Real Number Types

Video Transcription

Hi, and welcome. My name is Anthony Varela. And today in this tutorial, I'd like to talk about real number types. So we're going to talk about different types of real numbers, we'll look into the characteristics of those different number types, and then we're really going to focus on two types of real numbers-- rational numbers and irrational numbers.

So let's get started by defining what a real number is. A real number is a number that can be placed on the number line, and it can be written as a decimal. So the first type that I'm going to talk about we call natural numbers.

Think counting. So what are the numbers that we use to count things? 1, 2, 3, 4, 5, 6. Pretty simple, right? So our natural numbers are numbers that we use to count, such as 1, 2, and 3. Now, notice that 0 is not a natural number, and that's because 0 represents the absence of something, so we can't count it.

So we include 0, however in our whole numbers. So notice from this picture that all natural numbers are whole numbers. Whole numbers, then, just includes 0. So examples of whole numbers, then, would be 0, 1, 2, 3, 4, 5, 6-- so on and so forth.

So then we get into our integers. And notice again that all natural numbers, all whole numbers, are all considered integers. So what is an integer, exactly? Well, an integer includes opposites or negatives. So examples of integers, then, would be negative 2, negative 1, 0, 1, 2-- so on and so forth.

So now we get into our rational numbers. And taking a look at the word, "rational," that helps me remember what a rational number is. I see the word "ratio." So a rational number is a ratio of two integers.

So we can write this as a over b, where both a and b are integers. And I should point out that because b is in the denominator of the fraction, there is one value that b cannot be, and that is 0. But notice, still, that natural numbers, whole numbers, and integers are all considered rational numbers.

Well, there's one other type of real number that I'd like to introduce, and that is the irrational numbers. Now, notice one thing from our picture. Irrational numbers do not include our natural numbers, they don't include our whole numbers, they don't include our integers, they don't include our rational numbers, but they are all within this universe of real numbers.

So the easiest way, I think, to define irrational numbers is to say it's not rational. So an irrational number cannot be written as a ratio of integers, like our other numbers could be. So now let's talk more about rational numbers and irrational numbers and take a look at some other characteristics.

So I know that a rational number is a ratio between two integers, a over b. And so let's make one up. How about 3/8? I know that it's a rational number. It's a ratio of integers 3 and 8.

So writing this out as a decimal, we see that this equals 0.375. And I notice this as being a terminating decimal pattern. And that just means, when you're writing 3/8 as a decimal, you'll eventually stop writing digits. 0.375-- I'm done. So that is a characteristic of some rational numbers, that there is this terminating decimal pattern.

How about this rational number? I know it's rational. It's my ratio of two integers, 2 an 11. But writing this out as a decimal, this equals 0.1818 dot dot dot dot dot, which means that the 1 8 actually goes on and on and on forever. So my decimal pattern isn't terminating, but it's still a rational number.

So some rational numbers are characterized by what we call a repeating decimal pattern. And notice that I can write this as with a horizontal bar going over the digits that repeat forever. So this is a characteristic of some rational numbers, that we have this repeating decimal pattern.

Well, how does this compare, then, with irrational numbers? So a very common and popular irrational number is the number pi. And one thing you'll notice right away is why do I have to use a special symbol to represent the exact value of pi? Why can't I just write that out as a decimal?

Well, we can surely approximate it with a decimal. This would be 3.14159 dot dot dot, which means that there's digits following that. And this is what we call, then, a non-terminating decimal pattern. These digits go on and on and on and on forever, and there's really no discernible repeating pattern, like there are with some rational numbers.

So a characteristic, then, of irrational numbers, is that we have this non-terminating decimal pattern. And there are folks out there who would love to memorize all of these digits of pi. And you can spend the rest of your days writing out all of the decimal digits in pi, because it's a non-terminating decimal pattern.

Well, here's another very common and very popular irrational number-- the square root of 2. Let's see what that looks like as a decimal. 1.4142135 dot dot dot, which means that there are an infinite number of digits following that. And unlike some rational numbers, where we might find a group of decimal digits that repeat, we won't find that in irrational numbers. So there is a non-repeating decimal pattern with irrational numbers as well.

So characteristics of the decimals in irrational numbers is that they are non-terminating. They go on and on and on forever. And they're non-repeating. You're never going to find a set of digits together that repeats.

So now let's practice identifying a number as either rational or irrational. So here I have a set of numbers, and we're going to split them up into two groups-- rational on the left, irrational on the right. So what I remember is that all rational numbers can be expressed as a ratio between two integers-- a over b. So I'm going to look for a ratio.

Well, I see that right here with 5/7. 5 is an integer and 7 is an integer, so I know that 5/7 is a rational number. But let's take a look, just for fun, at what 5/7 looks like as a decimal. This is 0.714285 dot dot dot dot dot forever and ever and ever.

So I might be tempted to say, that's definitely an irrational number, right? Because the decimal pattern is non-terminating. And so far, it looks like it's non-repeating.

However, you can go ahead if you want to and do the long division and write down all these decimal digits, and you'll find out that 714285 repeats. The whole thing does. So 5/7 is 0.714285714285714285.

We have a repeating pattern. So that's kind of cool. Not all lengthy decimal numbers are automatically irrational. Be careful about that.

So I'm going to look for another ratio. I see 3 pi over 2. It's certainly a ratio, right? But 3 pi is not an integer, because pi is not an integer. So although this is a ratio, it's not a ratio of two integers, so I can conclude this is an irrational number.

Well, let's see. Negative 8 I know is an integer, and all integers are rational numbers. So great. I'm going to put that over here. Now I have this decimal number, 32.17. And this terminates, right? I'm stopping at 7 and there's nothing after 7, so I know that this is a rational number. That was a characteristic of rational numbers.

So now I have some square roots. And I remember from our previous example, the square root of 2 was irrational. So I'm going to guess that the square root of 5 is irrational as well. And it certainly is.

You can go ahead and type that into your calculator. Take a look at the decimal pattern. You'll find that it's non-terminating and non-repeating. So I'm tempted to say that's the same thing with the square root of 16. That's got to be irrational, right, too?

Well, be cautious about square roots. Many, many, many square roots are irrational, but not all of them. 16 is a perfect square, which means that the square root of 16 is 4. So the square root of 16, then, is a rational number. Be careful about those radicals.

So let's review our notes. What did we talk about today? We talked about real numbers-- numbers that can be expressed as a decimal and placed on the number line. We talked about our natural numbers. These are our numbers that we use to count. Our whole numbers, which are all of our natural numbers, and we're including 0.

Then we talked about integers. These are our natural numbers and their opposites, including 0. And then we really focused on rational numbers-- numbers which can be represented as a ratio of integers, a over b. And irrational numbers-- numbers which can not be represented as a ratio of integers, a over b.

Some characteristics to remember about rational numbers is you're going to find either a terminating decimal pattern or a repeating decimal pattern. And with irrational numbers, you're going to find both a non-terminating decimal pattern and a non-repeating decimal pattern. Well, thanks for watching this tutorial on the real number types. Hope to catch you next time.