1.2 Symbols and sets of Numbers

Real Numbers

( All numbers that correspond to points on a number line ) .

The two statements below are both true .

2 = 2 states that " two is equal to two . " TRUE 26 states that " two is not equal to six . " TRUE

If two numbers are not equal , one number is larger than the other .

3 < 5 states that " three is less than five . " 2 > 0 states that " two is greater than zero . "

Rational Numbers

All positive and negative integers, fractions and decimal numbers.

Helpful Hint

Every real number is either . rational number or an irrational number

Translating Sentences

Example: eight is great than one (8 > 1)

Order Property for Real Numbers

For any two real numbers a and b , a is less than b if a is to the left of b on a number line . a b a < b or also b

Helpful Hint

Notice that 2 > 0 has exactly the same meaning as 0 < 2. Switching the order of the num bers and reversing the direction of the inequality symbol does not change the meaning of the statement . 3 < 5 has the same meaning as 5 > 3 , Also notice that when the statement is true . the inequality symbol " points " to the smaller

Integers

The Integers to the left of 0 are called negative integers ; integers to the right of 0 are called positive -integers . The integer 0 is neither positive nor negative .

absolute value

The absolute value of a real number a , denoted by lal , is the distance between a and 0 on a number line .

Natural Numbers

The set of natural numbers is { 1 , 2 , 3 , 4 , 5 , ... ) .

whole numbers

The set of whole numbers is { 0 , 1 , 2 , 3 , 4 , ... ) .

These symbols may be used to form a statement

The statement might be true or it might be false.

number line

To draw a number line , first draw a line . Choose a point on the line and label it 0. To the right of 0 , label any other point 1. Being careful to use the same distance as from 0 to 1 , mark off equally spaced distances Label these points 2 , 3 , 4 , 5 , and so on . Since the whole numbers continue indefinitely , it is not possible to show every whole number on this number line . The arrow at the right end of the line indicates that the pattern continues indefinitely .

Using a Number Line to Order Numbers

We begin with a review of the set of natural numbers and the set of whole numbers and how we use symbols to compare these numbers . A set is a collection of objects , each of which is called a member or element of the set . A pair of brace symbols { } encloses the list of elements and is translated as " the set of " or " the set containing . "

Identifying Common Sets of Numbers

Whole numbers are not sufficient to describe many situations in the real world . For example , quantities less than zero must sometimes be represented , such as tempera tures less than 0 degrees .

Equality symbol

a = b a is equal to b .

Inequality symbols :

a = b a is not equal to b a < b a is less than b a > b a is greater than b a ≤ b a is less than or equal to b a > b a is greater than or equal to b .

On a number line , we see that a number

right = larger left = smaller

a review of these symbols . The letters a and b are used to represent quantities Letters such as a and b that are used to represent numbers or quantities are called

variables

Irrational Numbers

√2 = 1.414213562 ... ( decimal number does not terminate or repeat in a pattern ) . * = 3.141592653 ... ( decimal number does not terminate or repeat in a pattern ) .