Café Math : A Question of (First) Principles
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A Question of (First) Principles

Sun, 13 May 2012 6:16:51 GMT


1. Foreword

Yesterday my friend Fiza told me about an annoying question that her students asked her, namely, Can we prove that minus multiplied by minus gives plus ? Mmm... that's the kind of question with no clear answer. As this is merely a postulate... But the conversation took an interesting turn as we talked about how to construct the negative numbers from the non-negative ones.

2. The Construction

In what I know of how the system of numbers is constructed, the negative numbers are introduced by the following trick,

Step 0. In the beginning you start with a set $A$ of positive numbers that you can add and multiply but you can't subtract because there is no negative number in your system.

Step 1. You generalize your system $A$ by taking as new numbers, the set couples of numbers from $A$: $$ B = \{\, (a_1, a_2) \,|\, a_1, a_2 \in A \,\} $$

Step 2. Ok nice, but how do you add, multiply etc... well, \begin{align*} (a_1, a_2) + (b_1, b_2) &= (a_1 + b_1, a_2 + b_2) \\ (a_1, a_2) \times (b_1, b_2) &= (a_1 a_2 + b_1 b_2, a_1 b_2 + a_2 b_1) \end{align*}

Step 3. There is too much numbers in $B$... so we introduce the number system $C$ elements of $C$ are equivalence class of elements of $B$ under the following equivalence relation

Two elements in $B$ say $(a_1, a_2)$ and $(b_1, b_2)$ are said to be equivalent if, $$ a_1 + b_2 = a_2 + b_1, $$ So for example in our new system $(1, 2) = (3, 4)$ because $1 + 4 = 2 + 3$, but $(1, 2)$ is different from $(4, 3)$ because $1 + 3 \neq 2 + 4$.

This very much resembles the calculus of fraction. If it's too abstract... remember, $$ 1/2 = 2/4 = ... = 100/200 = ... = 30/60 = \text{etc...} $$ This is the same thing here, A couple $(a_1, a_2)$ is said to be a representative of a number in our new system of number. But another couple $(b_1, b_2)$ is also a representative of the same number, if it is equivalent to $(a_1, a_2)$. Just like $6/3$ and $10/5$ are both representatives of the same number $2$.

Step 4.

Fiza. Where are the original numbers from $A$ in this system ?

Samuel. A given number $a$ in $A$ is represented by the couple, $(a, 0)$. Consequence: The number $(a, b)$ represents the difference $a - b$...

Fiza. Ho ! I get it ! this is where those funny laws of addition and multiplication comes from and why you said that $(a_1, a_2)$ is equivalent to $(b_1, b_2)$ if, $a_1 + b_2 = a_2 + b_1$ ha ha

Samuel. Yes very clever of you.

Fiza. And so about the original question ? I fell we've digressed a bit far ;-)

Samuel. Sure but that was interesting, wasn't it ?

Fiza. Sure it was ;-)

Samuel. Ok now we can prove your original statement, the product of two negative numbers is positive, $$ (0, a) \times (0, b) = (00 + ab, 0b + 0a) = (ab, 0) $$ mmm no need to say more.

Fiza. Ok nice and simple as I like it ;-)

3. Exercises

Some exercises for those who want to learn more. As a student, I always read math book with a paper and a pen. When I found something unclear or mysterious, I always turned to my paper and pen and tried to figure out the thing myself. I recommend that practice warmly for those who sometime feel uncomfortable with some parts of a math text. By doing that, not that the Magic evaporates... instead I think it's the way one becomes a math Wizard oneself.

Reading the above text I've found some of the point that deserve to be worked with pen and paper.

  1. Can you recover the law of addition and multiplication, \begin{align*} (a_1, a_2) + (b_1, b_2) &= (a_1 + b_1, a_2 + b_2) \\ (a_1, a_2) \times (b_1, b_2) &= (a_1 a_2 + b_1 b_2, a_1 b_2 + a_2 b_1) \end{align*} from the comment "The number $(a, b)$ represents the difference $a - b$..." like Fiza did ?

  2. Prove that the notion of addition and multiplication is invariant by equivalence. Giving a precise meaning to that statement is part of the exercise.

  3. Give the definition of the subtraction in the new system. Hint: solve the equation $(a_1, a_2) + (x, y) = (b_1, b_2)$ for $(x,y)$.

  4. We supposed that positive numbers are represented in the new system by $(a, 0)$ and negative one by $(0, b)$. I said that a number $(a, b)$ is considered positive whenever $a > b$. Now can you give an invariant definition of the order relation, given that you can always compare two numbers from the original system $A$.

Have FUN ;-)

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