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A real number of the form p/2", where p is an integer and n is a nonnegative integer, is known as a dyadic rational number. Prove that there is a dyadic rational number between any two distinct real numbers.

A real number of the form p/2", where p is an integer and n is a nonnegative integer, is known as a dyadic rational number. Prove that there is a dyadic rational number between any two distinct real numbers.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Title: Understanding Dyadic Rational Numbers** A real number of the form \( p/2^n \), where \( p \) is an integer and \( n \) is a nonnegative integer, is known as a **dyadic rational number**. **Problem Statement:** Prove that there is a dyadic rational number between any two distinct real numbers. In this context, a dyadic rational number represents a fraction whose denominator is a power of two, making them useful in various computational and theoretical applications. The task is to demonstrate that for any two distinct real numbers, at least one dyadic rational number exists between them, highlighting their density in the real number line. To approach this proof, consider the construction of dyadic rationals as a subset of rational numbers and explore concepts such as number line density and binary representation of numbers.
Transcribed Image Text:**Title: Understanding Dyadic Rational Numbers** A real number of the form \( p/2^n \), where \( p \) is an integer and \( n \) is a nonnegative integer, is known as a **dyadic rational number**. **Problem Statement:** Prove that there is a dyadic rational number between any two distinct real numbers. In this context, a dyadic rational number represents a fraction whose denominator is a power of two, making them useful in various computational and theoretical applications. The task is to demonstrate that for any two distinct real numbers, at least one dyadic rational number exists between them, highlighting their density in the real number line. To approach this proof, consider the construction of dyadic rationals as a subset of rational numbers and explore concepts such as number line density and binary representation of numbers.
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