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August 28th, 2018, 11:55 AM   #1
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Question Prove that at least one number must be negative. abc = -1000

So I am having this question in my math textbook that says:

Assume $\displaystyle abc = -1000$
Prove that at least one number must be negative.

So the only proof method that is availible for the chapter is to do
contrary evidence (the opposite to prove that it can't be true). So
do it become this then?

Prove that more than one number must be positive.

Do I need to find a abstract example or can I show a concrete example that doesn't work?

Case 1: a and b is positive and c is negative, then abc will be a negative number.

Case 2: a, b and c is positive, then abc will be a positive number.

If I am correct I need to have that more than one number must be positive. If so, I only have two cases.

So this seems on the right track to me, but I haven't really proven anything, I guess.
So I need to know how I prove my cases because right now they are just statements. So how can I continue with that?
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August 28th, 2018, 12:19 PM   #2
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I think it's easier than you make it.

Assume that all 3 numbers are non-negative.

Then $a b c$ is non-negative. This is a property of multiplication of real numbers.

We are told to assume $abc = -1000 < 0$, i.e. $abc$ is negative.

Thus we have a contradiction between our two assumptions.

We are told to assume $abc$ is negative so it must be that all 3 numbers are not non-negative and thus at least 1 is negative.

This can all be stated in a lot fewer words than above.
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August 28th, 2018, 12:28 PM   #3
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I see. Seeing it from that perspective made the problem much easier
and less text to write too
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