Lecture 2: Properties of Real Numbers : Lecture 2: Properties of Real Numbers Josh C. Shott
Objectives : Objectives List the axioms of equality
List some of the field properties of real numbers
List some properties that apply to subtraction and division of numbers Algebra Lecture 2 by Josh C. Shott
Axioms of Equality : The real numbers must satisfy the axioms of equality as part of a structure of a mathematical system.
These axioms for the most part are common sense, and sort of the “paperwork” we must get out of the way before we can dig in to real mathematics.
For any real numbers, x, y, z, we must assume the following laws are true: Axioms of Equality axioms of equality Reflexive Law x = x Algebra Lecture 2 by Josh C. Shott Symmetry Law If x = y, then y = x Reflexive Law If x = y, and y = z then x = z
Field Properties : Field Properties field properties The properties of real numbers that apply to addition and multiplication.
If x, y, and z are real numbers, then the following properties are true: closure law 0 is the additive identity Algebra Lecture 2 by Josh C. Shott commutative law associative law identity law 1 is the multiplicative identity
Field Properties : Field Properties distributive law -x is the additive inverseor opposite Algebra Lecture 2 by Josh C. Shott zero property inverse law is the multiplicative inverse
or reciprocal If xy = 0, then either x = 0 or y = 0.
Practice Examples : Practice Examples If 1/12 of a foot equals 1 inch and 1 inch equals 2.54 centimeters, then 1/12 of a foot equals 2.54 centimeters.
3 + 5 = 5 + 3
2 + 5 is a real number Algebra Lecture 2 by Josh C. Shott Identify the axiom of equality or the field properties that verify the following:
Practice Examples : Practice Examples If 1/12 of a foot equals 1 inch and 1 inch equals 2.54 centimeters, then 1/12 of a foot equals 2.54 centimeters.
3 + 5 = 5 + 3
2 + 5 is a real number Algebra Lecture 2 by Josh C. Shott Identify the axiom of equality or the field properties that verify the following: Transitive Law Commutative Law of Addition Additive Inverse Property Associative Law of Multiplication Multiplicative Identity Property Closure Property Under Addition Distributive Law Over Addition
Arithmetic Review : When multiplying a x b, if a and b are:
SAME SIGN The result is positive
DIFFERENT SIGN The result is negative Arithmetic Review Multiplying Real Numbers Division of Numbers Algebra Lecture 2 by Josh C. Shott Subtraction of Numbers Formally, subtraction is defined by addition. Formally, division is defined by multiplication
Properties of Subtraction and Division : For all real numbers a, b, and c (where division by 0 is observed): Properties of Subtraction and Division Algebra Lecture 2 by Josh C. Shott
Properties of Subtraction and Division : For all real numbers a, b, c, and d (where division by 0 is observed): Properties of Subtraction and Division Algebra Lecture 2 by Josh C. Shott Reducing Multiplication of fractions Adding fractions (same denominator) Adding fractions (different denominator) Dividing fractions
Practice Examples : Practice Examples Algebra Lecture 2 by Josh C. Shott Use the properties of addition and subtraction to determine whether or not the following problems are performed correctly:
Practice Examples : Practice Examples Algebra Lecture 2 by Josh C. Shott Use the properties of addition and subtraction to determine whether or not the following problems are performed correctly:
Practice Examples : Practice Examples Algebra Lecture 2 by Josh C. Shott Use the properties of addition and subtraction to determine whether or not the following problems are performed correctly:
Slide 14 : ? Thank you for your time Algebra Lecture 2 by Josh C. Shott