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.
A decimal expansion is the representation of a number in decimal form, with a decimal point and digits after the decimal point.
A terminating decimal expansion is one that ends after a finite number of digits, meaning the decimal representation of the number does not go on forever.
A non-terminating decimal expansion is one that continues on forever without repeating or ending. This means the decimal representation of the number has an infinite number of digits after the decimal point.
In decimal expansion, "9's" refer to the repeating digit 9 after the decimal point. This can occur in both terminating and non-terminating decimal expansions.
For a decimal expansion to be terminating, the denominator of the fraction must only contain factors of 2 and/or 5. If the denominator contains any other factors, the decimal expansion will be non-terminating. To determine if the repeating digits will be 9's, you can divide the numerator by the denominator and look for any repeating patterns, specifically patterns of 9's.