Basic Math: Lesson 3 - Operations On Numbers - Part 3
This lesson consists of providing you with a Self-Tutorial on the typical operations done on numbers. These include: absolute value, opposites, addition, subtraction, multiplication, division and simple exponents. Order of Operations is covered in detail. I also explain how to use your graphing calculator to help you perform all these operations.
Step 1: The Reciprocal
If we're given some value "x" and we want to find what's called the reciprocal, well first off I need to rewrite this as a fraction. "X" is really "x divided by one." OK. The reciprocal means make your numerator the denominator and the denominator the numerator. In other words you can just switch these two around. So one "divided by x" is the reciprocal of x.
Step 2: An Example
Now here's another example of what I just talked about. What do you think the reciprocal of "a" divided by "b" is going to be? Very easy just switch these two around and you are going to get "b divided by a". They just flip. That's all that reciprocals are about.
Step 3: Multiplying A Number Times Its Reciprocal
Why is it important to learn about reciprocals? Well there's a really neat thing that happens when you multiply a number times its reciprocal. Here's a way to illustrate that:
If you five times its reciprocal, one over five, or one-fifth the answer's actually one. It's always one. Here's another case: negative three-sevenths times its reciprocal--flip it--negative seven-thirds the answer's also one.
Step 4: The Number Zero
Now there is one number that doesn't have a reciprocal. Do you know which one it is? Zero. Why's that? Well let's rewrite this as zero divided by one. First. So that's still zero. I want to take its reciprocal, so we kind of switch things around a bit and I get one divided by zero. But you may think "That's it's reciprocal!" But we just talked about the fact that you're not allowed to divide by zero. So zero is unique in that it does not have a reciprocal.
Step 5: Divison As Multiplication
One final thing I want to mention about reciprocals is that any division now can be re-written as a multiplication. So something like "a" divided by "b" really becomes "a".
Now with variables I really don't want to use the times sign so I'll just put a dot right there, raised dot, and that's the same thing as multiplying by the reciprocal of "b", which is one over "b". They're the same thing: "a" divided by "b" is "a" times one over "b". Why do we want to know this? Well it turns out that because we can now state every division as really a multiplication that means now that any rule for multiplication is now a rule for division. That's kind of neat. Also, it may be difficult to divide very long, complicated expressions but we can again multiply one divided the reciprocal might be much easier to do. So that is another advantage of doing the re-write.
Step 6: Calculator Keys
The division key, seen here, is used to perform divisions. It is displayed on the calculator screen as a forward slant, like this. Now the reciprocal of a number can also be found using the reciprocal key which is right here. But just to let you know this is rarely used in practice so if you want to find the reciprocal of something just go one divided by the number.
Step 7: Solving The Problem With Your Calculator
I'd like to evaluate this problem using a calculator. How do I go about doing that? What we need to do is slap some parentheses on this thing. OK, what I mean by that is that I need to just put parentheses around the numerator and around the entire denominator. What I don't mean is that you get yourself a set of parentheses and you smack them around a bit. No, I don't condone violence towards mathematical symbols, don't go there, please. So let's take a look at this in detail, using the calculator.
Let's try the problem I just put on the board in the calculator. So put the parentheses around it and type in the following: clear, parentheses, four plus two, close parentheses, divided by, and an open parentheses, five minus four, close parentheses and enter. The answer is six.
Step 8: Another Problem Solved
Alright, let's try another problem. Put this one in the calculator and do the following:
clear, negative three, divided by, now the entire denominator should be placed in another set of parentheses. So, parentheses two, and parentheses again, four plus eight, close parentheses and then close again, another set of parentheses and finally enter. The answer is negative point one two five.
Step 9: Negative Values
Let's say we have some kind of division here, values, and I want to throw in some negatives. So let's say, how about that? If I have a negative "x" what does that equal to? Well, I can actually move this negative down there with the "y", it's the same thing. Or I can put it in the middle like this. This is negative of the entire division. Let's go one step further, let's actually just go nuts and put negatives everywhere, let's put them over here. All what I just did are exactly the same thing. Another thing we can do, and this is a separate problem, if you have a fraction of items and they both have negatives on them what does that tell us? Well remember one of the rules of dividing is that if it is a negative divided by a negative what's the answer? Positive. So you really don't need the negatives at all. So if there is something with two negatives in it, it's positive.
Step 10: The Actual Test
Now, one thing I want to stress is going back to the single negatives for final answer on a test. Let's say you have some kind of fraction as a result and you want to write down the answer, where do we put the negatives? Should it be on the top, should it be on the bottom should it be in the middle, what's the deal? Well they're all correct as I said earlier but traditionally you should avoid negatives in the denominator for a final answer. I mean while you're working though a problem different processes you're going though if there is a negative downstairs, that's fine. But when all is said and done, box or circle your answer, this is my final answer, you don't want this down here. My recommendation move it to the outside of the whole entire fraction. It's very clear it, means I want the negative over the whole, entire thing. And that's the best way to go.
Step 11: A Value Divided By Itself
Now let's say you have some value divided by itself. What does that really equal to? Well, the rule is that anything divided by itself (make sure it is not zero, you can't divide by zero) the result's one. Now one final equation I want to talk about is "x" divide by one which we talked about earlier but let's formalize this the rule is that if you have a value divided by one, what is it equal to? Itself.
