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Lesson 3: Irrational numbers

# Classifying numbers: rational & irrational

Given a bunch of numbers, learn how to tell which are rational and which are irrational. Created by Sal Khan.

## Want to join the conversation?

• is there such a thing called 'fake' numbers?
• I suspect you mean "fake" in that there are other numbers that are "real".

As Mr. Mark pointed out, there are imaginary numbers, but don't read anything into the name "imaginary", like that they are not useful because they are somehow "made-up". Imaginary numbers are super powerful and useful - they allow us to extend the 1 dimensional real number line into the two dimensional complex number plane, and with that we can solve problems that we can't with just the real numbers alone.

Many disciplines use complex numbers, but perhaps the one that affects you, me, and pretty well everyone on a daily basis is electronic engineering. Without complex numbers, the quantum analysis of transistor development would not be possible, meaning pretty much every electronic device you own would not exist.

Now what we call the real numbers weren't always called the real numbers. Mathematicians only started to call them real when the concept of the imaginary number was introduced. At that time, most mathematicians poo-poo-ed the idea of the properties of these new numbers (the square root of negative one? Oh no-no-no-no-no!) so they called them "imaginary" as an insult, and that they only worked with REAL numbers. Well, it did not take long before the merits of imaginary numbers became apparent, but sadly the name did not change. I think it is sad because now, when students first hear of and begin to learn how to use these numbers, a sort of barrier is made in the students mind because at some level they think that these abstract imaginations must be more difficult - and since it is human nature to resist the difficult, shazzam! - the student makes it difficult in their own mind. Imaginary numbers are not at all difficult, just a wee bit different, so, when you get to them, worry not! Onward ho!
• Is infinity rational or irrational?
• Infinity is neither rational nor irrational. Rather, it's an abstract concept that we use in math. It doesn't have a numerical value; it just represents something that is larger than any number. So while we can represent a rational number (like 100) or an irrational number like pi, we cannot do the same for infinity. Thus, infinity can't be classified as either rational or irrational.
• I can divide an irrational number by 1, that's going to give me the same number, why isn't it rational?
• Because a rational number is a number than can be expressed as the fraction of two integers, not just any two numbers. 1 is an integer, of course, but the irrational number you are dividing by one most surely isn't.

(Good question though..!)
• Where do you get the 750 and so on? How do you solve ratios an easier way that has fractions? What is the formula?
• I didn’t really understand your question. If your looking for a way to identify rationals and irrationals: a rational number is a number that can be expressed as an integer by an integer. Any operation between irrational and rational will give an irrational number(unless the rational is zero). But don’t forget PEMDAS(Parentheses, Exponents, Multiplication/Division, Addition/Subtraction)
• Can somebody please tell me a list of what can be a rational number? I feel I am sort of getting it, but I am still a bit rough in some parts.
• Rational numbers are all numbers that can be written as the ratio (or fraction) of 2 integers. This is the basic definition of a rational number. Here are examples of rational numbers:
-- All integers. Numbers like 0, 1, 2, 3, 4, .. etc. And like -1, -2, -3, -4, ... etc.
-- All terminating decimals. For example: 0.25; 5.142; etc.
-- All repeating decimals. For example: 0.33333... where 3 repeats forever. Or 2.45454545... where the 45 repeats forever
-- All fractions where each number is an integer, like: 5/4; 42/113; etc.

Hope this helps.
• Sal is saying √8/2 is irrational but, if you divide 8 by 2 you get 4 and √4 = 2
So how's it an irrational number? Or is it a rule that you can't divide first?
• Order of Operations (Parentheses, Exponents, Multiplication/Division, Addition/Subtraction = PEMDAS) states you need to "take care of" exponents prior to dividing. So since √8 is the same as 8^(1/2) 8 has an exponent (other than 1) on it. You need to take care of that before you divide.

Hope that helps.
• why do we need rational and irrational numbers for real?
• Pi (3.14159...) is a very common irrational number. Pi is necessary to find areas of many shapes. Also, right triangles involve irrational numbers. Right triangles are important to make sure buildings are safe, cars protect their occupants in crashes, and people can travel great distances.
• what does rational and irrational numbers mean please be specific and keep it simple
• Rational numbers are numbers that can be expressed as a fraction or part of a whole number.
(examples: -7, 2/3, 3.75)

Irrational numbers are numbers that cannot be expressed as a fraction or ratio of two integers. There is no finite way to express them. (examples: √2, π, e)
• How do I know if a square root is a perfect square or not? Thanks
• Well, the basic logic to square every number starting from one until you reach the number. If the number is not super big, you can just try squaring some numbers and from 1 to 20, you can just memorize it.
• Could you multiply 2 irrational numbers together and get a rational number as your answer?

For example: √5 * √5.
• Yes, your example speaks for itself because its answer is 5 which is rational.

## Video transcript

Which of the following real numbers are irrational? Well, irrational just means it's not rational. It means that you cannot express it as the ratio of two integers. So let's see what we have here. So we have the square root of 8 over 2. If you take the square root of a number that is not a perfect square, it is going to be irrational. And then if you just take that irrational number and you multiply it, and you divide it by any other numbers, you're still going to get an irrational number. So square root of 8 is irrational. You divide that by 2, it is still irrational. So this is not rational. Or in other words, I'm saying it is irrational. Now, you have pi, 3.14159-- it just keeps going on and on and on forever without ever repeating. So this is irrational, probably the most famous of all of the irrational numbers. 5.0-- well, I can represent 5.0 as 5/1. So 5.0 is rational. It is not irrational. 0.325-- well, this is the same thing as 325/1000. So I can clearly represent it as a ratio of integers. So this is rational. Just as I could represent 5.0 as 5/1, both of these are rational. They are not irrational. Here I have 7.777777, and it just keeps going on and on and on forever. And the way we denote that, you could just say these dots that say that the 7's keep going. Or you could say 7.7. And this line shows that the 7 part, the second 7, just keeps repeating on forever. Now, if you have a repeating decimal-- in other videos, we'll actually convert them into fractions-- but a repeating decimal can be represented as a ratio of two integers. Just as 1/3 is equal to 0.333 on and on and on. Or I could say it like this. I could say 3 repeating. We can also do the same thing for that. I won't do it here, but this is rational. So it's not irrational. 8 and 1/2? Well, that's the same thing. 8 and 1/2 is the same thing as 17/2. So it's clearly rational. So the only two irrational numbers are the first two right over here.