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Section 1.2 SUBSETS OF THE SET OF REAL NUMBERS Set – A collection of objects. Each object is called a member or an element of the set. Sets are enclosed in braces. {1, 2, 3} is a set containing the numbers 1, 2, and 3. Ellipsis: Three dots at the end of a group of numbers meaning that the pattern that has been established continues indefinitely. Natural numbers: The counting numbers {1, 2, 3 … } Whole numbers: The natural numbers plus zero {0, 1, 2, 3 … } (Zero looks like a hole!) Integers: The negative natural numbers and the whole numbers {… -2, -1, 0, 1, 2, … } Rational numbers: any number that can be written as a fraction. 2/3 5 (because 5 = 5/1) BEWARE: Zero can never be on the bottom of a fraction. 5/0 is undefined. Rational numbers: written as decimals either terminate (0.5) or repeat (0.33333…) Irrational numbers: written as decimals they never end and never repeat. Irrational numbers look strange! (π, e, …) BEWARE: 9 looks strange but it equals 3 so it is a rational number. 2 is irrational. EQUALITIES AND INEQUALITIES = equal to 5 = 5 or ≠ not equal to 5≠6 < less than 2<3 ≤ less than or equal to 2 ≤ 3 or > Greater than 5>2 ≥ greater than or equal to 5 ≥ 2 or 5=2+3 2≤2 5≥5 Note: The arrow always points to the smaller number. On the number line, smaller numbers are to the left. Section 1.2 THE REAL NUMBER SYSTEM NOTE: All of the numbers that we will discuss before the last few weeks of class are real numbers. Any natural number is also a whole number, an integer, a rational number and real. Any whole number is also an integer, a rational number and real. Any integer is also a rational number and real. Any rational number is also a real number. Any irrational number is also a real number. The Real Numbers Rationals Irrationals Integers Whole numbers Natural numbers