The Real Number System

Real Numbers

• The set of all rational and the set of all irrational numbers together make up the set of real numbers.

• Any and all kinds of numbers fall under real numbers.

Rational Numbers

Rational numbers are numbers that can be written

as fractions. That is, the form a/b where a and b are

both integers and b ≠ 0.

Examples of Rational Numbers

• -6

• 8 2/5

• .05

• -2.6

• 5.3333333

• -8.12121212…

• √16

Irrational Numbers• Irrational Numbers – numbers that

are not repeating or terminating decimals.

• Examples:

• .01001000100001…• √2 = 1.414213562…• 3.14159…

Whole Numbers, Natural Numbers, and Integers

• Whole Numbers include the following:• 0,1,2,3,4,5,6,7,8,9,10,…..

• Natural Numbers include the following:• 1,2,3,4,5,6,7,8,9,10,….. Does not

include 0.

• Integers include the following:• …-3,-2,-1,0,1,2,3,…

Classifying Real Numbers

• Directions: Classify the following numbers as natural, whole, integer, rational, and/or irrational.

• 8

• This number is a natural number, a whole number, an integer, and a rational number.

• 0.33333

• This repeating decimal is a rational number because it is equivalent to 1/3.

• √17

• √17 = 4.123105… It is not the square root of a perfect square so it is irrational.

• -28/2

• Since -28/4 = -14, this number is an integer and a rational number.

• -√121

• Since -√121 = -11, this number is an integer and a rational number.

Solving Equations

• a2 = 49

• To undo the square, take the square root of both sides. Then, you have this.

• √a2 = √49

• a = √49 or a = -√49

• a = 7 or a = -7

• Hence, the solutions are 7 and -7.

• d2 = 55

• Take the square root of both sides.

• √d2 = √55

• d = √55 or d = - √55

• d = 7.41 or d = - 7.41

• Hence, the solutions are 7.41 and -7.41.