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Worked example: rational vs. irrational expressions

Sal shows how to determine whether the following expressions are rational or irrational: 9 + √(45), √(45)/ (3*√(5)), and 3*√(9). Created by Sal Khan.

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  • mr pants teal style avatar for user Wrath Of Academy
    How do we know that an integer plus an irrational number yields an irrational number? Is there another video on that?
    (15 votes)
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    • leafers ultimate style avatar for user nathan
      We'll do a proof by contradiction. This just means that we show that the false to our statement presents a contradiction.
      First, let us assume that an irrational number plus a rational number makes a rational number and make this lead to a contradiction.

      If a is rational, b is irrational, and c is rational, we will try to prove that:

      a + b = c


      is rational. If this is true, a = x/y and c = e/f for integers x, y, e, and f. So:

      a + b = c 
      x/y + b = e/f
      b = e/f - x/y
      b = ey/(fy) - xf/(fy)
      b = (ey - xf)/(fy)

      Since the right hand side of the equation is rational, then so is b. But we said that b is irrational! This leads to a contradiction and so the sum must be irrational. Let me know if you need anything clarified.
      (42 votes)
  • hopper jumping style avatar for user Sam D
    Sal cancelled out 3√5/3√5 to get 1. But the order of operations, PEMDAS states that we do the powers before division. So, what happened here? Can anyone please explain me. Thanks!
    Sam D
    (10 votes)
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    • mr pink red style avatar for user andrewp18
      It doesn't matter because since the numerator and denominator are the same, even if you did use PEMDAS to approximate √5 AND THEN divided them out, you would still get 1. And besides, when we have the square root of a non-perfect square, we leave the answer in radical form (not decimal form), because the decimal form goes on forever like the digits in π. This is because it is irrational. Comment if you have any questions.
      (13 votes)
  • leafers sapling style avatar for user Calla Jones
    Do negative square roots exist?
    (5 votes)
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    • male robot donald style avatar for user David
      @CallaJones
      By definition, the square root of a negative number does not exist. it instead is called an imaginary number or complex number.

      Originally there were only positive integers but over time the concepts of fractions, zero, decimals, negative numbers, irrational numbers, and then certain transcendental numbers (pi or e) were developed to make the number system complete. Leonard Euler invented the idea that we can represent sqrt(-1) with an imaginary number called "i".


      For example, the square root of -16 can be expressed as 4i.
      sqrt(-16) = sqrt(16) x sqrt(-1) = 4i
      (17 votes)
  • blobby green style avatar for user Janno Clay
    Pi is an irrational number and is the ratio of the circumference (c) over the diameter (d), therefore c/d = Pi. Does this mean that either c or d or both must be irrational or can the quotient of two rational numbers be irrational?
    (8 votes)
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  • duskpin ultimate style avatar for user Thunder Lambright
    Would you consider Infinity Rational or Irrational?
    (4 votes)
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    • stelly blue style avatar for user Kim Seidel
      Infinity is not a number. It is the concept that there is no largest number. If you think you have found the largest number, you can add 1 and get a still larger number.
      Since infinity is not a number, it is not classified as rational or irrational.
      (6 votes)
  • leafers ultimate style avatar for user cutiecritter412
    Hmm. Could one multiply irrational numbers to get rational numbers? her it an example.

    sqrt(3)*sqrt(3)
    sqrt(3*3)
    sqrt(9)
    3
    3/1
    (7 votes)
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  • male robot donald style avatar for user mansi rawat
    how do you find square root of 2 ? i know it is irrational but like i want to know the method in detail.
    (3 votes)
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  • piceratops ultimate style avatar for user Mark Bloggs
    Hi guys,

    Could anybody help with a question I encountered that I find quite confusing.
    "Write as a single fraction . . .
    [SQRT(x)] + 1/SQRT(x) . . . . "
    Two seperate terms, SQRT(x) and 1/SQRT(x).
    I understand many of the rules but I can't get my head around this one.
    Thanks for your time guys.
    (2 votes)
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  • scuttlebug blue style avatar for user The Travelling Twit
    What about the question (√2 - √3) ^ 2 ?

    Write in Racial Form?

    I know that it is not very relevant but there is nowhere else to ask it!

    This this the activity that I am stuck on: https://www.khanacademy.org/math/in-in-grade-9-ncert/xfd53e0255cd302f8:number-systems/xfd53e0255cd302f8:simplifying-expressions/e/multiplying-and-dividing-irrational-numbers?modal=1
    (1 vote)
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    • stelly blue style avatar for user Kim Seidel
      Radical form means you use the radical symbol where needed rather than exponential form.
      -- Radical form: √2
      -- Exponential form: 2^(1/2)

      How to do: (√2 - √3)^2
      Did you use the hints?
      You need to multiply 2 binomials, which means you use FOIL. (a-b)^2 = (a-b)(a-b) = a^2-ab-ab+b^2 = a^2-2ab+b^2

      Remember to simplify the radicals. For example: √2√2 = √4 = 2

      Give it a try. Comment back with questions.
      (4 votes)
  • aqualine ultimate style avatar for user DeltaWolf (Caden)
    I want an answer without having to go through the video. what is rational and irrational. please make it simple. happy st patrick's day!
    (1 vote)
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    • starky ultimate style avatar for user Patrick
      A rational number is any number that can be defined as a ratio, is a terminating decimal (a number that eventually runs out of nonzero decimals), and is a repeating decimal (a number that has an infinite amount of decimal places, but can be written as a ratio or fraction). An irrational number is a number that isn't described as one of the descriptions above, i.e. pi, or the square root of 15. They aren't rational because neither pi nor sqrt(15) can be determined as a ratio, and, when they are in decimal form, go on forever, but you don't know what the next number will be.
      (4 votes)

Video transcript

Let's think about whether each of these expressions produce rational or irrational numbers. And just as a reminder, a rational number is one-- so if you have a rational number x, it can be expressed as the ratio of two integers, m and n. And if you have an irrational number, this cannot happen. So let's think about each of these. So 9 is clearly a rational number. You can express 9 as 9/1, 18/2, or 27/3. So it can clearly be expressed as the ratio of two integers. But what about the square root of 45? So let's think about that a little bit. Square root of 45. That's the same thing as the square root of 9 times 5, which is the same thing as the square root of 9 times the square root of 5. The principal root of 9 is 3, so it's 3 times the square root of 5. So this is going to be 9 plus 3 times the square root of 5. So the square root of 5 is irrational. You're taking the square root of a non-perfect square right over here. Irrational. 3 is rational, but the product of a rational and an irrational is still going to be irrational. So that's going to be irrational. And then you're taking an irrational number and you're adding 9 to it. You're adding a rational number to it. But you add a rational to an irrational, and you're still going to have an irrational. So this whole thing is irrational. Now let's think about this expression right over here. Well, the numerator can be rewritten as the square root of 9 times 5 over 3 times the square root of 5. Well, that's the same thing as the square root of 9 times the square root of 5 over 3 times the square root of 5. Well, that's the same thing as 3 times the square root of 5 over 3 times the square root of 5. Well, that's just going to be equal to 1. Or you could view it as 1/1. And 1 is clearly a rational number. You could write it as 1/1, 2/2, 3/3, really any integer over itself. So this is going to be rational. Now, let's do this last expression right over here. 3 times the principal root of 9. Well, what's the principal root of 9? Well, it's 3. So this is going to be 3 times 3, which is equal to 9. And we've already talked about the fact that 9 can clearly be expressed as the ratio of two integers-- 9/1, 27/3, 45/5, all different ways of expressing 9.