شماره های مثبت و منفی - II
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Positive and Negative Numbers - II
More on positive and negative numbers. So in the last video we talked about adding and subtracting positive and negative numbers. Here we’ll talk about multiplication and division. The rules for multiplication and division of positive and negative numbers are considerably simpler than the rules for addition and subtraction.
First, let’s talk about the Sign rules. The sign rules for multiplication. We multiply a positive times a positive, we get a positive. If we multiply a negative times a negative the two negatives cancel and we get a positive. The only time we get a negative is when we do positive times negative in either order.
It could be positive times negative or negative times positive. Either one would result in a negative. So, notice when we multiply same signs we get a positive product, when we multiply different signs we get a negative product. The sign rules for division, you’ll notice these are quite similar. Positive divided by positive, that equals positive.
Negative divided by negative, the negatives cancel, we get positive. Positive divided by negative or negative divided by positive, we get negative. So, once again, same signs gives us a positive quotient. Different signs gives us a negative quotient. We can combine all this simply by saying, if we multiply or divide numbers with the same sign, the result is positive.
If we multiply or divide numbers with different signs, the result is negative. Right there, those are the sign rules. If you know that,. Everything about positives and negatives with multiplication and division is much easier. Multiplying and dividing positive and negative numbers.
Firs of all, always use the Sign Rule. Before you do any arithmetic, use the Sign Rule to determine what the sign of the product or the quotient will be. Determine the sign first. Once you’ve determined the sign, treat both factors as positive and simply perform ordinary multiplication and division as you would with positive numbers, and at the end, give the appropriate sign.
So, you’re gonna separate out, figuring out the sign from the actual arithmetic of the multiplication or the division. For example, pause the video here and find the answers of these three. Okay. The first one we see is a negative times a negative, so that’s definitely gonna be a positive, that’s gonna be positive six times seven which of course is 42.
The second one, we have a negative divided by a positive so that will be negative, whatever that quotient is, it’s a negative. So, we’ll take that negative sign outside. 65 divided by 5 is 13, so that just becomes negative 13. In the final one, we have negative divided by negative, so we know that will become positive, so we can cancel the negatives.
We just get 30 over 12, cancel a factor of 6, and it goes down to 5 over 2 in simplified form. Another topic related to positive and negative signs is Absolute Value. Many folks need to update their understanding of this mathematical idea. The naive definition of absolute value is it makes everything positive. This is pretty much what folks learn in grade school, and some people never update this understanding.
Well, what’s the problem with this? Certainly. The absolute value of six is six. The absolute value of negative 14 is 14. So in those two cases, yes, the number becomes positive. Well notice, the absolute value of zer is zero.
Zero’s not a positive number. So, this would be one example of a number, which, when we take the absolute value of it, it does not become positive. So that’s problem number one, with the it makes everything positive, way of looking in an absolute value. In much more sophisticated definition is: the absolute value of a number gives the distance of the number from the origin.
First of all, this is consistent with our previous understanding. On a number line the point six is a distance of six from zero. The point negative 14 is a distance of 14 from zero. And of course, the point zero is a distance of zero from zero. So, this is consistent with all this arithmetic. But, this new understanding has important implications in Algebra.
For example, the absolute value of X is the distance of X from the origin. The absolute value of x-5 is the distance of X from +5. The absolute value of X plus 3 is the distance of X from -3. This is very important. So, if we’re given an equation like this, the absolute value of X minus 1 is greater than 4, what this is telling us is the distance between X and 1 is greater than positive 4.
So we think of a number line, put 1 in the middle. Then go out four units on either side, so that wouldn’t be allowed. So, starting at -3 or at +5, that wouldn’t be allowed because that’s a distance of exactly 4 so we have to go to the left of -3 or to the right of +5. That would be the allowed region. That, that would be all the points that have a distance of more than four from the point, positive one.
And notice we could also write this as X is less than -3 or X is greater than positive 5. So, both of these blue inequalities say exactly the same thing. Here’s a practice problem. Pause the video, and then we’ll talk about this problem. Consider the positive integers from one to a hundred, so we have a hundred integers.
If N is a number in that set, then for how many numbers N is it true that N minus, the absolute value of n minus 30 is greater than 20. Well, that would be a very hard problem to start plugging in values and trying to figure it out, much easier with this updated understanding of absolute value. So, what we’re saying is that the distance between N and 30 is greater than 20. So, right here is 30.
We’re gonna go a distance of 20 on both sides. So the point at 10, that’s not gonna work cause that’s exactly a distance of 20. The point at 50, that’s not gonna work, that’s exactly a distance of 20. So we need the points less than 10. So that would be 1 through 9, or the points greater than 50. That would be 51 through 100.
Well 51 through 100, that would be 50 points. And of course 1 through nine, that’s nine points, 50 + 9, that would be 59 points there. There are 59 points on this number line that satisfy this condition. In summary, when we multiply or divide like signs, the result is positive.
When we multiply or divide unlike signs, the result is negative. An absolute value can be interpreted as the distance from zero. The distance from the origin
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