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Proposition 0.1. Let a be an integer and b a natural number. There is an integers q and r with 0

Proposition 0.1. Let a be an integer and b a natural number. There is an integers q and r with 0

College Algebra
College Algebra
7th Edition
ISBN:9781305115545
Author:James Stewart, Lothar Redlin, Saleem Watson
Publisher:James Stewart, Lothar Redlin, Saleem Watson
ChapterP: Prerequisites
SectionP.2: Real Numbers
Problem 1E: Give an example of each of the following: (a) A natural number (b) An integer that is not a natural...
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Proposition 0.1. Let a be an integer and b a natural number. There is an integers q and r with 0 <r < b such that: a = bq +r Here q is called the quotient and r is called the remainder. Here's a hint: think about all possible q and r 2 0 that make the equation hold (without the assumption r < b) and then use well ordering to find the smallest such r. This proposition is often called the division algorithm because it is tells you exactly what one gets from old fashioned long division of natural numbers - a quotient and a remainder. However there's a subtlety here - the proposition says that q and r exist, but not that they are unique - in other words the division problem might have more than one right answer. Obviously that's not what we expect. Here's how that is phrased precisely (make sure you understand why, then prove it): Proposition 0.2. Let b a natural number and q1, q2, r1, r2 integers with 0 < ri < b and 0 < r2 < b such that qib+ r1 = q2b+ r2 then qi = q2 and ri = r2.
Transcribed Image Text:Proposition 0.1. Let a be an integer and b a natural number. There is an integers q and r with 0 <r < b such that: a = bq +r Here q is called the quotient and r is called the remainder. Here's a hint: think about all possible q and r 2 0 that make the equation hold (without the assumption r < b) and then use well ordering to find the smallest such r. This proposition is often called the division algorithm because it is tells you exactly what one gets from old fashioned long division of natural numbers - a quotient and a remainder. However there's a subtlety here - the proposition says that q and r exist, but not that they are unique - in other words the division problem might have more than one right answer. Obviously that's not what we expect. Here's how that is phrased precisely (make sure you understand why, then prove it): Proposition 0.2. Let b a natural number and q1, q2, r1, r2 integers with 0 < ri < b and 0 < r2 < b such that qib+ r1 = q2b+ r2 then qi = q2 and ri = r2.
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