At 1:44 it is said that you should multiply both sides by the reciprocal. Reciprocals only exist for rational numbers that are nonzero. What would happen if the rational number happens to be zero? A very subtle counterexample. Can anyone see how to reword the statement to make it true?

There's now a correction at the beginning of this video indicating that it only works for non-zero rationals. (At 1:55 the division by `a` would be invalid if `a` were zero.) Zero times an irrational is of course the rational zero again.

So what is an irrational number times another irrational number?

A irrational number times another irrational number can be irrational or rational. For example, √2 is irrational. But: √2 • √2 = 2 Which is rational. Likewise, π and 1/π are both irrational but: π • (1/π) = 1 Which is rational. However, an irrational number times another irrational number can also be irrational: √2 • √3 = √6 Which is irrational. Comment if you have questions.

an irratinal number can also be expresed as irrational number/1 . so i am confused. is my question valid? because it becomes a ratio of 2 numbers.

You do not have to stop there, you could divide an irrational by any whole number, √/2/2 and √3/3 are common ones you will see in Math. However, the division of a irrational by a rational will still result in an irrational number. The question is valid, but the answer is not the one you thought. You can divide an irrational by itself to get a rational number (5π/π) because anything divided by itself (except 0) is 1 including irrational numbers. The issue is that a rational number is one that can be expressed as the ratio of two integers, and an irrational number is not an integer.

What about irrational times irrational?

√2 and √3 are both irrational. √2•√2=2, which is rational. √2•√3=√6, which is irrational. So a product of two irrationals can be either rational or irrational.

1:28 in this what is that dot for?

The dot is the same as a multiplication symbol. Check this out. https://www.khanacademy.org/math/algebra/introduction-to-algebra/modal/v/why-aren-t-we-using-the-multiplication-sign

Why do we have to assume? Isn't that true? Why do we only use variables while proving?

We assume because we do not know that it is true. In this case, it is not true. By assuming the contrary, if there is a way to contradict that, then we know it is not true. We use variables when proving so we can generalize the proof.

What do you mean? If you meant "1+0!=?" instead of "1+!=?" then your answer would follow: 1+ (0!) = ? 1 + (1) = ? ? = 2

if pi = 22/7 then why pi is considered an irrational numaber?

22/7 is a close approximation of pi which can be useful for some calculations, but it does not equal pi.

But 22/7 is irrational and is Pi, but Pi times 7 gives rational so irrational time rational can be rational?

22/7 is not irrational, because you just wrote it down as 22/7. An irrational number cannot be written as a fraction of two integers. Additionally, 7pi is not rational either.

Main content

Algebra 1

Course: Algebra 1 > Unit 15

Lesson 2: Sums and products of rational and irrational numbers

Proof: product of rational & irrational is irrational

Name: Proof: product of rational & irrational is irrational
Uploaded: 2013-12-24T01:50:44Z
Description: The product of any rational number and any irrational number will always be an irrational number. This allows us to quickly conclude that 3π is irrational.

Google Classroom

The product of any rational number and any irrational number will always be an irrational number. This allows us to quickly conclude that 3π is irrational. Created by Sal Khan.

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J Helston
Posted 10 years ago. Direct link to J Helston's post “At 1:44 it is said that y...”
At
1:44
it is said that you should multiply both sides by the reciprocal. Reciprocals only exist for rational numbers that are nonzero. What would happen if the rational number happens to be zero?

A very subtle counterexample.

Can anyone see how to reword the statement to make it true?
Button navigates to signup pageComment on J Helston's post “At 1:44 it is said that y...”
(29 votes)
Answer
- Wrath Of Academy
  Posted 10 years ago. Direct link to Wrath Of Academy's post “There's now a correction ...”
  There's now a correction at the beginning of this video indicating that it only works for non-zero rationals. (At
  1:55
  the division by a would be invalid if a were zero.) Zero times an irrational is of course the rational zero again.
  Comment on Wrath Of Academy's post “There's now a correction ...”
  (10 votes)
x=[-b±√(b^2-4ac)]/2a
Posted 8 years ago. Direct link to x=[-b±√(b^2-4ac)]/2a's post “So what is an irrational ...”
So what is an irrational number times another irrational number?
Button navigates to signup pageComment on x=[-b±√(b^2-4ac)]/2a's post “So what is an irrational ...”
(7 votes)
Answer
- andrewp18
  Posted 8 years ago. Direct link to andrewp18's post “A irrational number times...”
  A irrational number times another irrational number can be irrational or rational. For example, √2 is irrational. But:
  √2 • √2 = 2
  Which is rational. Likewise, π and 1/π are both irrational but:
  π • (1/π) = 1
  Which is rational.
  However, an irrational number times another irrational number can also be irrational:
  √2 • √3 = √6
  Which is irrational.
  Comment if you have questions.
  Button navigates to signup page
  (28 votes)
maaz
Posted 7 years ago. Direct link to maaz's post “an irratinal number can a...”
an irratinal number can also be expresed as irrational number/1 . so i am confused. is my question valid? because it becomes a ratio of 2 numbers.
Button navigates to signup pageButton navigates to signup page
(4 votes)
Answer
- David Severin
  Posted 7 years ago. Direct link to David Severin's post “You do not have to stop t...”
  You do not have to stop there, you could divide an irrational by any whole number, √/2/2 and √3/3 are common ones you will see in Math. However, the division of a irrational by a rational will still result in an irrational number. The question is valid, but the answer is not the one you thought. You can divide an irrational by itself to get a rational number (5π/π) because anything divided by itself (except 0) is 1 including irrational numbers.
  The issue is that a rational number is one that can be expressed as the ratio of two integers, and an irrational number is not an integer.
  Button navigates to signup page
  (7 votes)
MrLogic642
Posted 6 years ago. Direct link to MrLogic642's post “What about irrational tim...”
What about irrational times irrational?
Button navigates to signup pageButton navigates to signup page
(4 votes)
Answer
- kubleeka
  Posted 6 years ago. Direct link to kubleeka's post “√2 and √3 are both irrati...”
  √2 and √3 are both irrational.
  √2•√2=2, which is rational.
  √2•√3=√6, which is irrational.
  
