The Real Number System

1. The Real Number System NUM-L1 Objectives:To recognize and understand the way in which each subset of numbers fits into the real number system. Learning Outcome B-4

2. The simplest and smallest subset of real numbers is called natural numbers (N), or sometimes referred to as "counting" numbers. For example, if you were to count the number of children in families in Manitoba, some families would have 1 child, others 2 children, still others 3 children, and so on. This set begins with 1 and goes to infinity, and is written as N = {1, 2, 3,…}, or expressed graphically as Theory – Natural Numbers

3. This subset of numbers arose from the need for the number zero. In the example of counting children, you would no doubt find families with no children. As a result, you would have to include the number zero. This slightly larger set of numbers is called whole numbers and includes all natural numbers and zero. This subset can be written as W = {0, 1, 2, 3,...) or expressed graphically by placing a 0 in the outer shell of the diagram. Theory – Whole Numbers

4. The third subset within the real number system is the set of integers (I). Integers include all positive and negative whole numbers and can be written as I = {... -3, -2, -1, 0, 1, 2,3,...}. These numbers stem from the need of assigning negative values to numbers. For example, integers are used to express temperatures (e.g., -30°C) and profits and losses in business (e.g., a loss of $426.00 can be written as -$426.00). Graphically, this set of numbers adds an outer ring of negative numbers to the diagram. Theory – Integers

5. This subset can also be referred to as fractional numbers (both positive and negative), and may be written as Q={ a\b | a∈ I, b ∈ I, b ≠ 0}. This statement means that rational numbers include every number which can be written in the form: integer / integer, where the denominator is not equal to 0 (as division by 0 is impossible to do). If you were to begin building this set, it would look similar to the following: Examine the patterns created here. Notice that every positive and negative fraction can be created, as well as all integers (as certain fractions reduce to integers.) Theory – Rational Numbers

6. Graphically, this subset of numbers adds another ring to our diagram of real numbers, where Q includes all integers. This outer ring represents all the fractions that do not reduce to integers Theory – Rational Numbers

7. To write rational numbers (Q) in the form of decimals, divide the top of the fraction (numerator) by the bottom of the fraction (denominator). • You will find that these decimals either: • terminate, that is, come to an end (work out evenly at some point) • Examples • or • repeat, that is, a digit or a pattern of digits repeats • Examples Theory – Rational Numbers

8. The fifth and final subset of the real number system is the set of irrational numbers or Q'. These numbers are decimals that do notterminate and donotrepeat. These numbers cannot be expressed as fractions. • Examples: • 0.101101110… • (b) -3.727722777222... • (c) • (d) • (e) pi = π = 3.141592654... Theory – Irrational Numbers

9. Irrational numbers can be written as • decimal numbers, as in examples • (a) and (b) above • e.g., 0.101101110..., -3.727722777222..., • square roots of non-perfect squares as in example (c) above, e.g., • (Note: etc. are perfect squares which work out evenly. They are natural numbers.) • · cubic roots of non-perfect cubes as in example (d) above, e.g., Theory – Irrational Numbers

10. · cubic roots of non-perfect cubes as in example (d) above, e.g., • Note that is a natural number whose value is 2. • higher order roots behave according to the same principles as square roots and cubic roots, e.g., is irrational. • specialized numbers like π = 3.1415... e = 2.71828... are also irrationals that cannot be expressed as fractions. • These special numbers are non-terminating decimals and cannot be written in root form. They are sometimes called transcendental numbers. They have distinctive uses. (Recall π in formulas involving circles.) You will use these numbers in future modules. Theory – Irrational Numbers

11. ·Together, the two subsets Q (decimals that terminate or repeat) and Q' (decimals that do not terminate or repeat) create the set of real numbers. The real number system comprises these subsets of numbers: N (Natural), W (Whole), I (Irrational), Q (Rational), and R (Real). As you move from the centre of the model outward, the outer set includes the one(s) inside it. For example, W includes all of N, and I includes all of W and N. Theory – Real Number System