Proof that there is an irrational number between any two rational numbers | Algebra I | Khan Academy

Missed the previous lesson?

Algebra I on Khan Academy: Algebra is the language through which we describe patterns. Think of it as a shorthand, of sorts. As opposed to having to do something over and over again, algebra gives you a simple way to express that repetitive process. It's also seen as a "gatekeeper" subject. Once you achieve an understanding of algebra, the higher-level math subjects become accessible to you. Without it, it's impossible to move forward. It's used by people with lots of different jobs, like carpentry, engineering, and fashion design. In these tutorials, we'll cover a lot of ground. Some of the topics include linear equations, linear inequalities, linear functions, systems of equations, factoring expressions, quadratic expressions, exponents, functions, and ratios.

About Khan Academy: Khan Academy offers practice exercises, instructional videos, and a personalized learning dashboard that empower learners to study at their own pace in and outside of the classroom. We tackle math, science, computer programming, history, art history, economics, and more. Our math missions guide learners from kindergarten to calculus using state-of-the-art, adaptive technology that identifies strengths and learning gaps. We've also partnered with institutions like NASA, The Museum of Modern Art, The California Academy of Sciences, and MIT to offer specialized content.

For free. For everyone. Forever. #YouCanLearnAnything

Subscribe to Khan Academy’s Algebra channel:

[victorserras]

Sal making my real analysis life a bit easier. Thank you!

[kcwilliamson]

I think it's better to use this as proof that there is an irrational between any two real numbers instead of any two rational numbers. It is then a bigger proof that also contains the other.

[ivanlo]

when they ask this in the exam, there is just not much way to pop these things out from scratch, without any hints

[garou]

But how do you define r2 away from r1 if in the field of rational there is no next number, being dense. You have to step out of the field of rational to 'insert' an irrational. In fact you are not inserting it in the field of rational, which remains unchanged, but you are just inserting it in the field of real number...

[edemsauce]

Also there are infinitely many rational number between any two rational numbers.

[YourMJK]

With the same methode you can also prove that there is always a rational number between two rational number (the one in the middle):
r₁ < r₁ + (r₂-r₁)/2 < r₂

This implies that there are infinitly many rational numbers between two rational numbers

[daltongrowley]

if .9 repeating is rational can there be an irrational number between that and 1?

[skysalazar]

Hi so rational and irational = irational And irational and rational = irational ??? Please help

[aabidhussainmir]

Sir thank you very much.
You are the best.


Love from Kashmir India

[juanantoniovaldirimendez]

Same logic apply to rational on irracional numbers right?

[patarthenry]

If we used the square root of 2 as a measure standard then all rational numbers would be irrational when measured.

[swarnendusarkar]

It can be done using Archimedean property

[TikhoStudios]

But r2 and r1 could be negative numbers

[uchihaitachifan]

neat but why. why would you do this were in the universe would this logic of mathematics exist or do anything except here were we give value and language to math.

[corythecreeperplaysmc]

Are any numbers irrational? If there are infinitely many digits, they would have to repeat and some point. I guess we have to change the definition of an irrational number from a number that never ends and never repeat to a number that doesn't equal to any fraction.