s Law’s
 proportionality_constant

 K=F/(-X)

 be for the (math) constructs (=formulas,
 (that) I created in Capital letters).

E.g.
Please let
 the proportionality_constant
 K=k*[units]
 the (spring’s) force
 F=f*[N]
 & the displacement distance
 -X=-x*[m]

 which are also ONLY the number(_value_variable)s
 k=K/[unit]
 f=F/[N]
 -x=-X/[m]
 because the (Capital_letter) constructs
 (made
 of “number” multiplied by “unit”)
 are (then) divided by their unit
 (thus leaving ONLY their number(_value)). 

So again,
 the Hooke’s law’s
 proportionality_constant, (when) including units, is 
 k*[unit]=f*[N]/(-x*[m]).

We can ignore the (small_letter) number_variables
 thus leaving
 the (derived) unit
 k*[unit]=f*[N]/(-x*[m]).

This_time (=That_instance, or example, had)
 NO ERROR occurred
 when deriving
 the (proportionality_)constant’s [units];
 because it is algebraically sound=fit
 with (NON_ambiguous) consistent_syntax.

I.e.
NO ambiguous,
 double_meaning symbols.

The (3) math constructs
 eliminated
 the inconsistent syntax.
 

Edited November 23, 2023 by Capiert

[swansont]

3 minutes ago, Capiert said:

I have seen
 Physic’s tables
 where the units
 are stated as “per”.

Give an example 

[Capiert]

Proposal:

Now to summarize;
 & also be able
 to use either
 small (or large case, Capital) letters
 for proportionality_constants
 (&/or their variables, e.g. &/or their constructs),
 we might need
 an alphabetical subscripted syntax (=symbology, symbols).

E.g.
Please let
 the Hooke’s_spring’s
 proportionality_constant
 k=n_k*[u_k]
 where n is its number value
 & is its [unit],
 both multiplied together.

Their subscript _k
 denotes
 that they belong
 to the proportionality(_constant) (construct).

Hooke’s_Law‘s
 proportionality_constant(’s
 symbology: as variables
 including both numbers & units)
 could then look like
 k=n_k*[u_k]=n_F*[N]/(-n_x*[m]).

& where units:

[N]=[Newton]
[m]=[meter].

Disclaimer:

What a (complicated) mess;
 (that’s why I (would) prefer Capital_letter constructs (=formulas)
 & small letter number_value variables
 (to get around that (ambiguous_syntax) problem);
 but it should work right.

Again,
 only the numbers(‘ variables) are
 k=K/[unit]
 f=F/[N]
 -x=-X/[m].

Thus,
 only the number_variable
 (for the spring’s proportionality_constant), is

 k=(F/[N])/(1/(-X/[m]), rearrange

 k=(F/(-X))*[m/N],    <---That (whole) is (ONLY) a number(‘s value)!;
 although [units] are (also) present.
 k=(F/(-X))*[m/N], *[N/m]
 k*[N/m]=(F/(-X)),   <---Those are hybrids=mixtures: of number(_variable)s; & [units].

& (that proportionality_constant now has the correct [units]=[N/m]),
 k*[N/m]=K, swap sides
 as mixture_hybrid
 K=k*[N/m].

As Swansont said
 NO numbers are needed
 to obtain the units.

 

57 minutes ago, swansont said:

57 minutes ago, swansont said:

  16 minutes ago, Capiert said:

I have seen
 Physic’s tables
 where the units
 are stated as “per”.

Give an example 

E.g.

Temperature
T
/K

20
30
40
 

 

I saw it in only 1 book
 by 1 (specific) author
 (he did all his tables that way)
 & (I) thought it was very peculiar
 (& (I) tried to figure it out
 but could NOT
 back then)
 because NOBODY else did that;
 but from the style
 (it looked like)
 he was trying to do something very fundamental.

He wrote his units
 with a slash before them (units)
 under the symbol.

Thus,

Name
Symbol
/Unit

number 1
number 2
number 3.

So I interpreted that (2_Liner)

Symbol
/Unit

to (elegantly) mean
=Symbol/Unit
 (if it were written
 on ONLY 1 line).

 

Edited November 23, 2023 by Capiert

[studiot]

I think it is very important that you separate numbers , units, symbols and the objects which they refer to.

That is you think separately about them, each in their own right.

 

Let us take an example of this:

Making bread.

Bread has ingredients.

The ingredients are flour, yeast and water.

There are 3 ingedients.

Here the objects referred to are ingredients.

The number 3 is called a quantifier or a coeffiecient.

There are no units involved or if you prefer the unit i 'number or count of'

But there are several types of number and 3, each with their own special features.

So we could have said:

There are 3 ingredients

1st ingredient,, 2nd ingredient, 3rd ingredient.

Or we could have said even more

First ingredient flour
Second ingredient yeast
Third ingredient water.

Which tells us even more.

First second and third are numbers, just a different kind of number.
~They are cvalled ordinal numbers, as they show order.

We could also write a 1 on the jug of water,  a 2 on the tub of yeast and a 3 on the bag of flour.

Then the instructions might read

Mix the contents of 2 with the contents of 3, than add the contents of 1.

We are now using the numbers 1, 2 and 3 as symbols (you mentioned symbols)

 

We can make the description even more use full if we add units so

Mix 2 of yeast with 400 of flour and then add 250 of water is not very helpful.

But

Mix 2 teaspoons of yeast with 400 grammes of flour and then add 250 mililitres of water is very helpful.

Much more helpful than

Mix 2 teaspoons of  2 with 400 grammes of 3 and then add 250 mililitres of 1.

Although both are strictly correct and Mathematicians and computer engineers like the last as it is shorter.

 

I have another example to explore but let us see what you make of this one first.

