Since 0.199999 …=0.200000 …and 0.399999 …= 0.400000 …(see Problems 41 and 42), we see that certa

step 1 Okay, so we know that the rounding off property of numbers suggests that, let's say if we have,

Since 0.199999 …=0.200000 …and 0.399999 …= 0.400000 …(see...

Since $0.199999 \ldots=0.200000 \ldots$ and $0.399999 \ldots=$
$0.400000 \ldots$ (see Problems 41 and 42), we see that certain rational numbers have two different decimal expansions. Which rational numbers have this property?

Key Concepts

Rational Numbers

Rational numbers are numbers that can be expressed as the ratio of two integers. They have decimal representations that either terminate or eventually repeat a block of digits indefinitely. This fundamental property distinguishes them from irrational numbers, which have non-terminating, non-repeating decimal expansions.

Decimal Representation

Decimal representation is the expression of a number in the base-10 numeral system. It provides a way to express both the whole number and fractional parts of a number, and can be either finite (terminating) or infinite (repeating). This concept is crucial in understanding how numbers can have more than one, sometimes counterintuitive, representation.

Terminating Decimals

Terminating decimals are decimal numbers that have a finite number of digits after the decimal point. They can sometimes also be represented as an alternative, infinite repeating decimal, such as a number ending in an infinite sequence of 9’s. Recognizing terminating decimals is key to understanding when a rational number might have a dual decimal representation.

Non-Uniqueness of Decimal Expansions

Non-uniqueness of decimal expansions refers to the phenomenon where certain rational numbers have two different decimal forms. This occurs particularly with terminating decimals, where one representation has a finite number of digits and the other ends with an infinite sequence of 9’s. This concept emphasizes that the same numerical value can be expressed in more than one way in the decimal system.

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