1: Introduction and Notation
Wisdom is the quality that keeps you from getting into situations where you need it.
-- Doug Larson
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- 1.1: Basic Sets
- It has been said that “God invented the integers, all else is the work of Man.” This is a mistranslation. The term “integers” should actually be “whole numbers.” The concepts of zero and negative values seem (to many people) to be unnatural constructs. Indeed, otherwise intelligent people are still known to rail against the concept of a negative quantity – “How can you have negative three apples?” The concept of zero is also somewhat profound.
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- 1.2: Definitions - Prime Numbers
- You may have noticed that in Section 1.1 an awful lot of emphasis was placed on whether we had good, precise definitions for things. Indeed, more than once apologies were made for giving imprecise or intuitive definitions. This is because, in Mathematics, definitions are our lifeblood. More than in any other human endeavor, Mathematicians strive for precision. This precision comes with a cost – Mathematics can deal with only the very simplest of phenomena.
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- 1.3: More Scary Notation
- It is often the case that we want to prove statements that assert something is true for every element of a set. For example, “Every number has an additive inverse.” A statement that begins with the English words “every” or “all” is called universally quantified. It is asserted that the statement holds for everything within some universe. Statements that say something about a few (or even just one) of the elements of our universe are called existentially quantified.
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- 1.5: Some Algorithms of Elementary Number Theory
- An algorithm is simply a set of clear instructions for achieving some task. The Persian mathematician and astronomer Al-Khwarizmi1 was a scholar at the House of Wisdom in Baghdad who lived in the 8th and 9th centuries A.D. He is remembered for his algebra treatise Hisab al-jabr w’al-muqabala from which we derive the very word “algebra,” and a text on the Hindu-Arabic numeration scheme.
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- 1.6: Rational And Irrational Numbers
- In this section, we will "correct" the definition of rational numbers given in Section 1.1. The problem is that there are many expressions formed with one integer written above another (i.e. fraction bars) that represent the exact same rational number (e.g. 3/6 and 14/28 are considered distinct by the given definition despite both representing 1/2). To eliminate this problem with our definition of the rationals we need to add an additional condition that ensures that such duplicates don’t arise.
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- 1.7: Relations
- One of the principal ways in which mathematical writing differs from ordinary writing is in its incredible brevity. If one can prove a truly interesting, novel result in a single page – they’ll probably hand over the sheepskin. How is this great brevity achieved? By inserting single symbols in place of a whole paragraph’s worth of words! One class of symbols in particular has immense power – so-called relational symbols.