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### Course: 8th grade (Illustrative Mathematics)>Unit 8

Lesson 2: Lesson 3: Rational and irrational numbers

# Intro to rational & irrational numbers

Learn the difference between rational and irrational numbers, learn how to identify them, and discover why some of the most famous numbers in mathematics, like Pi and e, are actually irrational. Did you know that there's always an irrational number between any two rational numbers? Created by Sal Khan.

## Want to join the conversation?

• Is Sal saying there are more irrational numbers than rational numbers?
• Wrath,
Actually, Sal was saying that there are an infinite number of irrational numbers. And there is at least one irrational number between any two rational numbers. So there are lots (an infinite number) of both.

And saying one thing that is infinite is more than another infinite thing is questionable because you can't add to infinite. Infinite goes on forever. It is a hard concept to completely comprehend.

For instance, there are an infinite number of decimals between 0 and 1.
And there are an infinite number of decimals between 0 and 2.
And there are numbers between 0 and 2 that are not between 0 and 1. But "goes on forever" so you can't really say there are more numbers between 0 and 2 than between 0 and 1. Both go on forever.

Infinite is a concept of "going on forever" and is not something that can be added to to or multiplied. And you can't say that one infinite is more than another infinite, even though logically you might think there are twice as many numbers between 0 and 2 that there are between 0 and 1. It is also not correct to say that the numbers between 0 and 1 and the numbers between 0 and 2 are the same. One infinite is not greater than, less than, or even equal to, another infinite. Infinite goes on forever.

(Edited)
There are some math concepts that do compare infinities. See the Peter Collingridge comment below. The infinity of irrational numbers is more than the infinity of positive integers. So I was partly incorrect. That is not to say the the infinity of irrational numbers is larger than the infinity of rational numbers.

The concept of infinity is very hard to grasp.
• I'm getting stuck on the irrational number part. An irrational number is any number that doesn't divide into a fraction? Anything that is pi?
• anything that doesn't have a pattern, goes on forever, and has a decimal is basically an irrational number
• Can some one explain what a rational number is i am still confused
• A rational number is a number that can be express as the ratio of two integers. A number that cannot be expressed that way is irrational.
For example, one third in decimal form is 0.33333333333333 (the threes go on forever). However, one third can be express as 1 divided by 3, and since 1 and 3 are both integers, one third is a rational number.

Likewise, any integer can be expressed as the ratio of two integers, thus all integers are rational.

However, numbers like √2 are irrational because it is impossible to express √2 as a ratio of two integers.

The first irrational numbers students encounter are the square roots of numbers that are not perfect squares. The other irrational number elementary students encounter is π.
• Sal had a list of intriguing irrational numbers. What are they, and how can they be applied?

e?
square root of 2?
golden ratio?

Thanks.
• You would probably not need to apply those numbers in Algebra 1 however they are quite useful. For example, "e" is the basis of calculus and appears in a lot of limits and functions. The square root of 2 is the hypotenuse of a right-angled triangle with both sides 1 and can be seen through the exact value of certain trigonometric functions. The golden ratio is a number that people claim is spread all throughout nature and can be seen through many series such as the Fibonacci numbers.
Hope this helps.
• Is a two digit, repeating decimal ( 4 example: 0.12121212...) a rational number
or an irrational number?
• 0.1212... is definitely a rational number. The fraction is 4/33
• pie is an irrational number, which means it cannot be expressed in p/q form, where p and q are integers, but pie = 22/7 pls explain.
• Pi does not equal 22/7. It is just an approximation for Pi. 22/7 is a repeating decimal. Pi is a never ending and never repeating decimal.
22/7 = 3.142857142857...
Pi = 3.1415926535897932384...
As soon as you get to the 3rd decimal digit, the numbers are different.

Hope this helps.
• How is 0.3 = 1/3
• Use the long division method and see for yourself:)
• How do you know what numbers are rational
• A another trait of rational numbers is their decimal form. It will be either a terminating decimal (a decimal with a few decimal places, then stops), or a repeating decimal (a decimal with digits going on forever, but in a pattern so you know what comes next.
Fun Fact: if a number is not divisible by nine, then the repeating digit is what the remainder is, so if the remainder is 4, the decimal goes 44444444444444... forever.
• At about , Sal said that a rational number plus an irrational number equals an irrational number. what about an irrational number plus an irrational number?
• It depends.
Something like Sqrt(2) + Sqrt(3) is still irrational, but obviously something like Pi + (-Pi) = 0 is rational.
• are there any more irrational numbers?
• Thank you you guys make school so easy and I accually home school thanks!

