Abstract

Legend has it that the first person in ancient Greece who discovered that there are numbers that cannot be written as fractions was thrown overboard from a ship. Centuries later, while we regularly use numbers that cannot be written as fractions, those numbers that can be written as fractions remain powerful tools. What makes fractions so special? We explore how we can recognize the decimal representation of fractions and how fractions can be used to approximate any real number as closely as we wish.

Highlights

  • Legend has it that the first person in ancient Greece who discovered that there are numbers that cannot be written as fractions was thrown overboard from a ship

  • What makes fractions so special? We explore how we can recognize the decimal representation of fractions and how fractions can be used to approximate any real number as closely as we wish

  • A set of numbers is dense in the real numbers if for any two di erent real numbers, there is a number from the set in between them

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Summary

The Real Numbers

“ I am thinking of a number between negative and ,” Jordan says gleefully. If Jordan’s number is − , and you have already figured out that − is too low and − is too high, you know the answer is −. During that class, you realize that you have been assuming Jordan will always pick an integer. Jordan tells you your guess is high, so you know the answer is somewhere on the number line between and. Jordan says is still high, so you know the answer must be on the number line between and. Continuing with your strategy, you guess , , , , This seems like it is taking a long time!

DECIMAL EXPANSIONS
REPEATING DECIMAL
RATIONAL NUMBERS
For any rational number
IRRATIONAL NUMBER
CONCLUSION
YOUNG REVIEWERS

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