Main content
Algebra 1
Unit 15: Lesson 2
Sums and products of rational and irrational numbers- Proof: sum & product of two rationals is rational
- Proof: product of rational & irrational is irrational
- Proof: sum of rational & irrational is irrational
- Sums and products of irrational numbers
- Worked example: rational vs. irrational expressions
- Worked example: rational vs. irrational expressions (unknowns)
- Rational vs. irrational expressions
© 2023 Khan AcademyTerms of usePrivacy PolicyCookie Notice
Worked example: rational vs. irrational expressions
CCSS.Math:
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.
Want to join the conversation?
How do we know that an integer plus an irrational number yields an irrational number? Is there another video on that?(13 votes)
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.(40 votes)
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(7 votes)
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.(12 votes)
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?(6 votes)
Do negative square roots exist?(4 votes)
@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(13 votes)
Would you consider Infinity Rational or Irrational?(3 votes)
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.(5 votes)
- 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(5 votes)
Yes you did it!(1 vote)
- 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)
There is a long division method to find out square roots of numbers up to any decimal place you want. You can search it up.(2 votes)
- 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)
find a common denominator. You can multiply anything by 1 will be equal to the same thing. Keep in mind that 1 can be 432tvx^2/432tvx^2
or anything where the topand bottom are equal(2 votes)
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)- 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)
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)
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.