Is it true or false? If you think about this for a couple of seconds then you\u2019ll get into a bit of a pickle.<\/p>\n
Now consider the following statement:<\/p>\n
The happiest donkey in the world.<\/em><\/p>\n Is it true or false? Well it\u2019s not even a sentence; it doesn\u2019t make sense to even ask if it\u2019s true or false!<\/p>\n Clearly we\u2019ll be wasting our time trying to write proofs of statements like the two listed above\u2014we need to narrow our scope to statements that we might actually have a chance of proving (or perhaps refuting)! This motivates the following (informal) definition.<\/p>\n \u2726\u00a0Definition 0.1\u00a0<\/b>A proposition is a statement to which it is possible to assign a truth value (\u2018true\u2019 or \u2018false\u2019). If a proposition is true, a proof of the proposition is a logically valid argument demonstrating that it is true, which is pitched at such a level that a member of the intended audience can verify its correctness.<\/p>\n Thus the statements given above are not propositions because there is no possible way of assigning them a truth value. Note that, in Definition 0.1, all that matters is that it makes sense to say that it is true or false, regardless of whether it actually is true or false\u2014the truth value of many propositions is unknown, even very simple ones.<\/p>\n \u270e\u00a0Exercise 0.2\u00a0<\/b>Think of an example of a true proposition, a false proposition, a proposition whose truth value you don\u2019t know, and a statement that is not a proposition.<\/p>\n Results in mathematical papers and textbooks may be referred to as propositions, but they may also be referred to as theorems, lemmas or corollaries depending on their intended usage.<\/p>\n These are not precise definitions, and they are not meant to be\u2014you could call every result a proposition if you wanted to\u2014but using these words appropriately helps readers work out how to read a paper. For example, if you just want to skim a paper and find its key results, you\u2019d look for results labelled as theorems.<\/p>\n It is not much good trying to prove results if we don\u2019t have anything to prove results about. With this in mind, we will now introduce the number sets and prove some results about them in the context of four topics, namely: division of integers, number bases, rational and irrational numbers, and polynomials. These topics will provide context for the material in Part I, and serve as an introduction to the topics covered in Part II.<\/p>\n We will not go into very much depth in this chapter. Rather, think of this as a warm-up exercise\u2014a quick, light introduction, with more proofs to be provided in the rest of the book.<\/p>\n Later in this chapter, and then in much more detail in Chapter 2, we will encounter the notion of a set; a set can be thought of as being a collection of objects. This seemingly simple notion is fundamental to mathematics, and is so involved that we will not treat sets formally in this book. For now, the following definition will suffice.<\/p>\n \u2726\u00a0Definition 0.3\u00a0<\/b>(to be revised in Definition 2.1.1) A set is a collection of objects. The objects in the set are called elements of the set. If X is a set and x is an object, then we write x \u2208 X (LATEX code<\/a>: x \\in X) to denote the assertion that x is an element of X .<\/p>\n The sets of concern to us first and foremost are the number sets\u2014that is, sets whose elements are particular types of number. At this introductory level, many details will be temporarily swept under the rug; we will work at a level of precision which is appropriate for our current stage, but still allows us to develop a reasonable amount of intuition.<\/p>\n In order to define the number sets, we will need three things: an infinite line, a fixed point on this line, and a fixed unit of length.<\/p>\n So here we go. Here is an infinite line:<\/p>\n The arrows indicate that it is supposed to extend in both directions without end. The points on the line will represent numbers (specifically, real numbers, a misleading term that will be defined in Definition 0.25). Now let\u2019s fix a point on this line, and label it \u20180\u2019:<\/p>\n \u00a0 This point can be thought of as representing the number zero; it is the point against which all other numbers will be measured. Finally, let\u2019s fix a unit of length:<\/p>\n \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 This unit of length will be used, amongst other things, to compare the extent to which the other numbers differ from zero.<\/p>\n \u2726\u00a0Definition 0.4\u00a0<\/b>The above infinite line, together with its fixed zero point and fixed unit length, constitute the (real) number line.<\/p>\n\n
Number sets<\/h3>\n
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