Puzzle about Units

[mathenthusiast]

Is this statement true: 1^oC = 33.8^oF? If so, multiplying by 2 on both sides preserves equality... so 2^oC = 67.8^oF. But 2^oC=35.6^oF, so either 1^oC is not 33.8^oF or multiplying by 2 does not preserve equality . This seems like a valid argument, so what's the untrue premise?

 

[topsquark]

Never trust a statement when you have different units on either side. The temperature 1 C is equivalent to the temperature 33.8 F, but the numbers are not equal.

-Dan

 

[mathenthusiast]

So you're saying this (1^oC = 33.8^oF) is just abuse of notation in some way, not actually true as with other units of measurement 1ft = 12in?

 

[Steven G]

This is a great question!
What you have is a linear relation between F and C. The function F = (9/5)C + 32 has the 32. This means that F and C are NOT proportional. That means if you double F, C does not double. Note if we do double F and C we get 2F = (18/5)C + 32 which is NOT true. To make this equation true you must also double the 32. This is true no matter what we multiply F and C by.
Now if you have a linear function G = 7H (note that there is no constant like in G=7H -22). Here if you tripled G and H , you get 3G = 21H. This time this equation is true! So we say that G and H are proportional.
In the end if you have a linear equation of the form A = kB for some constant k, then you can do what you tried above!
I hope this is clear

 

[Steven G]

So you're saying this (1^oC = 33.8^oF) is just abuse of notation in some way, not actually true as with other units of measurement 1ft = 12in?

So you're saying this (1^oC = 33.8^oF) is just abuse of notation in some way, not actually true as with other units of measurement 1ft = 12in?

YES! The reason is F = (9/5)C + 32. That is 33.8F= (9/5)*1F + 32. This equation DOES NOT say that 33.8F = 1F !
I never realized how I abuse all of this when I say 33.8C = 1F. The correct phrase should be that 33.8F corresponds to 1F.

 

[mathenthusiast]

Thanks for the analysis. well said

 

[Deleted member 4993]

YES! The reason is F = (9/5)C + 32. That is 33.8F= (9/5)*1F + 32. This equation DOES NOT say that 33.8F = 1F !
I never realized how I abuse all of this when I say 33.8C = 1F. The correct phrase should be that 33.8F corresponds to 1F.

You are being real abusive now ..... go to the corner for 33.8 minutes (where the temperature is kept at 33.8 ^oF or 1^oC)

 

[Deleted member 4993]

So you're saying this (1^oC = 33.8^oF) is just abuse of notation in some way, not actually true as with other units of measurement 1ft = 12in?

This abuse comes from the fact that if plotted F vs. C, you'll get a straight-line that does not go through the origin. That is:

0 ^oC \(\displaystyle \ne\) 0 ^oF​

However, for inch \(\displaystyle \to\) ft. conversion, the straight line goes through the origin and we have 0 in = 0 ft.

 

[Steven G]

This abuse comes from the fact that if plotted F vs. C, you'll get a straight-line that does not go through the origin. That is:

0 ^oC \(\displaystyle \ne\) 0 ^oF​

However, for inch \(\displaystyle \to\) ft. conversion, the straight line goes through the origin and we have 0 in = 0 ft.

Hence the two variables must be proportional, that is in the form y=mx. I consider mentioning the 0=0 fact but felt that maybe it might go over the OP's head. In the end this might be the most interesting thread that I ever read. Very interesting how one can make a mistake in their own field and not even realize it. I had debates with my sister-in-law that kgs is not a measure of weight and lost every time. Her whole every day life it has been treated as a weight in her home country. Some arguments you just can't win!

 

[mathenthusiast]

I get that the Celsius temperature variable is not in a linear relation with the Farenheit temperature variable. It is pretty clear why proportional variables (variables in linear relation with one another) behave algebraically different from variables like Celsius vs Farenheit . But there is still a kind of semantic puzzle about how we can truly say on the one hand that 1^oC and 33.8^oF pick out the very same thing (which happens to be a temperature, to topsquark's point above), but they are nevertheless not interchangeable "salva veritate" in mathematical statements, which are "extensional" in the sense of only caring about the referents of expressions, not their meanings. (Contrast mathematical contexts with so-called "intensional" contexts, which are sensitive to the meanings of expressions in addition to their referents ... e.g. "Karl believes that Phosphorous is the morning star" could be true while "Karl believes that Hesporous is the morning star" is false, even though the expressions "Phosphorous" and "Hesporous" both refer to the planet Venus. )

By analogy, if we model the distance of an object from the origin traveling at, say, (9/5)m/h with a starting point of... i don't know, say, 32m away from the origin, this would be written as: D = 9/5*T + 32. We would not be tempted to say that 1hour = 33.8miles. But unlike the temperature case, these are different "dimensions" (time vs distance), so both sides of the equals sign clearly have different referents.

 

[topsquark]

Two things. The first is a bit nitpicky.
1. The relationship between the temperature limits is linear, but it is not directly proportional. ie. F = aC + B but not F = DC, where both of these are linear but the second is also directly proportional.

2. Note that when I made my first post I refered to the property of temperature, not the units directly. We may have many units to describe temperature and the numbers for the boiling point of water can take on as many as four values in the systems I'm most familiar with. But the temperature is the same for all of them. This kind of thing is pretty commonly seen in Physics but not so much in Mathematics which doesn't typically make use of properties on this level.

FYI: I find it to be occasionally useful to put units into variables in Mathematics. It can be a way to do a quick check on the Math or make an equation a bit easier to do. So if I'm trying to solve an equation for a variable I might do the following: Solve for x: y = ax^2 + bx + c. Note that we need the units on both sides to be the same. This means I may temporarily assume that y and the coefficients on the RHS have units, say meters: x and y are in meters, the "a" in front of the [math]x^2[/math] term has the unit 1/m, the "b" in front of the x term is unitless, and the constant "c" has units of m. Armed with this I can use the quadratic equation to find that [math]x = \dfrac{-b \pm \sqrt{b^2 - 4a(c - y)}}{2a}[/math]. As one way to check my work I can use the unit concept: x has to be in meters or if not then I've made a mistake.

-Dan

 

[Steven G]

I need to repeat what of on board Physicist said above: F and C are linear but since then do not cross the origin they are NOT proportional

 

[mathenthusiast]

By a linear function (relation), I mean a function with the following property: f(cx) = cf(x) for all real numbers x,c, which is a common understanding of linearity, excluding the more general affine functions of the form fx = mx + b, in which b is non-zero.