Intro

Meeting the Infinite

Algebra of Constants

Tricky Question

Solution

Analysis

Here is an
application of a simple formula

$PV=nRT$
in a non-trivial way, see a
Tricky Question.

Rather than abolishing uncertainty the calculation through approximation accepts uncertainty.

However approximation is not quite correct because infinite magnitudes are used which are exacting.

Variation of proportion mathematics is replace by the algebra of constants.

A non-zero constant multiplied or divided by another non zero contant is itself a non-zero constant.

Let
$k=\mathrm{constant}$

$k\ne 0$

Generalize multiplication of constants.

$k{}^{n}=k$ where $n\in \mathbb{R}$ and $n$ is constant.

For example.

$k\times k=k$

$k\xf7k=k$

It was suggested to me that this is not taught because teaching

$\frac{k}{k}=k$

would confuse people.

I will argue that people are already confused by its omission.

The algebra states that a constant multiplied
by another constant is itself a constant.

3 times 4 is 12 is a contant

2 ÷ 9 = is a constant

To this end I am attempting to make explicit what has been done implicitly so that the structure can easily be seen, hence the tool or technique is accessible.

This thinking used to be taught in a different way through proportion.

$P\propto T$

The proportion notation is inherently implicit as the contant multiplied through definition is implied - you can not see the constant being multiplied.

When actual calculations are done with the proportion notion the constant is re-introduced.

Using absorbing constant multiplier k algebra a potentially more general and hence useful way can be taught. For example Gauss's congruence notation(which I intensly dislike) also hides the constant algebra in the same way the proportion notation does. So the constants generalized algebra has applications in number theory.

Showing relationships in maths is important. For example given the ideal gas equation what can be found?

Sealed chamber of gas.

See Gas as moving particles experiment.

$PV=nRT$

Using what we know about the problem, by identifying what variables are constant we can identify relationships.

- As the Volume is constant let V=k.
- R=k as the universal gass constant is constant
- n=k as the number of molecules is the same as the space is sealed.

$Pk=kkT$

$Pk=kT$

$P=kT$

The relationship between P and T is linear.

There really is not an infinite volume or infinite pressure, but this view is limiting.

A small number when added to a large number can be treated similarly to adding a finite number to an infinite number.

For example lets say you travelled to another plannet, bottled some air and then came back to earth. While earths air is a finite quantity adding the new air to the earths air does not change the pressure much.

Instead of a full bottle of air, we now get 0.5 bottle of new air. Adding this to the earths air changes the pressure even less.

In a limiting sense we could keep reducing how much of the other plannets bottled air s being added to earth; this increases the model of the earth having an infinite amount of air and the bottled air being finite.

Because the ratio of the bottled air to earths air approaches zero.

Another analogy is that if you have a million dollars and add one dollar you still have a million dollars.

The trick here is to know to treat a finite arithmetic situation with infinite arithmetic.

Pressurised gas is released into an unsealed room. The gas expands to fill the room.

No new energy has been added to the room, the gas was simply released.

If in the gases expansion the gas molecules sped up (grabbing energy from somewhere to do this) what has happened to the room?

P and V are constant into PV = nRT formula.

$PV=nRT$

$k\times k=n\times k\times T$

$k=n\times k\times T$

$k\xf7k=n\times T$

$k=nT$

Since n increases then T must decrease as their multiplier is constant.

Hence the room gets colder.

The pressurised gas is finite. As the room is unsealed the volume
is infinite. Adding new finite gas means it will blend
in to the infinite volume.

ie infinite pressure + finite pressure = infinite pressure
and hence P is constant.

Technically new energy was added to the system. But as this new energy is finite and the energy of the system is infinite then infinite + finite = infinite and the energy is the same.

Locally the amout of substance does change in the room. Over time this will dissipitate but at the event the amount of substance n is changed.

Then use simple algebra to come to the conclusion that the temperature decreases.

I was watching a documentary on tv about
the world's largest telescope where the
explained that they used CO_{2}
in their cooling system to keep the tellescope
room at constant temperature.

Metals and other materials expand and contract with temperature changes. To keep micro (10^-6) tollerences they needed to keep the room at a constant temperature.

The documentary mentioned the use of gas in cooling, as an class exercise a fellow student teacher wanted to teach pressure, when I saw the equation I realized that it may help explain why the gas does the cooling.

A first thought may be that because the gas is going fast energy is being added to the system, and hence the opposite effect of heating is ocuring. The gas equation explains this, drawing the opposite conclusion.

Discussions with Bill a teacher ar RMIT.

Addition can also be generalised.

$x+k=x$

Other issues were need for more worked examples and more general uses.

Bill raised the issue of infinity not being practical to problems. However I disagreed. Consider number of air molecules in room - effectively infinit in model even though the room is finite. Infinity is every where in nature. Other examples: vertical line in graph gradient = infinity.

Need for development of symbolic mathematics to describe the problem in more detail. Perhaps use calulus notation V + change in V.