I've written a rule for multiplying any two positive or negative numbers over here on the board. The rule says,
to multiply two real numbers, multiply absolute values. So, that is, you just forget the signs and multiply the number part;
that product, or the answer, is positive if both numbers have the same sign, and negative if the two numbers have different signs.
So, I'll just simply multiply absolute values. If the two numbers I'm multiplying have the same sign, the answer is positive.
And if the two number I'm multiplying have different signs, the answer is negative. In all cases, you see,
the absolute value of the answer is simply the product of the two absolute values. That is, we just forget the signs and multiply: 3 x 5 = 15.
You already know that. -3 x -5 is also 15. 3 x -5 is -15, as is -3 x 5.
So, multiplication is a pretty easy operation. You simply multiply the numbers; the answer is positive if the two numbers have the same sign,
and it's negative if the two numbers have different signs. Let's look at some examples. First, I want to multiply 2 x 3.
Now, I'm going to just show you how we justify this rule real quickly: 2 x 3 means 3 + 3, that's the definition of multiplication.
It means write down 3, 2 times, and add: 3 + 3 = 6. Now, 2(-3), by the same definition, is -3 + -3.
That is, I want to multiply; I want to write down -3, 2 times, and add: -3 + -3 = -6.
-2 x 3: I'll solve that problem by using the commutative property of multiplication to rewrite this as 3 x -2;
now, that's -2 + -2 + -2. And when I add those three (-2)s up, I also get -6.
So, you see, I can justify this rule I have for multiplying positive and negative numbers,
simply by going back to the definition for what multiplication is, that multiplication is repeated addition.
So, 3 + 3 = 6; -3 + -3 = -6; -2 + -2 + -2 = -6.
So, you see, in each case, we simply multiply the two numbers; we get a 6.
If both numbers have the same sign, the answer is positive. If the two original numbers had different signs, then the answer is negative.
Now, how do we justify -2 times -3? It's difficult this way because I can't write -3 down a negative 2 number of times;
or, I can't write -2 down a negative 3 number of times. But what I can do is this: let's look at 3(-3);
2(-3); 1(-3); 0(-3). And now, -1(-3); and -2(-3).
And let's see if we can see a pattern: 3(-3); if I multiply those, I get -9.
2(-3); I multiply those, I get -6. 1(-3) = -3; and 0 times any number is 0.
Now, if I just look at this pattern right here: -9, -6, -3, 0, and ask myself, if I want to continue that pattern,
what would the next number be? The answer is 3; and then after that, 6. So, according to this pattern right here,
the product of -3 and -1 would have to be positive 3; and the product of -2 and -3 would have to be positive 6.
So, in any case, that's what we're going to use for our rule. We'll multiply absolute values.
If the two numbers have the same sign, the answer is positive. If the two numbers have different signs, the answer is negative.
Now, if I multiply three numbers at a time: 3(-2)(4). I multiply two of them first: 3(-2) = -6.
And then, that's multiplied times 4: multiply absolute values; 6 x 4 = 24.
The two numbers have different signs, so the answer is negative. Likewise for this problem: -1(-2) = 2;
and then, that's multiplied times -3. I multiply absolute values and get 6. The two numbers have different signs,
so the answer is negative. So, see, applying my rule for multiplication of positive and negative numbers is really pretty simple.
The next problem we're going to do involves an exponent: (-2)³
I know, by definition, that this means -2(-2)(-2).
Now, I'll multiply -2(-2) = 4; 4(-2) = -8.
So, (-2)³ turns out to be -8.
And I use my definition for this exponent as being repeated multiplications here, to give me -8.
Here's our next example; simplify (let's work inside the parentheses): -3 + 2 = -1; 4(-1) = -4.
Now, on to our next example: I'm going to work inside the brackets, right here. This is -7 + 2[-5 + (-9)].
Now remember, this is a subtraction problem, but I think in terms of addition. So, -5 + (-9) is going to be -14.
Now, I continue here. I have addition and multiplication. I want to multiply before I add. -7 + -28;
and -7 + -28 = -35, when I add.
So, there's a combination of subtraction, multiplication, and addition, with positive and negative numbers.
Another more complicated problem, here: -3(4 - 7) - 2(-3 - 2).
Inside the parentheses is where I'll go first: 4 - 7. I think to myself, 4 + (-7), that's -3;
-3(-3) - 2[-3 + (-2)]. This is -5. Now, I'll multiply -3(-3), that will be positive 9;
9 - 2(-5), and 9 - (-10) is the same as 9 + 10, which will give me 19.
So, I apply my rule for addition, subtraction, and multiplication with positive and negative numbers,
along with my rule for order of operations to work some more complicated problems. Here's our next one:
let's subtract -3 from the product of -5(4). So, I'm going to subtract this -3 from the product of -5(4).
So, here's the product of -5(4). And from that, I want to subtract a -3.
So, notice that I have to put my own subtraction sign in for this word "subtract." So, subtract -3 from the product of -5(4).
Now, that will be -20 + 3 because subtracting -3 is the same as adding positive 3,
and that will be, of course, -17; so, a nice translation of a problem written in words into an equivalent problem written in symbols.
You see I have subtraction, multiplication, and then also addition in the same problem.