Grasping The Abstract: An exercise from mathematics

[Pythagorean]

I believe that grasping abstract ideas or 'objects' is important in philosophy and to help me do so I am taking up some algebra. Algebra I think is one of the best ways to polish one's thinking skillls, so here goes.


Algebraic equations 'Transposition'

Transposition is the process of moving a quantity from one side of an equation to the other side by changing its sign of operation.

Division is the operation opposite to multiplication.
Addition is the operation opposite to subtraction.

Rule: A term may be transposed from one side of an equation to the other if its sign is changed from + to -, or from - to +.

Rule: A factor (multiplier) may be removed from one side of an equation by making it a divisor in the other. A divisor may be removed from one side of an equation by making it a factor (multiplier) in the other.

Example:

If R=WC , solve for W,C and L.
L

I will lay out the solution and then ask how this was done.

Solution:

R=WC , original formula
L


LR= W. Step one.
C

In this first step W,C and L are transposed.

This is the first step. My question is how was this done?

My answer. I first change the sign in front of L from division into multiplication, which if I perform the same act on the opposite side of the equation (i.e. R times L) thus cancels out the L on the right side and transposes it onto the left side with the R so that I now have:

LR=WC

My next step is to change WC by changing the sign in front of C into a division sign which cancels out C on the right side of the equation by simultaneously dividing LR by C in transposition I am left with:

LR=W
C

I find that step difficult because I don't know exactly what to start transposing first. I don't know the order to begin with. (Help!)

Now back to the results of step one:

LR=W
C

Now my next step is similiar to the first step. I take the division sign in front of C and change it to a multiplication sign while simultaneously multiplying C by W and am left with:

L times R = C times W, or LR=CW

(If someone could tell me why the C is transposed before the W in this step please tell me.)

Then I change the multiplication sign in front of the W to a division sign and carry the W over to the other side of the equation thus isolating C on the right side, like this:

LR=C
W

Next I change the multiplication sign in front of the R into a division sign which again cancels out the R's on the left side and transposes it into a division operation on the right side like this:

L=WC
R

What I need to know is how does one know which act of transposition is to take place first or second or at any time? If anyone can help that's great.



[/IMG]

-- Pythagorean

[Ragnell]

The mathematician of this forum steps up... and drops his notecards. (stunned silence w/ cricket chirps)

Anyhow, here we go

(If someone could tell me why the C is transposed before the W in this step please tell me.)

It generally {perhaps I shouldn't put that word in here} should not matter (at very least, when there are no multiplication-denoting parenthases [which there are not in your problem] And in this case it should not matter).
Note that if you say LR/C=W and you divide W from both sides it would turn out like so... LR/CW=0, to the which you can safely move the C over from the left side to the right, ending with LR/W=C.
I believe the people who construct the books do it that way (with no 0 shown) so as not to be unsightly, which proffering a 0 not in the final answer in an equation that is not quadratic tends to be, though I personally would not mind it; it comes out the same either way.

What's the image for?

[Pythagorean]

Thank you very much Ragnell, that was helpful, I appreciate it. As you say without parenthesis there is no order unless specified. I realize now (thanks to you) in that example, the book is solving for every possible combination to elucidate the concept of transposition.

The image is part of a problem (which I meant to post, I find visuals of "mechanics" to be enriching) which I can't seem to find the rule for. But I'm getting closer.

But here is the one I find hardest, you may like it. I like it but I always seem to get stuck by it.

H=P/AW

W=?

I'm stuck with H/W=P/A and I can't bring the H over to the right. And if I bring the W over to the right I am right back where I started.


Perhaps if you find the time you could do it online here and elaborate by writing operations out in English, it would be appreciated by me -and perhaps understood by others? I don't understand why I get stuck on this one problem?

Thanks again Ragnell.


-- Pythagorean

[perplexity]

Originally Posted by Pythagorean

But here is the one I find hardest, you may like it. I like it but I always seem to get stuck by it.

H=P/AW

W=?

I'm stuck with H/W=P/A .....

Think of it like this:

1. Given H=P/AW, multiply both sides by W, hence:

HW=PW/AW

2. Cancel W on the right hand side, hence:

HW = P/A

3. Divide both sides by H, hence:

HW/H = P/AH

4. Cancel H on the left hand side, hence:

W = P/AH

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So to transform H/W=P/A,

HW/W = PW/A,

hence

H=PW/A

hence

HA = PWA/A

hence

HA =PW

hence

HA/P = WP/P

hence

HA/P = W

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