negative times positive equals negative; positive times negative equals negative. WHY?
Dear Ed, do maths and stats people like this type of riddle? Let's have a bit of humor then. If I borrow $2 for 2 days, I get $4 in hand, and if I saved $2 for 2 days, I also get $4 in hand :) There's something not right with this thread, at this moment, but RG has fixed such problem for me recently. Thanks RG.
there are many proofs , however none of them explains it easily. Perhaps the best way to understand the statement can be to seek the help of logic where double negation is positive . Truth tables are derived from logical arguments If you need a detailed explanation I can send you one .
Miranda, a riddle it may seem, but it is not. The question seeks to explore one's depth of understanding of the statements beyond memory work. I myself took a lot of reading before I understood (not fully) the meaning of the statements. I tried to ask this to over a thousand students going to five years ago, and less than one percent gave an answer with confidence.
It is the same as my question on 1/0 and 0/0, when less than 1 percent also could give a correct answer.
https://www.researchgate.net/post/Are_the_quotients_of_1_0_and_0_0_the_same
Ed
I researched my answer to this question from math books, the answer which convinced me is similar to
Reveendran’s- logic, however with a little grammatical twist, from the Beatle’s song line-
Oh Darling, please believe me, I’ll NEVER do you No harm.
I am however uneasy with the word double negation because mathematically, it means negations multiplied by 2.
Negation x negation I believe is negation raised to the second power.
Dear Eddie thanks for Beatle’s song analogy.
By the way "negation raised to the second power" does not always produce positive result! For instance in complex number theory, i = squareroot (-1), but when it raised to the power of two, we get ,i2 = -1
Back to analogy, there is a famous Arabic proverb: "The enemy of your enemy is your friend"..The proverb suggests that two parties can or should work together against a common enemy.
Dear friends, Prof Kamal, Ed, Pahlaj, Sribas, Mahmoud, it's very interesting that in language double negatives are positive, as in the example Mahmoud used ('enemy of your enemy'). So, " I' m not unhappy." may mean the same as "I'm happy". But we try to avoid the use of double negatives in research questionnaire statements, and in multiple choice questions.
Thanks dear Cecilia. 'I'm not unhappy' only means the absence of unhappiness. It MAY not mean that happiness is present. (Generally little children are happy, but when they become a teenager, they have lost this happiness. But they would have gained more verbal skills.) So back to Ed's question, in language, a double negative doesn't always become a positive.
(+i)*(+i)=i2=-1!, so your obervation is not in general valid!
Language (mathematical or otherwise) is as imperfect as its inventor, and is continually evolving. Negative raised to the second power may not always become positive. Double negative may not always be negative raised to the second power?
This is a more rigorous proof (with a little algebraic manipulation) in found in the literature. Let a and b be positive numbers. Then -a and -b are negatives. we can prove that their product is positive. Let's multiply them together and add (-a)(b),
that is (-a)(b) + (-a)(-b)
we can factor out the -a, giving us -a(b - b) = -a(0) = 0.
So, (-a)(b) + (-a)(-b) = 0
However, remember that a negative times a positive is a negative, so in that equation, we can replace (-a)(b) with -(ab) and get
-(ab) + (-a)(-b) = 0
Finally, add ab to both sides to cancel that -(ab) on the left side:
(-a)(-b) = ab, Q.E.D.
NB.: With the same logic we can prove that a negative times a positive is a negative, etc.
Thanks Kamal and Mahmoud for your simple and precise response of this question. I fully agree with you.
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