Question: "Explore whether or not irrational numbers are 'dense'. In other words, between any two irrational numbers is there always another irrational number?"
My answer until I got stuck: An irrational number is a decimal that is non-terminating and non-repeating. So if you have one non-terminating and non-repeating decimal then it is logical to expect that there cannot be another irrational number in between ANY TWO.
Now "between any two" implies between one irrational number and the next one in the set. So if there is one non-terminating and non-repeating decimal number, then the next one in the set is just one up from that. (right?) So how could there be another number in between such specific numbers that go on forever?
However my first thought was that in between (square root of 3) and (square root of 5) there is (Pi), so there will always be infinite amounts of irrational numbers between any two irrational numbers.
Which logic is correct (if any of it is)? Help my brain hurts!
My answer until I got stuck: An irrational number is a decimal that is non-terminating and non-repeating. So if you have one non-terminating and non-repeating decimal then it is logical to expect that there cannot be another irrational number in between ANY TWO.
Now "between any two" implies between one irrational number and the next one in the set. So if there is one non-terminating and non-repeating decimal number, then the next one in the set is just one up from that. (right?) So how could there be another number in between such specific numbers that go on forever?
However my first thought was that in between (square root of 3) and (square root of 5) there is (Pi), so there will always be infinite amounts of irrational numbers between any two irrational numbers.
Which logic is correct (if any of it is)? Help my brain hurts!