  So a product of two irrationals can be either rational or irrational.
  Button navigates to signup page
  (6 votes)
rahul chhonkar
Posted 6 years ago. Direct link to rahul chhonkar's post “if pi = 22/7 then why pi ...”
if pi = 22/7 then why pi is considered an irrational numaber?
Button navigates to signup pageComment on rahul chhonkar's post “if pi = 22/7 then why pi ...”
(2 votes)
Answer
- Ian Goodwin
  Posted 6 years ago. Direct link to Ian Goodwin's post “22/7 is a close approxima...”
  22/7 is a close approximation of pi which can be useful for some calculations, but it does not equal pi.
  Comment on Ian Goodwin's post “22/7 is a close approxima...”
  (9 votes)
tenzin zomkyi
Posted 6 years ago. Direct link to tenzin zomkyi's post “1:28 in this what is that...”
1:28
in this what is that dot for?
Button navigates to signup pageButton navigates to signup page
(3 votes)
Answer
- Darth Vader
  Posted 6 years ago. Direct link to Darth Vader's post “The dot is the same as a ...”
  The dot is the same as a multiplication symbol. Check this out.
  https://www.khanacademy.org/math/algebra/introduction-to-algebra/modal/v/why-aren-t-we-using-the-multiplication-sign
  Button navigates to signup page
  (4 votes)
Kimberly
Posted 6 years ago. Direct link to Kimberly's post “Couldn't m/n divided by a...”
Couldn't m/n divided by a/b equal a rational number, x?
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(4 votes)
Answer
Jake McCulloch
Posted 7 years ago. Direct link to Jake McCulloch's post “But 22/7 is irrational an...”
But 22/7 is irrational and is Pi, but Pi times 7 gives rational so irrational time rational can be rational?
Button navigates to signup pageComment on Jake McCulloch's post “But 22/7 is irrational an...”
(2 votes)
Answer
- Fai
  Posted 7 years ago. Direct link to Fai's post “22/7 is not irrational, b...”
  22/7 is not irrational, because you just wrote it down as 22/7. An irrational number cannot be written as a fraction of two integers.
  
  Additionally, 7pi is not rational either.
  Comment on Fai's post “22/7 is not irrational, b...”
  (4 votes)
😎 Tommy 😎
Posted 6 years ago. Direct link to 😎 Tommy 😎's post “Why do we have to assume?...”
Why do we have to assume?
Isn't that true?
Why do we only use variables while proving?
Button navigates to signup pageButton navigates to signup page
(3 votes)
Answer
- Nobel Zhou
  Posted 6 years ago. Direct link to Nobel Zhou's post “We assume because we do n...”
  We assume because we do not know that it is true. In this case, it is not true. By assuming the contrary, if there is a way to contradict that, then we know it is not true. We use variables when proving so we can generalize the proof.
  Button navigates to signup page
  (2 votes)
LauraP
Posted 4 months ago. Direct link to LauraP's post “1+!=?”
1+!=?
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(3 votes)
Answer
- Ayush_K.
  Posted 2 months ago. Direct link to Ayush_K.'s post “What do you mean? If yo...”
  What do you mean?
  
  If you meant "1+0!=?" instead of "1+!=?" then your answer would follow:
  
  1+ (0!) = ?
  
  1 + (1) = ?
  
  ? = 2
  Comment on Ayush_K.'s post “What do you mean? If yo...”
  (2 votes)

Video transcript

What I want to do with this video is do a quick proof that if we take a rational number, and we multiply it times an irrational number, that this is going to give us an irrational number. And I encourage you to actually pause the video and try to think if you can prove this on your own. And I'll give you a hint. You can prove it by a proof through contradiction. Assume that a rational times an irrational gets you a rational number, and then see by manipulating it, whether you can establish that all of a sudden this irrational number must somehow be rational. So I'm assuming you've given a go at it. So let's think about it a little bit. I said we will do it through a proof by contradiction. So let's just assume that a rational times an irrational gives us a rational number. So let's say that this-- to represent this rational right over here, let's represent it as the ratio of two integers, a over b. And then this irrational number, I'll just call that x. So we're saying a/b times x can get us some rational number. So let's call that m/n. Let's call this equaling m/n. So I'm assuming that a rational number, which can be expressed as the ratio of two integers, times an irrational number can get me another rational number. So let's see if we can set up some form of contradiction here. Let's solve for the irrational number. The best way to solve is to multiply both sides times the reciprocal of this number right over here. So this, let's multiply times b/a, times b/a. And what are we left with? We get our irrational number x being equal to m times b. Or we could just write that as mb/na. So why is this interesting? Well, m is an integer, b is an integer, so this whole numerator is an integer. And then this whole denominator is some integer. So right over here, I have a ratio of two integers. So I've just expressed what we assumed to be an irrational number, I've just it expressed it as the ratio of two integers. So now we have x must be rational. And that is our contradiction, because we assumed that x is irrational. And so therefore, since this assumption leads to this contradiction right over here, this assumption must be false. It must be that a rational times an irrational is irrational.