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Classifying numbers: rational & irrational

We can write any rational number as the ratio of two integers. We cannot write irrational numbers, such as the square root of 8 and pi, in this way. Learn other forms, such as decimals, in which these types of numbers can appear. Created by Sal Khan.

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  • leaf yellow style avatar for user ayano.chan
    is there such a thing called 'fake' numbers?
    (31 votes)
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    • leaf blue style avatar for user Stefen
      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!
      (54 votes)
  • hopper cool style avatar for user surinder khan
    Is infinity rational or irrational?
    (20 votes)
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    • blobby green style avatar for user Richard He
      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.
      (46 votes)
  • spunky sam red style avatar for user Ángel Venegas
    I can divide an irrational number by 1, that's going to give me the same number, why isn't it rational?
    (15 votes)
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    • leaf green style avatar for user Tjeerd Soms
      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..!)
      (30 votes)
  • blobby green style avatar for user Dani Berger
    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.
    (9 votes)
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    • stelly blue style avatar for user Kim Seidel
      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.
      (28 votes)
  • leaf green style avatar for user Sophia Nyquist
    Where do you get the 750 and so on? How do you solve ratios an easier way that has fractions? What is the formula?
    (18 votes)
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    • piceratops ultimate style avatar for user 1140858
      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)
      (4 votes)
  • blobby blue style avatar for user Jellybean
    This makes absolutely no sense to me. Help me, somebody! Halp!
    (8 votes)
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    • blobby blue style avatar for user RainbowSprinkles🏳️‍🌈
      OK, let's start from the beginning. :D
      We're told that "an irrational number is a number that cannot be expressed as a ratio of two integers."
      So what this means is, it's a number that you can't express as a generic fraction with two integers (whole numbers, including negative numbers and zero). Obviously, this means all rational numbers can.

      So we can say "0.5 is rational because we can express it as 1/2, and 3 is also rational because we can express it as 3/1."

      But, to use the example from the video, pi can't be written as a/b (with an and b being integers). In fact, when it's written as a decimal, it goes on and on with no pattern to it: 3.14159...

      That's the second way to determine if a number is rational or not: does the decimal terminate or repeat eventually. All this means is "does it eventually stop" (like, say, 26.62986413 is long but it does end) and "is there a pattern to the decimal" (like 9.191919... or 7.7777...)

      So, a number with a decimal that goes on forever randomly, it's irrational.
      If you have a number where the decimal stops and/or repeats itself, then it's rational.

      I hope this helped! ^^
      (18 votes)
  • blobby green style avatar for user Pianoman🎹
    Just curious to know, what if you write the pi as pie over pie(pi/pi) like a fraction? Would it become a rational number then?!
    (9 votes)
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  • hopper jumping style avatar for user Sam D
    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?
    (8 votes)
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    • piceratops ultimate style avatar for user rmeissner
      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.
      (14 votes)
  • hopper cool style avatar for user 19zreed
    why do we need rational and irrational numbers for real?
    (5 votes)
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    • male robot hal style avatar for user cbruns
      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.
      (9 votes)
  • aqualine ultimate style avatar for user BookQueen13
    Does anyone have a simple explanation on how to figure out whether or not its rational or irrational. I'm having a hard time understanding. Thank you!
    (5 votes)
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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.