Two rational numbers between 0.333...and0.444

Take the average. The average of two numbers is halfway between them.

We have 0.333… = 1/3 and 0.444… = 4/9. Then the average of these is

(1/3 + 4/9)/2 = (3/9 + 4/9)/2 = (7/9)/2 = 7/18

To get another number between 1/3 and 4/9, take the average of one of these and 7/18. For instance,

(1/3 + 7/18)/2 = (6/18 + 7/18)/2 = (13/18)/2 = 13/36

Or more simply, just chop off some of the non-terminating part and increment the last digit. For example,

• chop 0.333… down to 0.33, then increase the last digit by 1 to get 0.34

• chop 0.444… down to 0.44, then decrease the last digit to get 0.43