Proof that -a = -1(a)

In the elementary school and high school mathematics, we are usually treating some statements as facts (or axioms). In reality, many of these statements are actually theorems. This means that if you want to go deeper, it is good to read the proofs of these theorems.

Note that all theorems in mathematics, even the most obvious, have their mathematical proofs just as the one we will prove below.

In this proof, we show that the the product of any number and -1 is equal to the negative of that number. That is, if a is a real number then -a = -1(a). Please refer to The Axioms of Real Numbers to understand more about the explanation below including the numbered axioms.

Theorem

-a = -1(a)

Proof

Notice in the proof that on the left hand side, we have (-a). We manipulate the right hand side to become -1(a). 

(-a) = (-a) + 0: Any number added to 0 is equal to the number. (Existence of Additive Identity, Axiom 5A).

(-a) = (-a) + 0a: Theorem: Any number multiplied by 0 is equal to 0 (see Proof). Here, we replaced 0 with 0a because they are equal.

(-a) = (-a) + (1 + (-1))a: The sum of any number and its additive inverse is equal to 0. (Existence of Additive Inverses, Axiom 6A). In the equation, we replaced 0 with 1+ (-1).

(-a) = (-a) + (1a + (-1)a): The number a was distributed over 1 + (-1). (Distributive Property, Axiom 4)

(-a) = (-a) + (a + (-1)a): The product of any number an 1 is equal to that number. (Existence of a Multiplicative Identity, Axiom 5M). We simplified 1a to become a.

(-a) = ((-a) + a) + (-1)a: By Associative Property of Addition (Axiom 2A). In any addition expression, we can always regroup the addends and the sum will still be the same.  That was why we put together ((-a) + a).

(-a) = 0 + (-1)a: The sum of a number and its additive inverse is equal to 0. (Existence of Additive Identity, Axiom 5A). We changed (-a) + a with 0.

(-a) = (-1)a: Any number added to 0 is equal to that number. (Existence of Additive Identity, Axiom 5A). We replaced 0 + (-1)a with (-1)a.

And we are done.

If you have questions, please use the comment below. Enjoy studying!

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