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# 2.3.2: Some terminology

Before we look at some sample proofs, here is some terminology that we will use through- out our sample proofs and the rest of the course of Reasoning & Logic.

• The natural numbers (denoted N) are the numbers 0, 1, 2, . . .. Note that the sum and product of natural numbers are natural numbers.
• The integers (denoted Z) are the numbers 0, −1, 1, −2, 2, −3, 3, . . .. Note that the sum, product, and difference of integers are integers.
• The rational numbers (denoted Q) are all numbers that can be written in the form $$\frac{m}{n}$$ where m and n are integers and n
• The real numbers (denoted R) are numbers that can be written in decimal form, possibly with an infinite number of digits after the decimal point. Note that the sum, product, difference, and quotient of real numbers are real numbers (provided you don’t attempt to divide by 0).
• The irrational numbers are real numbers that are not rational, i.e., that cannot be written as a ratio of integers. Such numbers include $$\sqrt{3}$$(which we will prove is not rational) and $$\pi$$ (if anyone ever told you that $$\pi=\frac{22}{7}$$ , remember that $$\frac{22}{7}$$ is only an approximation of the value of $$\pi$$). Later you will learn that we can describe this set of irrational numbers as R − Q, that is: it is all the numbers that are in R but are not in Q.
• An integer n is divisible by m iff n = mk for some integer k. This can also be expressed by saying that m evenly divides n, which has the mathematical notation m | n. So for example, 2 | 8, but 8 ∤ 2. 2 | n iff n = 2k for some integer k; n is divisible by 3 iff n = 3k for some integer k, and so on. Note that if 2 ∤ n (i.e., n is not divisible by 2), then n must be 1 more than a multiple of 2 so n = 2k+1 for some integer k. Similarly, if n is not divisible by 3 then n must be 1 or 2 more than a multiple of 3, so n = 3k + 1 or n = 3k + 2 for some integer k.
• An integer is even iff it is divisible by 2 and odd iff it is not.
• An integer n > 1 is prime if it is divisible by exactly two positive integers, namely 1 and itself. Note that a number must be greater than 1 to even have a chance of being termed ‘prime’. In particular, neither 0 nor 1 is prime.