Adding / Subtracting negatives

Question

Suppose one of your students asks, "Why does -(-a) = a?" What could you say to help develop understanding for this student?

Answer 1

English grammar and most languages

the double negative is a positive

"I deny that I was not there" means you're claiming that you were there

the jury did not return a not guilty verdict. That means the jury came back with a guilty verdictN(that one's also sort of a triple negative, if you think about it. "guilty" is itself a negative, so - - - = -)

if rain& clouds are itself negative (weather), then to say" it's Not raining & there are no clouds" is the same as saying "it's a sunny day" which is positive (weather).

"There is no way that I failed to do it" means I did do it.

You take a covid19 test. It did Not come back negative, that means it came back positive

(one notable exception is "two wrongs do not make a right," although many disagree. the person retaliating against a wrong done by someone else might think they've come out ahead or at least even, as they "get even." So, in some sense, maybe an immoral or amoral sense, two negatives cancel each other out and you get a non-negative result.)

bankruptcy proceedings

you owe a debt of an "a" amount, say $2,000 on a credit card bill. It's a negative "asset," a liability, a debt, a debit on your accounting summary.

you can't pay, and you filed for bankruptcy to discharge or cancel the debt

that puts a negative sign in front of your debt, that's - - $2,000 = +$2,000. You just came out $2,000 ahead. You were in debt with -$2,000. Now you have zero debt. That's a gain of $2,000. zero - - $2,000.=+$2,000.

Bankruptcy made you $2,000 better off.

also

anything divided by itself is 1, such as a^-1/a^-1 = a^1 times a^--1 = a^1 times a^1

a^-1(a^1) = a^0 = 1

if a<0 then you're multiply two negative numbers together to get a positive number

if a=-1, then -1(-1) = +1

or

-1(-1) = -(-1) = +1 where there is an implicit -1 as the coefficient of the parenthetical term -(-1)

if you move a term from the denominator to the numerator it changes sign, from + to -, or from - to +.

then when you multiply two terms with the same base, it's the same as adding their exponents 1+(-1) =0

any base to the zero exponent = 1

another example

divide 16 slices of pizza among 16 people, each get 1 slice

16/16 = 1

4x4 = 16

= 4^2/4^2

and also

=(-4)^2/(-4)^2

= (-4)^2((-4)^-2)

= (-4)^(2-2)

= (-4)^0

=1