The discussion centers on the classification of decimal expansions, specifically those that terminate in an infinite string of 9's. It establishes that numbers ending with an infinite string of 9's are rational, leading to a countable infinity of such numbers. Conversely, it concludes that irrational numbers do not exhibit this property, resulting in an uncountable infinity of decimal expansions that do not terminate in 9's. This distinction is fundamental in understanding the nature of rational and irrational numbers.
PREREQUISITESMathematics students, educators, and anyone interested in the foundational concepts of number theory and decimal representation.
cragar said:Homework Statement
How many decimal expansions terminate in an infinite string of 9's?
How many dont?
The Attempt at a Solution
If we have a number terminate with an infinite amount of 9's then it will be a rational number.
So there would be countably many of these.
And since irrational numbers do not end with all 9's then their would be
uncountably many of them that do not.