Add new comment
Want facts and want them fast? Our Maths in a minute series explores key mathematical concepts in just a few words.
What do chocolate and mayonnaise have in common? It's maths! Find out how in this podcast featuring engineer Valerie Pinfield.
Is it possible to write unique music with the limited quantity of notes and chords available? We ask musician Oli Freke!
How can maths help to understand the Southern Ocean, a vital component of the Earth's climate system?
Was the mathematical modelling projecting the course of the pandemic too pessimistic, or were the projections justified? Matt Keeling tells our colleagues from SBIDER about the COVID models that fed into public policy.
PhD student Daniel Kreuter tells us about his work on the BloodCounts! project, which uses maths to make optimal use of the billions of blood tests performed every year around the globe.
No. There actually are an infinite number of rationals and irrationals between any two rationals or irrationals.
You claimed that sometimes two irrationals have no rationals between them. That's clearly false. For example, if you picked these two irrationals:
Then you can find a rational between them like this. Just truncate the larger number after the first digit where it differs from the smaller number:
That is guaranteed to be a rational that's between the two irrationals. If you prefer rationals written as fractions, then just write it over a power of ten:
And that fraction is guaranteed to be between the two irrationals. So you can never have zero rationals between them.
In fact, by truncating later and later, you can generate an infinite number of rationals, all of which lie in between the two given irrationals. And clearly, the above procedure works, no matter which two irrationals you choose, as long as they're different and positive. And it works with slight modifications for negatives, too.