Today we are going to talk about real numbers. Real numbers are numbers that can be expressed as a decimal and placed on a number line.
So today we're going to look at five different categories of real numbers. We'll talk about what each of those categories mean and what the numbers within those categories look like, and then we'll do some examples placing numbers into different categories.
So let's go over our five categories of real numbers, and we'll give some examples of numbers that fall into each category.
So our first and smallest set of numbers-- our first category-- are called natural numbers. And natural numbers are the numbers you learn when you first start counting. So 1, 2, 3, and on and on and on.
Then we've got whole numbers. And whole numbers are just our natural numbers and 0. So whole numbers would be 0, 1, 2, 3, and on and on and on.
Then we've got integers. And integers are our natural numbers, the opposite of our natural numbers, and 0. So integers would be negative 2, negative 1, 0, 1, and 2, and on and on in both the positive and the negative direction.
Then we've got rational numbers. And rational numbers are defined as numbers that can be written as a ratio of two integers a and b, where b cannot be equal to 0. So it may look like a fraction where the number in the denominator of the fraction cannot be equal to 0.
So rational numbers would include something like 3/4. 3/4 is the ratio of two integers, 3 and 4. Another example would be 2.6. So even though 2.6 is in decimal form, we could write it as a ratio of two integers, 26 and 10, or it would look like a fraction 26/10.
And our last category of numbers are called irrational numbers. And irrational numbers cannot be written as a ratio of two integers a and b. So probably the most common irrational number is pi. Pi is often approximated with 3.14 but is actually 3.14159 dot dot dot dot dot. It goes on forever. So there are no two integers that we could write that we could use to write the exact number of pi, so pi is considered a irrational number.
Another example would be the square root of 2. Square root of 2 is 1.41 dot dot dot. This also goes on forever. And so there are no two integers that we could use to write the exact value of the square root of 2, so it's considered an irrational number.
So let's look a little bit more between rational and irrational numbers and how we can decide which category it falls into. So let's look a little bit more closely at what distinguishes a rational from an irrational number.
Remember, our rational number will look like a fraction, and in decimal form it will either be a terminating decimal or it will have a repeating pattern to its decimal form. An irrational number cannot be written as a fraction, and its decimal form will both be non-terminating and non-repeating.
So for our first example, we have 1.333 repeating. And 1.333 repeating is going to be a rational number, again because its decimal form has a repeating pattern. We also know that it's rational if we recognize that 1.3 repeating is the same as 4 over 3, or 4/3. So since it can be written as a fraction we also know that it's going to be rational.
Our second example is the square root of 5. The square root of 5 is equal to approximately 2.23606. But this is a decimal that has a non-terminating decimal form, and it also has a non-repeating decimal form. So because of those two things, it's going to be considered irrational.
Our third example is the square root of 25. So here we have another square root, but the square root of 25 is actually just equal to 5. So because it's equal to 5, which can be written as 5/1, we know that it's a rational number because it can be written as a ratio of two integers. So square root of 25 is going to be rational.
So you cannot tell just because a number is a square root whether it's rational or irrational. You have to evaluate the square root to see what it looks like in decimal form.
Our fourth example is 5/2. And 5/2 is a fraction, so we know that it is going to be rational. It can be written as a ratio of two integers.
Our fifth example is negative 3. And negative 3 is also a rational number because it can be written as a ratio of two integers, negative 3/1.
And then our last example is 14.7142857, and this one has a non-terminating decimal form, so it may look like it is an irrational number. However, this number is actually equal to 103/7 as a fraction, so this is in fact a rational number.
And so if you were to plug in 103/7 into your calculator, you would see this decimal expansion. But if you had a calculator with a bigger window such as an online calculator, you were able to see more of its digits. You would see that even though it is non-terminating, it does have a repeating pattern. So this again is going to be a rational number.
So let's go over our key points from today. As usual, make sure you get them in your notes if you don't have them already so you can refer to them later.
The five categories of real numbers are natural, whole, integers, rational, and irrational. Rational numbers can be written as a ratio of two integers, and their decimal form has a terminating or repeating decimal pattern. Irrational numbers cannot be written as a ratio of two integers, and their decimal form has a non-terminating and non-repeating decimal pattern.
So I hope that these key points and examples helped you understand a little bit more about real numbers. Keep using your notes and keep on practicing, and soon you'll be a pro. Thanks for watching.
Natural numbers and their opposites, including zero.
Numbers which can be represented as a ratio of integers, a/b.
Numbers which cannot be represented as a ratio of integers.
Numbers that can be expressed as a decimal and placed on the number line.