Double negatives, or the meaning of -(-2)

In the extended number system of signed numbers, that is, the positive and negative numbers I see a lot of heart searching over the meaning of -(-2). This can be put to rest in one or both of two quite satisfactory ways:

1: Signed numbers are directed numbers, used for position, temperature, voltage etcetera. The basic question is “How far apart are the two numbers  A  and  B ?”, or more useful in a practical situation “How far is it from  A  to  B ?”.

This is a subtraction problem with direction and the answer is  B – A
For A=3 and B=7 we get
Distance from A to B = B – A = 7 – 3 = 4
For A=-3 and B=7 we get
Distance from A to B = B – A = 7 – (-3) = ???????????????
But a quick look at a number line shows that the distance is 10
So 7 – (-3) = 10
But 7 + 3 = 10 as well
Conclusion: -(-3) = +3

2: A simple and more abstract approach:
Starting with 7 – (-3) = ??????????????? we give a name to the unknown answer. Call it D.
Then using the basic fact that 12 – 4 = 8 is equivalent to 12 = 8 + 4 we have
7 – (-3) = D is equivalent to 7 = D + (-3)
7 = D + (-3) is equivalent to 7 = D – 3
7 = D – 3 is equivalent to 7 + 3 = D
which says that D = 10
So subtracting -3 is the same as adding 3

A meaningful example is as follows:
My friend from Anchorage calls me and says “It’s cold here this morning, -5 degrees”.
Down here in Puerto Rico it’s 68 degrees this morning.
How much warmer is it here than in Alaska?
 

 

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1 Comment

Filed under arithmetic, teaching

One response to “Double negatives, or the meaning of -(-2)

  1. Oh thank you – the heavens are clearer to me now – you see I never did believe my lawyer or the lesser half – TWO WRONGS DO INDEED MAKE A RIGHT – THANK YOU – I can now go out and shag her second best friend having already shagged her bestest best friend!!

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