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When Close Enough is Close Enough

E. Scheinerman
Journal / Anthology

American Mathematical Monthly
Year: 2000
Volume: 107
Issue: 6
Page range: 489-499

The student's intuition is that if two algebraic expressions such as Sqrt[2] + Sqrt[5 - 2 Sqrt[6]] and Sqrt[3] agree to 28 digits, then surely they are equal. We transform the student's intuition by asking: When does |a - b| < epsilon imply a = b? For exmple, if a and b are integers, then if we know that |a - b| is (say) les than 0.9, then we may conclude that a = b. ... [O]ne approach to making the student's intuition rigorous is to develop the theory of algebraic integers; see [16]. Instead, we present an equivalent theory based on eigenvalues of matrices; the proofs are sufficiently straightforward that they can be presented in an undergraduate linear algebra class. This will enable us to answer the question: When is close enough, close enough?

*Mathematics > Algebra > Linear Algebra
*Mathematics > Number Theory
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