Expert: Josh - 10/10/2005

Question
Hello:

I want to thank you for the reply.

In your answer you indicated the following: "This is a consequence of "the existance of multiplicative inverse", one of the twelve axioms in mathematics; which states..."
What are the other eleven axioms in mathematics?

I thank you for your follow-up reply.
-------------------------
Followup To
Question -
Hello:

I want to thank you for the reply and answer, but I am somewhat confused.
If I solve the equation as 1152y = 1152y, is this solution as correct as the others, 48y = 48y and 2y = 2y?

-------------------------
Followup To
Question -
Hello:

What is the answer for this equation: y/3 * 6 = 4y/8 * 4?

If y/3, 6, 4y/8, and 4 are multiplied by 24, The equation becomes 8y * 144 = 12y * 96 and this now equals 1152y = 1152y.

If only y/3 and 4y/8 are multiplied by 24 the equation becomes 8y * 6 = 12y * 4, this equals 48y = 48y

If the equation is simplified as 6y/3 = 16y/8 the equation becomes 2y = 2y

Are all of these calculations and answers correct?

I thank you for your reply.
Answer -
All values of "y" in the field of real numbers will satisfy this equation. Actually, I prefer to call this a trivial "identity".

It does not matter what common denominator you use, because we always cancel out common factors.

Writing "y/3 * 6 = 4y/8 * 4" as "6y/3 = 16y/8", you have not simplified anything. All you have done is applying the associative law of multiplication.

As I have made reference to this many times before, an expression like (a/b)*c is same as (a*c)/b.
Answer -
Good question. It's important to clear this up.
This point is this, solving 1152y = 1152y, is no different to solving 48y = 48y OR 2y = 2y for that matter.

When we solve equations, we always cancel out common factors. If we consider "2y = 2y" for instance, twice of "y" is equal to twice of "y", so the multiplier might as well be anything. Why not "3y=3y", or "1000y=1000y"? They are identical expressions. The thing is, it is common practice to reduce such expressions to a monic expression, like "y=y", with a leading coefficient of 1.

It does not matter which equation you pick, "48y = 48y" is same as "y=y".

Now, regarding the solution.
We call an expression like "y=y" a trivial identity, because it is not a real equation at all. Any value will satisfy the relation "y=y". The unknown quantity represented by "y" must be the same as the value of y. This is always true. That's why I said there is nothing to solve for. Any number you pick for "y" must satisfy this condition.

This is a consequence of "the existance of multiplicative inverse", one of the twelve axioms in mathematics; which states

"For a non-zero and finite value "y" in the field of real numbers (R), there exists an inverse, viz., "1/y", such that when "y" is multiplied by "1/y", it yields one."

We can express this as "y*(1/y)"=1, which of cause, follows from "y=y", if we divide both sides of the equation by "y".

Okay, I'm off for the weekend. Bye for now.

Answer
A few of these include,

i) associativity under addition;
ii) associativity under multiplication;
iii) commutativity under addition;
iv) commutativity under multiplication;
v) existance of zero;
etc.

Rather than recounting these axioms, please look up "axioms" and "mathematics" in a good search engine; or refer to an introductory book on linear algebra.