What is the value of $-(-6)$?

I know this is not absolute value, so what do the parentheses mean? If it was absolute value, it would have lines so I can rule that out.

And I guess I should mention, the question I was asked is "Enter the opposite of this number in the box." So, that's the context.

$-(-6)$

• Thanks. They are confusing. – awesomesauce Aug 11 '15 at 15:48
• The tag negative binomial is meant for something else (it's a distribution) – Karl Aug 11 '15 at 15:49
• @ManuelLafond: But if they're just normal parentheses, then the value is $+6$, not $-6$? – joriki Aug 11 '15 at 15:50
• $\binom{n}{1}$ is equal to $n$ for integers.We can extend $\binom{x}{1}$ to be $x$ for any complex number. In fact this is the only way to extend it so that it is a polynomial. You can do the same thing for any natural $k$. In other words make $\binom{x}{k}=\frac{x(x-1)\dots (x-k+1)}{k!}$ for any complex number $x$. – Jorge Fernández Hidalgo Aug 11 '15 at 15:52
• Rarely has a mere misplaced tag had such a fateful effect... – joriki Aug 11 '15 at 15:58

This means negative, negative $6$. As you probably know negative times negative is positive, so this is simply the value $6$ (positive six). The reason for the parenthesis is that the convention is that we do not have multiple operation signs in a row, e.g. signs like "$+$","$-$", and "$\times$". In a way you can say that every two negatives "cancel out", so

$$\require{cancel}-(-6) = \cancel{-}(\cancel{-}6) = \fbox{6}.$$

Now to answer the added question on what the "opposite number" is, I'm assuming that it is meant "the negative of the number." The question then is "What is the negative of $-(-6)$"? Since we have already resolved that $-(-6) = 6$, this question is the same as "What is the negative of $6$?", which is simply $-6$.

• I don't think that this is best viewed as an instance of "negative times negative is positive". That would imply that negation is multiplication by $-1$. Negation is best viewed as forming the additive inverse, an operation that's available in any additive group, on which no multiplication may be defined. In any such group, double negation is the identity operation. – joriki Aug 11 '15 at 16:07
• @joriki That would be the way I would describe it to someone who was used to higher levels of mathematics. I think this is the best way to explain it to someone who struggles with relatively basic mathematics such as fractions. It would not help this person to introduce them to abstract groups. – Eff Aug 11 '15 at 16:12
• I wasn't saying that you should introduce abstract groups or additive inverses -- but if we agree that negation is best viewed in this way, then describing it as "negative times negative is positive" is misleading; then you're not just not talking about complicated stuff, but talking about simple stuff the wrong way that will make it harder to build a basis for later understanding the complicated stuff. Without introducing abstract groups, you could say something like "as you probably know, changing the sign of a number twice gives you back the original sign". – joriki Aug 11 '15 at 16:15
• Positive or negative?!?!?!? – awesomesauce Aug 11 '15 at 16:20
• @awesomesauce What are you confused about from my answer? – Eff Aug 11 '15 at 16:24

I think a lot depends on context.

If by $-6$ you are referring to the additive inverse of $6$ then $-(-6)$ is the additive inverse of the additive inverse of $6$

$4-(-6)$ however is slightly different. Here the first $-$ means subtract defined as adding the additive inverse. To be precise $4+-(-6)$ where the first $-$ now refers to the additive inverse.

There may be other contexts I'm unaware of. Hope that helps.

• Thanks, but all it said was "Enter the opposite of this number in the box". So that is all it was. – awesomesauce Aug 11 '15 at 16:10
• $-(-6)=6$ showing why requires understanding how additive universes are defined. – Karl Aug 11 '15 at 16:31

Remember that a negative number times a negative number is negative.

If you see $-(x)$ that's a short way of writing $(-1) \times x$, that is, "minus one" or "negative one" multiplied by $x$. In this particular case we have $x = -6$, so then $(-1) \times (-6) = 6$.

The parentheses in this case are for clarity: compare $--6$. Without the proper typesetting, that can look like one very long dash, instead of two minus signs. (In other cases, parentheses override operator precedence, plus they also have other mathematical uses that are not relevant to your question here--as for "minus one" versus "negative one", that's another can of worms altogether).

• See my comment under Eff's answer -- explaining negation as multiplication is not a good idea in my view, since it exists in all additive groups and doesn't require multiplication to make sense. – joriki Aug 13 '15 at 14:50
• @joriki If this kid makes it so far as to study additive groups, the damage done by his elementary school teachers is already done, and it's not limited to the foundations necessary to understand additive groups. – Robert Soupe Aug 14 '15 at 17:00