Problem 43 - Calculus Early Transcendentals: Pearson New International Edition

Question

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?

Step-by-Step Solution

Answer

Rational numbers with repeating 9s or terminating decimals ending in 0s.

Step 1: Understanding Infinite Repeating Decimals

When you see a decimal like \(0.9999\ldots\) or \(0.199999\ldots\), it denotes a number with an infinite repetition of the digit 9.These kinds of decimals can sometimes be equivalent to a simpler form, such as 1, 0.2, or 0.4, etc.

Step 2: Recognizing Equivalent Forms

These decimals occur specifically when there is an overlap at an endpoint of the decimal representation for these numbers. For example, 0.9999... is equivalent to 1.0000..., and 0.19999... is equivalent to 0.2000.... This commonality arises due to the base-10 representation of numbers and can be seen through limit or algebraic methods.

Step 3: Identifying Rational Numbers with Two Decimal Expansions

Rational numbers that end in an infinite repeat of '9' have two decimal expansions. This is always the case for numbers whose fractional part can be expressed as \(\frac{k}{10^n}\). In simpler terms, these are numbers that can be written as a sum of an integer and \(\frac{a}{b}\) where \(b\) is a power of ten, such as \(\frac{1}{2}\) as 0.5000 and 0.4999\ldots.. This gives them equivalent decimal and infinite decimal forms.

Key Concepts

Infinite Repeating Decimals

Infinite repeating decimals are numbers with decimals that go on forever. Imagine you're writing a number, and you just keep writing the same digit over and over. That's what infinite repeating decimals look like. A good example is when you come across a number like \(0.9999\ldots\). The \'9\'s just go on without stopping. When you see such a number, it's fascinating to learn that it can actually be equal to a number that looks different. For instance, \(0.9999\ldots\) isn't just a pattern; it's actually equal to the whole number 1, even though it might not look that way at first glance! This equivalency is possible due to how numbers work in math.

Decimal Expansions

Decimal expansions refer to the way numbers are written in the decimal system. When a number is broken down into a series of digits following a decimal point, that's its decimal expansion. Consider \(0.4\), its decimal expansion might continue on as \(0.4000\ldots\). This format lets you see a number as a part of the base-10 system. Even more interesting is that some numbers can have two different decimal expansions! It might seem strange but remember \(0.19999\ldots\)? Well, that actually equals \(0.2000\ldots\). These decimal expansions, though written differently, point to the same exact value.

Equivalent Forms

Equivalent forms in mathematics are different ways of expressing the same value. Like a nickname for your name, they might appear different but mean the same thing. We've seen this with infinite repeating decimals. For example, \(0.9999\ldots\) is just another way to write the number 1. Not every decimal has an equivalent form, but those that do usually end in continuous 9's. These equivalent forms arise because of the properties of numbers in our base-10 system. Such equivalent forms are a part of the magic of simplifying and understanding numbers in math.

Base-10 Representation

Base-10 representation is the way we most commonly write numbers, using digits from 0-9. This system is also known as the decimal system. Each place has a value that is a power of 10. So, as you move left or right from the decimal point, each position means 10 times more or 10 times less respectively. When it comes to numbers like \(0.19999\ldots\) and \(0.2000\ldots\), their equivalency comes from the nature of this base-10 system. It allows these decimal numbers to be reinterpreted as equivalent forms, even though on the surface they seem different. This understanding helps us comprehend why some rational numbers can appear in two forms within our numeral system.