Is it correct to think of a negative number as a way of showing that you have a deficit in something?

Question

Is it correct to think of a negative number as a way of showing that you have a deficit in something? For example,

• Let $x$ be a real number such that $x=25$.

• Let $y$ be a real number such that $y=27$.

  Consider this equality: $$x-(-2)=y$$ The way I think of it is that $x$ is $y$ with a deficit of $2$ and in order for it to equal $y$ we need to remove that deficit and that's why: $$ -(-2)=+2$$ It's like we have a deficit in something and we want to remove it by substracting it (the deficit). Is my intuition correct?

Answer 1

From the perspective of Group Theory, a negative number is simply an (the) "additive inverse" of a "positive" number. Note: This is more or less an over-simplification, because in $\mathbb{R}$ (the set of all real numbers) we have an order relation $\le$, and things can get more complicated.

Let $(G,+)$ be a group (If you don't know what a group is, just think of it as a very simple algebraic structure that allows you to perform cancellation). One of the group axioms says that given any element $g$ in $G$, you can find some $\tilde{g}$ satisfying $$g+\tilde{g}=\tilde{g}+g=0.$$

Then you may ask what $0$ actually is. Again I am not going too deep, so it would be better to treat it as the ordinary $0$ so that the value of any number is unchanged after adding $0$.

Then $\tilde{g}$ is called an (the) inverse of $g$. We give it a notation $-g$. Another note that is perhaps hard for you. When you denote something (by using a notation), you have to make sure that it is unique (so you cannot have something like $\tilde{g}=2$ and $\tilde{g}=3$ simultaneously), and the article "the" has an implicit meaning of uniqueness.

Hence, what is $-2$? $-2$ is just the additive inverse of $2$. So if you have a $2$, after adding $-2$, you get back $0$. This is just something like a deficit of something: suppose you have $2$ apples, and after you have a deficit of $2$ apples, you have no apple left. Alternatively, if your friend got $2$ apples from you, then you have a deficit of $2$ apples. Hence if I add back $2$ apples to you, you have no deficit.