Mathematics Foundations/4.5 Decimal Representations

Decimal Representations 4.5 Decimal Representations 4.5.1 Introduction to Decimal Representations Every real number can be represented as a decimal expansion. This representation provides a concrete way to visualize real numbers and understand their properties. In this section, we explore how decimal representations connect to the fundamental nature of real numbers. 4.5.2 Finite and Infinite Decimal Expansions A decimal representation of a real number is an expression of the form: where is an integer and each for is a digit between 0 and 9. Finite decimal expansions terminate after a finite number of digits. For example: Infinite decimal expansions continue indefinitely. These can be further classified as: Repeating (periodic) decimals: A sequence of digits repeats indefinitely Non-repeating (non-periodic) decimals: No pattern of repetition occurs 4.5.3 Rational Numbers and Decimal Representations Theorem 4.5.1: A real number has a finite or repeating decimal expansion if and only if it is rational. Proof: (⇒) Suppose has a finite decimal expansion: . Then: for some integer . Thus, is rational. If has a repeating decimal expansion, we can express it as: where the bar indicates the repeating portion. Let: Multiplying by an appropriate power of 10 and subtracting yields a rational expression for . (⇐) Conversely, if is rational, the long division algorithm for must eventually repeat a remainder (as there are only possible remainders). This repetition in the division process creates a repeating decimal. 4.5.4 Irrational Numbers and Decimal Representations Theorem 4.5.2: A real number has a non-repeating decimal expansion if and only if it is irrational. Examples: (non-repeating) (non-repeating) (non-repeating) 4.5.5 Unique Representation Theorem Almost every real number has a unique decimal representation. The only exceptions are numbers with terminating decimal expansions, which can also be represented with an infinite sequence of 9's. Theorem 4.5.3: For any real number with a terminating decimal expansion: where There exists an equivalent representation: Example: Proof: Consider the geometric series: 4.5.6 Decimal Representations and the Real Number Line The decimal representation provides a way to visualize the density of rational and irrational numbers on the real line: Rational numbers correspond to points with repeating or terminating decimal expansions Irrational numbers correspond to points with non-repeating decimal expansions 4.5.7 Constructing the Real Numbers via Decimal Expansions An alternative construction of the real numbers (beyond Dedekind cuts or Cauchy sequences) uses infinite decimal expansions. This approach directly connects to the completeness property of the real numbers. 4.5.8 Computational Aspects Approximating Irrational Numbers: For any irrational number and any precision , there exists a rational number such that: This is achieved by truncating the decimal expansion of to a sufficient number of places. 4.5.9 Base Representations The concept of decimal representation extends to other bases. For a base , digits range from 0 to , and the positional values are powers of . Example: The binary (base 2) representation of is in base 2. 4.5.10 Exercises 1. Prove that using at least three different methods. 2. Find the decimal representation of and explain why it repeats. 3. Prove that has a non-repeating decimal expansion. 4. Convert (where the number of 0's between consecutive 1's increases by one each time) to a rational number. 5. If , find a fraction representation of .