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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

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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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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