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The context:

$4 - 4 = 0$

Ok, the first number is assumed as a positive and the operation here is a subtraction, so the second number is a positive also.

So I can summ it in the same way by specifying the second number as a negative: $4 + (-4) = 0$

If I did the same but making the first number a negative, $-4 + 4$, why is it equal to $0$ and not $-8$?

Why do I have to specify that the second number is a negative also? $-4 + (-4) = -8$

The question:

How can I add a number to a negative number without specifying it to be negative?

(I mean, by the same logic of specifying a negative, why don't I need to specify this: $-4 + (+4) = 0$ instead of assuming the one without a specific direction is always a positive?)

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    "If I did the same but making the first number a negative" – but that wouldn't be -4+4 as you have written, it would be -4-4 (which indeed is -8).2012-08-28

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Maybe this way of thinking about it will help you. Please correct me if this is not at all what you wanted to know. It seems to me as if you think about addition as "going in the same direction". If you have $(-4)$ and you add $4$, you seem to expect the resulting number is "more negative".

Think about it this way. Positive numbers represent balls, so a $4$ represents $4$ balls: $\bullet\bullet\bullet\bullet$

Negative numbers instead represent "holes" where the balls can fall in. Thus $-4$ is something like $\circ\circ\circ\circ$

Addition is then the operation that simply puts together everything you have. So $2+3$ would be $\bullet\bullet+\bullet\bullet\bullet=\bullet\bullet\bullet\bullet\bullet$

Similarly, $2+(-3)$ would be $\bullet\bullet+\circ\circ\circ=\bullet\bullet\circ\circ\circ$

But: Whenever a hole and a ball meet, the ball falls in the hole and clogs it, so the ball is not there anymore but the hole isn't either. Thus $\bullet\bullet\circ\circ\circ$ simplifies to $\circ$. Thus $2+(-3)=-1$. Nothing changes if you swap the numbers: $-3+2=-1$, because you add the same numbers of holes and balls.

So, if you want to have $-8$, that is $8$ holes, you would have to add $4$ holes and another $4$ holes. This means you need to sum $-4$ and $-4$.

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    @GregorBruns spot on!, I forgot it, I meant it to be $−(4+4)=−8$. Thanks for that, I wasn't understanding why Doug commented but that answers it. Unfortunately I can't edit the first comment, but this explains it.2012-08-29
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From the point of view of Group Theory:

Consider the Group $G$ with the set of integers and a binary operation $*$. There exists an identity element, $e$, with the property:

$ \forall a \in G, a *e = e *a = a $

Now, each element $a$ in G must have an unique inverse:

$ \forall a \in G, \exists{b}, b*a = a*b = e $

Write the inverse of an element $a$ as $-a$ (since its unique). Replace $*$ by $+$, and $e$ by $0$. We have:

$ \forall a \in G, \exists{-a}, (-a) + a = a + (-a) = 0 $

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If you want to add a positive number to a negative number without specifying the negative number as negative, I would suggest that you use entirely different symbols for negative numbers than positive numbers. In other words, you'll need to use something unlike Hindu-Arabic numerals, since 4 and -4 resemble each other too much (from one perspective one can even take the perspective that -4 doesn't represent a number, but rather the negative function "-" operating on 4 which equals what we call "negative four", but I digress). The following consists of one possibility along those lines, though many different symbolizations will work:

Let zero get represented by m, one by n, two by o, ..., thriteen by z. Then for fourteen we would write nm, for fifteen nn, sixteen by no, and so on.

Let negative one by l, negative two by k, ..., negative twelve by a, negative thirteen by lm, negative fourteen by ll, negative fifteen by lk, and so on.

So, then we can write q+j=0 for "four plus negative four equals zero" and p+k=n for "three plus negative two equals one".

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    This is a nice idea, thanks.2012-08-29
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There is no difference between subtracting a positive number and adding a negative number. In fact, subtraction is defined as the addition of the number's negative.

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    One very common alternative is to define "$a-b$" to mean "the unique solution to $b+x=a$, if such a thing exists". I'm pretty sure that's how I learned it (if in less lofty language) in grammar school. It is easy enough to show that this is _equivalent to_ adding an additive inverse (if enough such inverses exist!), but that doesn't make it the _same definition_. Once you're sophisticated enough to grasp abstract reasoning about inverses, your definition is more economical, but at this question's fairly basic level, the more elementary $b+x=a$ definition is more appropriate.2012-08-28