Lecture1: Overview

INTRODUCTION

ALGEBRA, Defined:

·        Came from two Arabic word: Al – means “the” and Jabara – means “reunion”

·        Literally, it is called as “the subject of reunion”, because of combining numbers and letters for computation.

·        A branch of mathematics which deals with numbers and letters that is use for computations.

IMPORTANCE OF ALGEBRA

·        This branch of mathematics have various importance in its application. But its main significance is, Algebra, serves as a preparatory subject for better understanding of higher mathematics.

NUMBERS vs. NUMERALS

·        Numbers – are abstract ideas of certain quantity.

·        Numerals – are symbols or representation of numbers.

REAL NUMBER SYSTEM

     ·        Zero = {0}

·        Positive/ Natural/ Counting numbers = {1, 2, 3, 4, …}

·        Whole numbers/ non-negative integers = {0, 1, 2, 3, 4, …}

·        Negative Integers = {-1, -2, -3, -4, …}

·        Integers = {… , -3, -2, -1, 0, 1, 2, 3, …}

·        Fractions = {…, ¼, 1/3, ½, …}

·        Decimals = {…, 0.25, 0.333, 0.5, …}

·        Rational numbers = {D, F, Z}

·        Irrational numbers = {sqrt 2, π, …}

·        Real numbers = {Q, Q’}

Additional definition:

·        Even numbers = {...,-4, -2, 0, 2, 4, …}

·        Odd numbers = {…, -5, -3, -1, 1, 3, 5, …}

·        Prime numbers = {1, 2, 3, 5, 7, 11, …}

·        Composite numbers = {4, 6, 8, 9, 10, …}

PROPERTIES OF EQUALITY

1.)  Reflexive: a = a                                             e.g.: 2 = 2

2.)  Symmetric: if a = b, then b = a                      e.g.: if x = 5, then 5 = x

3.)  Transitive: if a = b and b = c, then a = c        e.g.: if x = y and y = 3, then x = 3

4.)  Addition Property of Equality (APE)

If a = b, then a + c = b + c                           e.g.: if x = 2, then x + 4 = 2 + 4

5.)  Multiplication Property of Equality (MPE)

If a = b, then ac = bc                                    e.g.: if x = 5, then x(3) = 5(3)

6.)  Substitution

PROPERTIES OF REAL NUMBER

1.)  Closure – a real number added, subtracted, multiplied or divided to another real number, the result is always a real number.

e.g.:    Addition: 3 + 15 = 18

            Subtraction: 4 – 19 = -15

            Multiplication: 6 * 2 = 12

            Division: 27/3 = 9

2.)  Commutative – two numbers can be added/ multiplied in any order.

            Addition: a + b = b + a                         e.g.: 5 + 7 = 7 + 5

            Multiplication: ab = ba                          e.g.: 6(-4) = (-4)6

3.)  Associative – when three numbers are added/ multiplied, it makes no difference on either which two are added/ multiplied first.

Addition: (a + b) + c =a + (b + c)   e.g.: (2 + 5) + 8 =2 + (5 + 8)

Multiplication: (ab)c =a(bc)             e.g.: (5*2)*3 =5*(2*3)

4.)  Distributive – Multiplication distributes over addition.

a(b + c) = ab + ac                       e.g.: 4(3 + 6) = 4*3 + 4*6

5.)  Identity – the sum of a number and zero or the product of a number and one is equal to the number itself.

Addition: a + 0 = a                       e.g.: 3 + 0 = 3

Multiplication: a * 1 = a                e.g.: 8 * 1 = 8

6.)  Inverse – the sum of a number and its negative inverse is equal to 0 or the product of a number and its reciprocal is equal to 1.

Addition: a + (-a) = 0                   e.g.: 2 + (-2) = 0

Multiplication: a*(1/a) = 1             e.g.: 6*(1/6) = 1

7.)  Cancellation

ABSOLUTE VALUE

·        It is the number itself regardless of its sign. (||)

e.g.:

1.)   |-13| = 13                               2.)   |5| = 5

3.)   |-|-15|| = 15                         4. )   - |5|= -5