NOTE: I’m going to sacrifice some mathematical precision in some of this, but I think that will be a little better for our purposes. If you have questions, ask.
Fascinating Facts about Fractions between Fractions:
- One can always find a fraction between any two other fractions. (Indeed, if you add the two fractions and divide by 2, the new fraction will be exactly halfway between!)
- Taking this a step further, it can be seen that one can always find any number of fractions between two other fractions.
- As you might expect, this property has a name for math-types. We say that fractions (rationals) are dense. Note, then that whole numbers are NOT dense. (You can’t find a whole number between 2 & 3, e.g.)
- There’s another mind-blowing fact that follows from the above items. When it comes to fractions, there is no meaning to the phrase ‘next to’ ! [What is the closest fraction ‘next to’ 1/2 (or 1/3, or 2, or even 0)? THERE ISN’T ONE! What’s the smallest positive fraction (i.e. next to 0)? THERE ISN’T ONE!]
- There are several options for actually finding fractions between fractions when it’s not evident (what’s a fraction between 3/997 and 4/997, e.g.?) [I know, I know: Why would one want to? But play along, OK?] You could always do the old common denominator trick, adjust a little if needed and find one, but that’s could be a pain (not to mention traumatic), eh? It turns out that one slick method that will always work (believe it or not) is to simply add numerators and denominators. (In the example above, e.g. 7/1994 will work.) To a math type, that requires proof, but it does work.