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1-5: Multiplying Integers 1 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: 4 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. 6 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. 7 3. We want negative numbers to obey the distributive law. This states that a*(b+c) = a*b + a*c. 8 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. 9 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. 10 LET’s Practice