By Peter J. Eccles

This ebook eases scholars into the pains of collage arithmetic. The emphasis is on realizing and developing proofs and writing transparent arithmetic. the writer achieves this through exploring set thought, combinatorics, and quantity conception, themes that come with many basic rules and will now not be part of a tender mathematician's toolkit. This fabric illustrates how typical principles might be formulated conscientiously, presents examples demonstrating a variety of simple tools of evidence, and contains a few of the all-time-great vintage proofs. The ebook provides arithmetic as a consistently constructing topic. fabric assembly the desires of readers from a variety of backgrounds is integrated. The over 250 difficulties contain inquiries to curiosity and problem the main capable pupil but additionally lots of regimen workouts to assist familiarize the reader with the fundamental rules.

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Example text

The aim was to prove the universal implication ‘if q is an integer then 2q ≠ 101’. It is impossible to work through the integers one by one. The proof was achieved by considering two cases: integers q such that q 50 and integers q such that q 51. This argument works because these two cases are exhaustive: for every integer q either q 50 or q 51. Here is another example deriving a familiar property of the integers from the inequality axioms. 4 For non-zero real numbers a, a2 > 0. Constructing a proof.

2 Arithmetic The basic principles of mathematical reasoning apply throughout mathematics but in this book we are going to explore them mainly in the context of number theory or arithmetic since the basic ideas here are familiar from early schooldays. e, the numbers … , –2, –1, 0, 1, 2, 3, … , and their basic arithmetic properties (under addition and multiplication) and order properties. We will also make use of the rational numbers (or fractions) and the real numbers (or infinite decimals) and these will be discussed in more detail in Chapter 13.

For example, if a is an integer, the statement ‘a is even and a is odd’ is a contradiction since ‘odd’ is defined to mean ‘not even’. Having obtained a contradiction we know that our initial assumption must have been wrong and so the result we are trying to prove must be true. 1. Constructing a proof. e, 14m + 20n ≠ 101. We can sum this up as follows. A proof by contradiction is given by showing that if the goal were false then this would lead to something which we know not to be true, a contradiction.

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