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.



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-- Pythagorean