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June 1st, 2016, 10:01 AM   #1
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Question Multiplication signed numbers question

Hi,

I'm reading Dan Hamilton's Mastering Algebra (introduction) and on page 7 he has the following rule for multiplying a negative and positive:

a * (-b) = -a * b = -ab

He gives examples, such as:

2 * (-5) = -2 *5 = -10

and

40 * (-9) = -40 * 9 = -360

I don't understand why the negative is being moved to the first number and feel like it's going to confuse me and lead to errors with harder problems in the future.

Thanks,
Ryan
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June 1st, 2016, 12:27 PM   #2
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Think back to multiplication of two positive numbers. Rule # 1 is

$a * b = b * a.$

For example $3 * 7 = 21 = 7 * 3.$

That is a handy rule to know, right. You hardly have to memorize it: you are so used to it that you just think it natural.

Now think back to multiplication of three positive numbers. Rule #2 is

${(a * b) * c} = {a * (b * c)}.$

For example {(3 * 5) * 2} = (15 * 2) = 30 = {3 * (5 * 2)} = (3 * 10).

Another handy rule.

When we get to negative numbers, Rule 3 is $-\ a\ MEANS\ + {(-\ 1) * a}.$

Understanding that equivalency will let you avoid many problems. Now

$\{b * (-\ a)\} = \{b * [(-\ 1) * a]\}. $ Rule 3

$So\ \{b * (-\ a)\} = \{[b * (-\ 1)] * a\}.$ Rule 2

$So\ \{b * (-\ a)\} = \{[ (-\ 1) * b ] * a\}.$ Rule 1

$So\ \{b * (-\ a)\} = \{(-\ 1) * (b * a)\}.$ Rule 2

$So\ \{b * (-\ a)\} = \{(-\ (b * a)\}.$ Rule 3

Now see if you can prove to yourself that $\{b * (-\ a)\} = \{(-\ b) * a\}.$
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Last edited by skipjack; June 1st, 2016 at 01:59 PM.
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June 2nd, 2016, 05:43 AM   #3
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He is simply trying to tell you that it doesn't matter where the negative sign is.
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June 2nd, 2016, 07:53 AM   #4
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Thanks for the responses. I just thought it was odd he switched the negative to the first number with no explanation. I understand it ends up the same but thought maybe I was missing something.

Thanks again, especially for the equivalency examples.
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June 2nd, 2016, 10:48 PM   #5
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Anyone here who knows where I can ask questions?
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June 2nd, 2016, 10:59 PM   #6
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Church?
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June 3rd, 2016, 01:26 AM   #7
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Math Focus: tetration
Quote:
Originally Posted by Tphanis View Post
Anyone here who knows where I can ask questions?
Here i am at ure service!
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June 3rd, 2016, 08:50 AM   #8
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Quote:
Originally Posted by Tphanis View Post
Anyone here who knows where I can ask questions?
At the top of the topic list in each sub forum is a "New Thread" button. Click on that.
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