How to prove this basic fact : $(a) (-1) = -a$ . How to prove that an additive inverse can be expressed multiplicatively?

Question

As an exercise I try to prove this inequality principle : the sense of an inequality is reversed if each side is multiplied by the same negative number.

In steps $(2)$ and $(4)$ of the proof ( below) , I need to use the fact that :

$$(a) (-1) = -a$$

So I've asked myself the question : how to prove this fact ?

The number : $-a$ is defined as the opposite (or additive inverse) of the number $a$ . Opposites ( additive inverses) are not defined in terms of multiplication. So, though basic, this fact is not absolutely trivial.

It seems to me that I cannot use here the " rule of signs" since the sign of the number a is undetermined.

Which axioms or basic principles could be used in an algebraic system to prove this elementary fact?

Below, my attempt of proof regarding inequalities :

Answer 1

Let $R$ be a commutative ring and let $a,b\in R$. Let $0$ be the zero element of $R$ and $-a$ be the additive inverse of $a$. Here are some basic facts:

(1) $0a = 0$.

Proof: $0+0=0$. Multiplying with $a$ gives $a0 = a(0+0) =a0+a0$. Adding $-a0$ on both sides gives $0 = a0$.

(2) $-(-a) = a$.

Proof: $-(-a)$ and $a$ are both additive inverses of $-a$. By the uniqueness of add. inverses, $-(-a)=a$.

(3) $(-a)b = a(-b) = -ab$.

Proof: $ab + (-a)b=(a+(-a))b= 0b=0$ and so $(-a)b$ is inverse to $ab$. By the uniqueness of add. inverses, $(-a)b=-ab$.

(4) $(-a)(-b) = ab$.

Proof: $(-a)(-b) = -((-a)b) = -(-(ab))$ by (3) and $-(-ab) = ab$ by (2).

In particular, by (3), $a(-1) = -(a1) = -a$.

Answer 2

You can use the fact the\begin{align}a+(-1)a&=1\times a+(-1)\times a\\&=\bigl(1+(-1)\bigr)a\\&=0\times a\\&=0.\end{align}Of course, I am using here the fact that $0\times a=0$, which can be proved as follows:\begin{align}0&=0\times a-0\times a\\&=(0+0)\times a-0\times a\\&=(0\times a+0\times a)-0\times a\\&=0\times a+(0\times a-0\times a)\\&=0\times a+0\\&=0\times a\end{align}

Answer 3

Using distributivity of multiplication over addition,

$$a(-1)+a=a(-1)+a\cdot1=a(1-1)=a\cdot0=0$$ so that $$a(-1)=-a.$$

Answer 4

Three key points:

• Multiplication gives the same answer as repeated addition.
• Subtraction is addition of negatives
• Addition starts from 0.

These transform it as follows:$$a(-1)=0+\underbrace{(-1)+\cdots+(-1)}_{\text{a times}}=0-a=-a$$