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### Course: Algebra 1>Unit 15

Lesson 2: Sums and products of rational and irrational numbers

# Proof: product of rational & irrational is irrational

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.

## Want to join the conversation?

• At 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 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.
• 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.
• 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.
• Couldn't m/n divided by a/b equal a rational number, x?
• in this what is that dot for?
• 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.
• Why do we have to assume?
Isn't that true?
Why do we only use variables while proving?