Transcript Negative - Mr. McCarthy

1-5: Multiplying Integers
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Four (4) Multiplication Rules:
integers with like & unlike signs
1. Negative X Negative = Positive
2. Positive X Positive = Positive
• Negative X Positive = Negative
• Positive X Negative = Negative
• You can also use patterns to find products of more
than two negative factors.
Odd number of factors:
negative X positive X positive = NEGATIVE
negative X negative X negative = NEGATIVE
Even Number of Factors:
negative X negative X negative X negative =
POSITIVE
Mathematical argument takes a little getting used to.
This might look rather strange at first. Here's how the
reasoning goes:
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For all counting numbers a , b and c we have:
a+b=b+a
2) a×b=b×a
3) a×0=0
4) a(b+c)=ab+ac
(along with other variations of the distributive rule).
1. Zero times anything equals zero.
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Every number has exactly one additive inverse.
This means if N is a positive number, then -N
is its additive inverse, so that N + (-N) = 0.
Likewise, the additive inverse of -N is N.
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3. We want negative numbers to obey
the distributive law. This states that
a*(b+c) = a*b + a*c.
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4. Now, we are forced to accept a new law, that negative times
positive equals negative. This is because we can use the
distributive law on an expression like
2*(3 + (-3)).
This equals 2*(0), which is zero. But by the distributive law,
it also equals
2*3 + 2*(-3).
So 2*(-3) does the job of the additive inverse of 2*3, and
therefore 2*(-3) is the additive inverse of 2*3. But the
additive inverse of 6 is just -6. So 2 times -3 equals -6.
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5. Next, we are forced to accept another new law, that negative
times negative equals positive. It's a lot like the example in (4).
We use the distributive law on, say,
-3*(5 + (-5)).
This is again equal to zero. But by the distributive law, it also
equals
-3*5 + (-3)*(-5).
We know the first thing, (-3*5) equals -15 because of the law in
(4). So (-3)*(-5) is doing the job of the additive inverse of -15.
We know -15 has exactly one additive inverse, namely 15.
Therefore,
(-3)*(-5) = 15.
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LET’s Practice