[swansont]

14 hours ago, Capiert said:

I saw it in only 1 book
 by 1 (specific) author

And from that you extrapolate this into being a widespread problem. Which you interpret as “per” (despite it not making sense) but was not stated as you claimed.

Your real issues are with this one author and your propensity to declare things to be a problem.

[Capiert]

21 hours ago, studiot said:

I think it is very important that you separate numbers , units, symbols and the objects which they refer to.

Ok, but didn't I do that with my constructs?

E.g.
number (It(=the following variable k) is only a number)
k=K/[unit],
symbol (represents a composite=mixture.)
K=k*[unit],
unit (is stripped of all numbers.)
[unit]=K/k.

Those equations made a connection
 to each other.

You however
 want me to separate them
 from each other.
But I think
 that is already done,
 because of the equals sign =
 & which side
 of the equation
 they are on.

The (3) variables
 k, K & [unit]
 are all each alone
 (meaning "separate(d)").

That I avoid (really using) numbers
 by using variables (instead),
 has (at least) separated numbers (e.g. 1,2,3..)
 as NON_existent out from my syntax.

21 hours ago, studiot said:

That is you think separately about them, each in their own right.

Ok.
Only a  number e.g. 1.
Only a unit, e.g. [kilogram] (it has NO number).
Only a symbol, e.g. M.

(Warning: But I now know the Mass's symbol is M=1*[kilogram]
 (because I have seen the necessity)
 & so that (awareness of mine) might interfere with your (intended) lesson (for me).
E.g. I can NOT overcome my intelligence;
 NOR the lack of it.)

21 hours ago, studiot said:

Let us take an example of this:

Making bread.

Bread has ingredients.

The ingredients are flour, yeast and water.

There are 3 ingredients.

Here the objects referred to are ingredients.

The number 3 is called a quantifier or a coefficient.

Is there any difference
 between those 2 words?

21 hours ago, studiot said:

There are no units involved or if you prefer the unit i 'number or count of'

Any reason why i?
No units is fine;
 but you lost me with i.

21 hours ago, studiot said:

But there are several types of number

Types?
Strictly math "numbers" 1,2,3?
 each has its own name: one, two, three, ..
E.g. Ordinal (name & sequence), cardinal (value), irrational, ..

Or
do you mean (physic's) constructs,
 made of a number with unit? No?

21 hours ago, studiot said:

and 3, each with their own special features.

Good.
Then they are constructs?
(e.g. small formulas.?).

21 hours ago, studiot said:

So we could have said:

There are 3 ingredients

1st ingredient, 2nd ingredient, 3rd ingredient.

Or we could have said even more

First ingredient flour
Second ingredient yeast
Third ingredient water.

Which tells us even more.

First second and third are numbers, just a different kind of number.
~They are called ordinal numbers, as they show order.

=Sequence (order).

21 hours ago, studiot said:

We could also write a 1 on the jug of water, a 2 on the tub of yeast and a 3 on the bag of flour.

Then the instructions might read

Mix the contents of 2 with the contents of 3, then add the contents of 1.

E.g. Similar to a computing (pointer, pointing) language.

21 hours ago, studiot said:

We are now using the numbers 1, 2 and 3 as symbols (you mentioned symbols)

E.g. Names;
 NOT (number_)values!

Yes.
But there, ..
 they have no number value.
Instead they designate (dictate)
 the sequence (order).
Priority (order).

21 hours ago, studiot said:

We can make the description even more useful if we add units so

Mix 2 of yeast with 400 of flour and then add 250 of water is not very helpful.

Quantities without knowing what (kind)
 from "several" possibilities.

21 hours ago, studiot said:

But

Mix 2 teaspoons of yeast with 400 grammes of flour and then add 250 milliliters of water is very helpful.

Yes.

21 hours ago, studiot said:

Much more helpful than

Mix 2 teaspoons of 2 with 400 grammes of 3 and then add 250 milliliters of 1.

Yes, that is a bit (more) awkward for us (normal=typical) humans.

21 hours ago, studiot said:

Although both are strictly correct and Mathematicians and computer engineers like the last as it is shorter.

Why? Only 1 symbol, or minimal (& sequence advantages)?

21 hours ago, studiot said:

I have another example to explore but let us see what you make of this one first.

OK.
But, I doubt that I have fullfilled what you asked concerning separation.
& I took the liberty to eliminate the typos,
 if you don't mind
 (because they disturb=distract my thinking=concentration).

[Capiert]

21 hours ago, swansont said:

On 11/23/2023 at 1:43 AM, Capiert said:

I saw it in only 1 book
 by 1 (specific) author

And from that you extrapolate this into being a widespread problem.

If you will? Yes.
It seems possible to me.

(But) I DON'T need your wrath
 (if it goes against too much).

21 hours ago, swansont said:

Which you interpret as “per” (despite it not making sense)

Please explain.
I thought "per" (=divided by) is rather clear.

21 hours ago, swansont said:

but was not stated as you claimed.

Please explain.

21 hours ago, swansont said:

Your real issues are with this one author

I am rather greatful
 to that author.

Years ago,
 I hadN'T a clue what he meant.

With this thread
 & your team's help
 it (finally) dawned on me
 what he (might have) meant.

His style technique, inspired me to recognize that (so).

(That's a lot of years
 in between.)

I can thank him for the inspiration
 (if I could).

21 hours ago, swansont said:

and your propensity to declare things to be a problem.

It (Ambiguous (unit) syntax) was a problem for me
 no matter how you see it.
I just put the clues together.
& it is (now) solved for me.
(I am content with the results.)
Which is more than I can say
 before starting this thread.
I now have a solution method
 to deal with that problem (for me).
A work around.
Thus the problem has vanished.
You DON'T need it (the solution);
 but I do.

 

Edited November 24, 2023 by Capiert