Abstract
The collection of integers is defined to be the set of numbers
where the dots indicate that the pattern continues indefinitely in each direction. The integers are also sometimes called the whole numbers. We will use these terms interchangeably. In this chapter we will introduce definitions of the terms even, odd, divides and prime. Then we will develop the skills needed to prove statements about the integers relating to these terms. Direct proofs, indirect proofs, and proofs by contradiction are included.
Before we take to sea we walk on land. Before we create we must understand.
—Joseph-Louis Lagrange, 1736–1813
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- 1.
The term whole numbers does not have a standard definition. It is sometimes used to represent the positive integers (1, 2, …), sometimes the non-negative integers (0, 1, 2, …) and sometimes the entire collection of integers. We will use whole number interchangeably with integer to emphasize that we are not including fractions.
- 2.
The novel is called Uncle Petros and Goldbach’s Conjecture: A Novel of Mathematical Obsession by Apostolos Doxiadis.
- 3.
Sometimes, the connective “not” is called a unary connective since it is applied to a single statement, rather than to connect two or more statements.
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© 2015 Sylvia Forman and Agnes M. Rash
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Forman, S., Rash, A.M. (2015). Conjectures, Proofs, and Counterexamples. In: The Whole Truth About Whole Numbers. Springer, Cham. https://doi.org/10.1007/978-3-319-11035-6_2
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DOI: https://doi.org/10.1007/978-3-319-11035-6_2
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Publisher Name: Springer, Cham
Print ISBN: 978-3-319-11034-9
Online ISBN: 978-3-319-11035-6
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