Video3: Real Number System

Now we're gonna ask if 21 over 19 is a natural number. Is 1.105263158 is natural number? No. It is not natural. Is it a whole number? No, it's not. So it's not a whole number. Is it an integer? No, it's not an integer. Is it a real number? Yes. It is a real number. 6 over three. So of course we're going to put that into the calculator 6 goes at the top, the little fraction line. Or the division sign goes here in the middle. So 6 divided by three. So I want to take the calculator and we're going to do 6 divided by three. And I see it's the number two. So is two a natural number. Yes. Two is natural. Is to a whole number? Yes, two is a whole number. Is to an integer. Yes, it is an integer. And it is to a real number. Yes. It is a real number. Then we have pi. So on your calculators, pi is going to look like maybe I can find the pi button on here. Well, first of all, let me show you how the pie button looks. It's a squiggly line with two lines underneath, and sometimes it's a straight line with two lines underneath. So let's see, we can find the pi button on here. And I found it. This is the pi button. So let me hit clear. And then I'm going to hit the pi button and see what comes up. And I'm going to hit equals 3.141592 6 5 four, okay? This is called pi. Let's see, is this number a natural number? No. Pi is not natural. Is it a whole number? Is not, it's not a whole number. Is it an integer? Nope. Not an integer. Is it a real number? Yes, it is. A real number. Now, all of these vocabulary words that we learned so far natural whole integer and real. We did not include fractions and we did not include decimals. Now, we have to figure out a word that does include fractions and decibels. And that word is rational number. So now we have a word that describes the fractions and the decimals, a rational number, still includes our positive numbers. Still includes our negative numbers. Our positive whole numbers are negative numbers. Now, we have fractions that repeat, have a nice pattern. We have fractions at terminates. The term needs to mean stops or ends. We have decimals that repeat. Now we have decimals that terminates terminates means stops. The numbers do not go all the way across to calculator. They stop. Okay? So now let's look at our numbers to see the same numbers if they are rational. Are they fractions and decimals that repeat or terminate? And are they our positive and negative numbers? So let's go back to this side. Let's look at negative 8. Is negative 8 a rational number? Yes, it is rational because it's a negative number. Rational numbers include negative numbers. So I'm going to go to the end and put yes. It is rational. Is one over 8 a rational number. I don't know. Let me divide it again and see if it's a fraction that stops or has a nice repeating pattern. Yes, it does. It doesn't have all kinds of numbers that go all the way across the calculator. It stops at 0.125 stops. That means yes, it is rational. So I'm going to go here to the end and put yes. Rational. What about 8 over one? Let's take the calculator and make sure we do 8 divided by one again. We see that as the number 8, which is a whole number. So rational numbers, yes. It is rational. 0.8. Yes, it's a decimal that ends terminates. It just has a decimal and an 8, and then it stops. It doesn't have any more numbers. So yes. It is rational. What about square root 8? So now we're going to take the cavalier again. I'm going to hit this second button. Square root 8 no matter which is sometimes you can close the parentheses. In this case, it only has one number. You don't have to worry about it. Just hit equals. 2.828427 one two 5. It has numbers that go all the way across the calculator. There's no pattern. There's no numbers ever repeat. If it had 2.828828 8 two 8, that's a nice repeating pattern, it doesn't. This is like saying, I'm going to wake up at 2 o'clock in the morning, then 8 o'clock in the morning, 2 o'clock in the morning to talk about 4 o'clock at morning 2 o'clock 7 o'clock in the morning. This is not rational. This is a bunch of crazy numbers all over the place. So is square root 8 rational no. So instead of saying not rational, or swear, we're going to say it is irrational. That's how I and R. Irrational means not. Rational. If it has a bunch of crazy numbers, it's irrational. I don't want to say crazy. Numbers. All the way across the calculator. You just called irrational. All right, let's look at 8 squared. Remember, 8 squared just means 8 times itself 8 times 8. 64. Is 64. Let's see. Let me go back to square root two 25. So I will do the square root of two 25 equals 15. Is 15 rational? Yes. It is rational. I've already done 8 squared, which means 64 is 64 rational. Yes. It also is rational. 21 over 19. Let's try that again 21. Divided by 19. Again, we see a whole bunch of crazy numbers across the calculator. 1.105263 one 5 8. Now, this is not rational. It's a bunch of crazy mixed up numbers going across the calculator. So it is ear rational. That means not rational. 6 divided by three. 6 divided by three. I see is two, two is a nice rational number. So yes, it is rational. And then we have pi. So let me hit pi again. Which is right here. Equals 3.141592 6 5 four no. There's no nice repeating patterns. So pi is also ear. Rational. I'm gonna give you another example before you go. If you have one divided by three. You see how this is a nice repeating number, 0.33333. This decimal, even though it goes all the way across to calculator, it has a nice repeating pattern. It is rational.