108 Decimal Fractions
- Draw a number line in power-of-10 style that shows the rational numbers from 10 to
100,000. How many orders of magnitude is this? - Draw a number line in power-of-10 style that shows the rational numbers from 30 to
300,000. How many orders of magnitude is this? - How many orders of magnitude larger than 330 is 75,000,000? Here’s a hint: Express
the answer by saying “75,000,000 is between n and n+ 1 orders of magnitude larger
than 330,” where n is a whole number. - Write the following decimal expressions as combinations of integers and fractions.
Reduce the fractional part to lowest terms.
(a) 4.7
(b)−8.35
(c) 0.02
(d)−0.29 - Express the numbers from the solutions to Prob. 4 as ratios of integers, with the
denominator always positive. - Write the following ratios as decimal expressions.
(a) 44/16
(b)−81/27
(c) 51/13
(d)−45/800 - Convert the fraction 1/17 to another fraction whose denominator is a string of 9s.
- Convert the expression 2.892892892 ... to a ratio of integers.
- Suppose that somebody tells us there are two integers so large that it would take a
person billions of years to write either of them out by hand. We are also told that both
of these integers are prime numbers, and that the decimal expansion of their ratio (that
is, one of these primes divided by the other) is an endless sequence of digits. Is there a
repeating pattern to the digits in the decimal expansion? - Imagine that we come across a gigantic string of digits—miles long if we try to write
it out—and we can’t see any pattern. We use a computer to examine this number to a
thousand decimal places, then a million, then a billion, and we still can’t find a pattern.
Can we ever know for sure whether or not a pattern actually exists, so we can decide
whether or not the number is rational? Here’s a hint: This exercise is meant to force
your imagination into overdrive!