Section 1.2Properties of Real

Numbers

Common Core State Standards:MACC.912.N-RN.2.3: Explain why the sum or product of

two rational numbers is rational; that the sum of a rational number and an irrational number is irrational, and the product of a nonzero rational number and an

irrational number is irrational.

Warm UpWrite each number as a percent.

-15, -7, -4, 0, 4, 7{…, -2, -1, 0, 1, 2, 3, …}Add the negative natural numbers to the whole numbers

IntegersZ

0, 4, 7, 15{0, 1, 2, 3, … } Add 0 to the natural numbers

Whole NumbersW

4, 7, 15{1, 2, 3, …}These are the counting numbers

Natural NumbersN

ExamplesDescriptionName

Key ConceptsSubsets of the Real Numbers

This is the set of numbers whose decimal representations are neither terminating nor repeating. Irrational numbers cannot be expressed as a quotient of integers.

Irrational NumbersI

These numbers can be expressed as an integer divided by a nonzero integer:Rational numbers can be expressed as terminating or repeating decimals.

Rational NumbersQ

ExamplesDescriptionName

Key ConceptsSubsets of the Real Numbers

Rational Numbers

The Real Numbers

Irrational Numbers

Integers

Whole Numbers

Natural Numbers

The set of real numbers is formed by combining the rational numbers and the irrational numbers.

Example 1Your math class is selling pies to raise money to go to a math competition. Which subset of real numbers best describes the number of pies p that your class sells?

Example 2Classify and graph each number on a number

line.

Example 3Compare the two numbers. Use < and >.

a) -5, -8

b) 1/3, 1.333

c) 3, √3

Key ConceptsLet a, b, and c be real numbers.

Opposite - (additive inverse) the opposite of any number a is -a.

Reciprocal - (multiplicative inverse) the reciprocal of any nonzero number a is 1/a.

Property Addition Multiplication

Commutative

Associative

Identity

Inverse

Distributive

Example 4Name the property of real numbers illustrated by each

equation.

a) n · 1 = n

b) a (b + c) = ab + ac

c) 4 + 8 = 8 + 4

d) 0 = q + (-q)

Example 5Show each statement is false by providing a counterexample.a) The difference of two natural numbers is

a natural number.

b) The quotient of two irrational number is irrational.

c) All square roots are irrational.

MACC.912.N-RN.2.3: Explain why the sum or product of two rational numbers is rational; that the sum of a rational number

and an irrational number is irrational, and the product of a nonzero rational number and an irrational number is irrational.

Score Learning Progression

4 I am able to • use properties of real numbers to perform algebraic

operations

3 I am able to• graph and order real numbers• to identify properties of real numbers

2 I am able to • understand that real numbers have several special

subsets related in particular ways

1 I need prompting and/or support to complete tasks.

Section 1.2 - Rate Your Understanding