## Video transcript

So let's talk a little bit about rational numbers. And the simple way to think about it is any number that can be represented as the ratio of two integers is a rational number. So for example, any integer is a rational number. 1 can be represented as 1/1 or as negative 2 over negative 2 or as 10,000/10,000. In all of these cases, these are all different representations of the number 1, ratio of two integers. And I obviously can have an infinite number of representations of 1 in this way, the same number over the same number. The number negative 7 could be represented as negative 7/1, or 7 over negative 1, or negative 14 over positive 2. And I could go on, and on, and on, and on. So negative 7 is definitely a rational number. It can be represented as the ratio of two integers. But what about things that are not integers? For example, let us imagine-- oh, I don't know-- 3.75. How can we represent that as the ratio of two integers? Well, 3.75, you could rewrite that as 375/100, which is the same thing as 750/200. Or you could say, hey, 3.75 is the same thing as 3 and 3/4-- so let me write it here-- which is the same thing as-- that's 15/4. 4 times 3 is 12, plus 3 is 15, so you could write this. This is the same thing as 15/4. Or we could write this as negative 30 over negative 8. I just multiplied the numerator and the denominator here by negative 2. But just to be clear, this is clearly rational. I'm giving you multiple examples of how this can be represented as the ratio of two integers. Now, what about repeating decimals? Well, let's take maybe the most famous of the repeating decimals. Let's say you have 0.333, just keeps going on and on forever, which we can denote by putting that little bar on top of the 3. This is 0.3 repeating. And we've seen-- and later we'll show how you can convert any repeating decimal as the ratio of two integers-- this is clearly 1/3. Or maybe you've seen things like 0.6 repeating, which is 2/3. And there's many, many, many other examples of this. And we'll see any repeating decimal, not just one digit repeating. Even if it has a million digits repeating, as long as the pattern starts to repeat itself over and over and over again, you can always represent that as the ratio of two integers. So I know what you're probably thinking. Hey, Sal, you've just included a lot. You've included all of the integers. You've included all of finite non-repeating decimals, and you've also included repeating decimals. What is left? Are there any numbers that are not rational? And you're probably guessing that there are, otherwise people wouldn't have taken the trouble of trying to label these as rational. And it turns out-- as you can imagine-- that actually some of the most famous numbers in all of mathematics are not rational. And we call these numbers irrational numbers. And I've listed there just a few of the most noteworthy examples. Pi-- the ratio of the circumference to the diameter of a circle-- is an irrational number. It never terminates. It goes on and on and on forever, and it never repeats. e, same thing-- never terminates, never repeats. It comes out of continuously compounding interest. It comes out of complex analysis. e shows up all over the place. Square root of 2, irrational number. Phi, the golden ratio, irrational number. So these things that really just pop out of nature, many of these numbers are irrational. Now, you might say, OK, are these irrational? These are just these special kind of numbers. But maybe most numbers are rational, and Sal's just picked out some special cases here. But the important thing to realize is they do seem exotic, and they are exotic in certain ways. But they aren't uncommon. It actually turns out that there is always an irrational number between any two rational numbers. Well, we could go on and on. There's actually an infinite number. But there's at least one, so that gives you an idea that you can't really say that there are fewer irrational numbers than rational numbers. And in a future video, we'll prove that you give me two rational numbers-- rational 1, rational 2-- there's going to be at least one irrational number between those, which is a neat result, because irrational numbers seem to be exotic. Another way to think about it-- I took the square root of 2, but you take the square root of any non-perfect square, you're going to end up with an irrational number. You take the sum of an irrational and a rational number-- and we'll see this later on. We'll prove it to ourselves. The sum of an irrational and a rational is going to be irrational. The product of an irrational and a rational is going to be irrational. So there's a lot, a lot, a lot of irrational numbers out there.