Number sets. In math there are different types of numbers and different types of number sets. So in this video we're going to take a look at all of these different types of numbers. And we're going to start with real numbers.
Well, what are real numbers? Real numbers are just about any number that you guys can think of. Zero is a real number, whole numbers, decimals, repeating decimals, negative numbers. terminating decimals fractions square roots these are all examples of real numbers so what wouldn't be a real number infinity is is not a real number and if you look at our chart here real numbers would include all of the different types of numbers that we're about to talk about so let's start with natural numbers also known as counting numbers natural numbers are whole numbers and they're positive and they start with the number one.
If you think about it, when you're counting things, you start, if you were counting these apples, for example, down here, you would start counting by one, two, three. You wouldn't start with the number zero when you were counting. Zero is not a natural number or a counting number.
Now, whole numbers are the next type of number. And they're the exact same thing as counting numbers, except they do include the number 0. So they're all positive numbers, they're all whole numbers, and they start with the number 0. And if you look at our chart here, whole numbers include all of the natural numbers. The next type of number is integer.
And integers are also whole numbers. But they can be positive or negative, so they would be all of the positive whole numbers and their opposites. And integers do include the number 0. So the opposite of 5 is negative 5. Both of these numbers are integers. 6 and negative 6, these are both integers. So they're whole numbers, positive or negative.
Negative 3.2 is not an integer because it's not a whole number. So again, if you look at our chart, integers would include all of the whole numbers and all of the natural numbers. Okay, I'm confused.
If negative 3.2 is not a counting number, and it's not a whole number, and it's not an integer, well, what is it? Negative 3.2 is a rational number, and in 6th grade we pretty much only work with rational numbers. Well, what is a rational number?
Well, as its name implies, rational includes the word ratio. A rational number is any number that could be written or expressed as a fraction, as a ratio. Remember that fractions are a type of ratio, so rational numbers are any number that can be expressed or written as a fraction.
Let's take a look at some examples. 5 is a rational number because it can be written as 5 over 1. It can be written as a fraction. The square root of 9 is a rational number because the square root of 9 would be 3, and 3 can be written as a fraction. The integer negative 8 can also be written as a fraction.
1.2, a decimal, can also be written as a fraction. The next one is a terminating decimal. And it can be written as a fraction, 1, 1000. The repeating decimal, 0.333, you might recognize that as 1 third. And the last example can also be written. Negative 3.2 can also be expressed as a ratio.
Now this is kind of important. If a number, if a decimal terminates or it ends, such as this one, it's a rational number. If a decimal repeats, like 0.3333. Then it's a rational number, 0.9999 would be a rational number, which leads us into irrational numbers.
Irrational means they're not rational. You can't write these or express these as ratios. For example, pi is the ultimate. Irrational number, 3.14159.
It goes on and on forever. It doesn't repeat and it doesn't end. So it doesn't terminate, it doesn't repeat. Therefore, it's irrational. And the square root of 2 is...
also an irrational number. There is no number times itself that would give you two, so it would be an irrational number. So if you look at our chart, you can see irrational numbers here, and these are all rational numbers here. And rational numbers include all the integers, all the whole numbers, and all the natural numbers.
So let's take a look at a couple more examples of irrational numbers. So we mentioned pi, the square root of 2. In fact, the square root of any prime number is an irrational number. These would all be examples of irrational numbers.
And also, the square root of 18, any number like the square root of 10, there is no number times itself that will give you 10. or the square root of 12. These would also be irrational numbers. Now this is, you're actually looking at a Venn diagram, and you'll notice they don't intersect. And the reason being, there is no number that can be both rational and irrational. It's either one or the other. Okay, so just to recap then, so we've looked at all the real numbers.
These are all real numbers. All irrational and rational numbers are real. And a number is either irrational or it's rational. And rational numbers include all the integers, all the whole numbers, all the natural numbers.
Those are all examples of rational numbers. Okay, let's just answer a few questions. Now, 6 is an integer, true or false?
Well, the answer to that would be true. 6 is a whole number. And whole numbers are integers.
Negative 6 would have been an integer. 5, negative 5, those would have been integers. 5.2 would not be an integer because it's a decimal.
So it has to be a whole number. 0 is a counting number. True or false? Well, the answer to that is false.
When you're counting things, you don't start with the number 0. You would start with 1. 0 is a whole number. True. or false?
Well the answer there is true. The difference between whole numbers and counting numbers. Whole numbers include all the counting numbers and zero.
2.3467 is a rational number. True or false? Well the answer there would be true because the decimal terminates.
It ends and any decimal that ends is a rational number. Negative 3.5 is an integer. True.
or false. The answer there is false because it's a decimal. It's not a whole number. Integers are whole numbers. The square root of 7 is an irrational number.
The answer there is true. As we mentioned, the square root of any prime number is an irrational number, or any square root that doesn't come out. There's no number times itself that would give you 10, so those would both be irrational numbers. And our last question, the square root of 1 is an irrational number.
Well, the answer to that is false. Because the square root of 1 is 1, and we can write that as a ratio. We can say 1 over 1, so it would be a rational number.
So those are the different types of numbers and number sets.