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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