Multiplication is multiple adding. When we say 3x6=, we are saying to add 6, 3 times like in the addition problem 6+6+6. The first number in the multiplication problem is called the multiplier and it tells how many times to add the second number, in the multiplication problem. The second number in the multiplication problem is called the multiplicand, because it is the number being acted on. You can find the product of 3x6 using the Homemade Math's built in scratchpad. Drag 3 number 6 cards down into a blank area of the screen, count the total number of dots appearing on the 3 six cards. You should find that 3x6=18 . The order we multiply in makes no difference to our final answer. We can drag 6 number 3 cards down into a blank area of the screen and count the total number of dots appearing on them and it will still be 18.
Now lets look at 3x(-5)=. We are saying to add -5, 3 times like in the addition problem -5+(-5)+(-5)+(-5)=. We simply add 5, 3 times and then put a minus sign in front of our answer to show that it is negative. Since it doesn't matter which order we multiply in i find it easier to make the negative number the multiplicand in the problem and add the negative numbers to find the product. If we have the problem -3x5=, it's saying we have - 3 sets of 5 or to take 5 away 3 times and we start at 0. Since there is a -(minus sign) in front of the multiplier instead of no sign which means the same thing as +(plus sign) we are being told to do multiple take away(or difference) instead of multiple addition.
We can find the product of -3x5 on the number line by starting at 0, counting the graduations moving to the left in a negative direction. Count to 5 landing on -5, count 1,2,3,4,5 landing on -10, count 1,2, 3,4,5 and land on -15. Whenever you multiply a negative number by a positive number the product will be a negative number. If the multiplier is positive and the multiplicand is negative, you add the negative number, the number of times the multiplier tells you to. If the multiplier is negative and the multiplicand is positive, you perform multiple subtraction by taking the multiplicand away the number of times the multiplier tells you to. In this case you start at 0(zero) in your subtractions. Remember one number is the multiplier and tells how many times to add the other number or how many times to take the multiplicand away in the case that the multiplier is negative. Now what about a negative times a negative like in -2X-3=? WE get a Plus 6, -2X-3=6. -2 is the multiplier and tells you to add -3, -2 times. Because the multiplier -2 has a minus sign in front of it, it is saying to take -3 away 2 times like 0--3--3=6. Because there are 2 - signs they cancel and --3 is really +3. This double -- sign really being positive is easier to see in a problem like 5-(-9)=14. If we say 5-9=-4, we are saying what is the difference between 5 and 9. If we say what is the difference between 5 and -9 as in the problem 5--9=14 we just look at the number of units between these two values. Try it on the number line. One real life example of a negative times a negative: You have $24 and buy 2 hats for $5 each. You have $24-(2purchases X -$5)=_? |2X-5=-10 So $24-$10=$14. Now all of the sudden you find that you are unhappy with the hats and ask to return them for a refund. $14+(-2purchases X -$5)=_? You are reversing the direction of your purchases and now have your whole $24 back. Whenever you multiply a negative times a negative you get a positive. Whenever you multiply a negative times a positive you get a negative. Because the mathematicians of the past have excepted the convention that 1x-1=-1 and not that 1x-1=1, we have to except that -1x-1=1.
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Copyright 2008 Robert Lee Thomas
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