*by Eugene Chen of the Amador Valley Math Team*Repeating decimals may appear on math competitions. Sometimes, they rely on knowledge that must be learned from sources such as this page.
To start, we introduce some notation:
In short, if a number has a line on top of it, it is intended to be repeated infinitely.
For example, and .
A few things to note:
If is a digit, then . For example, and .
Furthermore, if is a two-digit string of digits, then . For example, and .
Some more shortcuts:
If are digits, then . Furthermore, if are digits, then . The number of nines and zeroes used depends on how many places repeat.
In general, if are strings of digits, , where the number of nines is the number of digits in string and is the number of digits in string , and where represents the numeral formed by putting the string to the left of string .
Exercise 1:
Write the following repeated decimals as common fractions.
a)
b)
c)
d)
e)
Problem 1:
Prove that if are digits, then .
Problem 2:
Prove that if are digits, then .
Challenge 1:
Prove that if are strings of digits, , where the number of nines is the number of digits in string and is the number of digits in string , and where represents the numeral formed by putting the string to the left of string . |