Find three irrational numbers between : (i) -1 and -2 (ii)1/5 and 1/4
Irrational numbers are numbers that cannot be expressed as rational numbers, i.e. fractions with numerator and denominator both whole numbers.
(1) Irrationals between -1 and -2:
`-sqrt(2),-sqrt(3),-pi/2` are all irrational.
(2) Between `1/5` and `1/4` :
`sqrt(1/23),sqrt(1/19),sqrt(1/17)` are all irrational.
** If you write them using the rules of simplifying radicals you will get `(sqrt(23))/23,(sqrt(19))/19,(sqrt(17))/17` where each numerator is irrational.**
3 irrats between -1 and -2 is asking the same as 3 irrats between 0 and 1.. the 1st that comes to mind is 2-sqrt(2).. the next two are 4-pi and 3-e.. to translate those three numbers to (-2,-1), we employ a linear combination of rationals to 'force' 2-sqrt(2), 4-pi, and 3-e into that region.. the simplest solution is: -1-2+sqrt(2), -1-4+pi, and -1-3+e. this equates with: -3+sqrt(2), -5+pi, and -4+e .. to push 'the solution' to the interval: (1/5,1/4), we need to construct a linear combination of rationals and irrationals as we did for (-2,-1). the length of the segment is critical as is the 'starting point'. so we must perform a linear transformation (rational based) on the solution set to conform to (1/5,1/4). 0.05 is the length of the interval. 0.2 is the starting point. so, 0.2+0.05i where i represents an irrational number. this is a 'new' result of my conjecture and soon-to-be theorems.