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Chapter I

Algebra ReviewMATH-020

Dr. Farhana Shaheen1

Chapter I

Algebra Review1. The Real Number System2. Sets3. Inequality & Interval Notation4. Integer Exponents5. Ratios, Proportions, and Percentages6. Simple and Compound Interest

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1.1 Real Number System

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STORY OF NUMBERS

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Who invented number systems?• The Mayans according to historians are first who invented the

number systems 3400 BC.

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Tally Marks: Numerals used for counting

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• After them independently Egyptians around 3100 BC invented their numeral system.

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ROMAN NUMERALS

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The Universal Numerals• The Universal Numerals are the numbers we use today! • Note that each Numeral has the number of angles equal to the

number it represents.

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How were numbers invented?

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STORY OF NUMBERS

• The story of numbers begins with• Natural numbers N= {1, 2, 3, 4, 5, ……} • Whole Numbers W = {0, 1, 2, 3, 4, 5, ……} • Integers Z= {-3, -2, -1, 0,1, 2, 3, 4, 5, ……} • Rational Numbers Q = {a/b: a,b are Integers}• Irrational Numbers Q’ = { ? } • Real Numbers= All Q and Q’

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Number Line• A Number Line is used to arrange all numbers along a line.

The points on the right are greater than the points on the left. The numbers on the Number Line are infinite, meaning they never end and keep increasing.

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SYMBOLS FOR NUMBERS

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Natural Numbers

The story of numbers begin with Natural Numbers, also known as Counting Numbers, which consist of 1, 2, 3, 4, 5, 6… • These numbers are infinite, that is, they go

on forever • Counting numbers do not contain 0, as the

number “0” cannot be “counted”13

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Whole Numbers

• Whole Numbers : are natural numbers, but they also contain the number “0” • They consist of 0, 1, 2, 3, 4, 5, 6…. and so on. • Note that Whole Numbers do Not contain Fractions like 2/3, 4/7 etc.

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Natural Numbers/Whole Numbers• Natural Numbers are also known as Counting Numbers,

which consist of 1, 2, 3, 4, 5, 6… • These numbers are infinite, that is, they go on forever • Counting numbers do not contain 0, as the number “0” cannot

be “counted”.• Whole Numbers are natural numbers, but they also contain

the number “0” • They consist of 0, 1, 2, 3, 4, 5, 6…. and so on

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Integers• Integers are just like Whole Numbers; however, they contain

negative numbers as well. • Negative Numbers are numbers smaller than 0. • Just like Whole Numbers, Integers do not contain Fractions.• Examples: -8, -5, 0, 4, 17, 23

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INTEGERS • A Number Line is used to arrange all numbers along a line.

The points on the right are greater than the points on the left. The numbers on the Number Line are infinite, meaning they never end and keep increasing.

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• Integers = { ..., -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, ... }• Positive Integers = { 1, 2, 3, 4, 5, ... } = Natural Numbers• Negative Integers = { ..., -5, -4, -3, -2, -1 }• Non-Negative Integers = { 0, 1, 2, 3, 4, 5, ... } = Whole Numbers

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SET OF POSITIVE AND NEGATIVE INTEGERS

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Adding and subtracting Integers

• 3 - 4 = ?

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Adding and subtracting Integers

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• Examples:• 7 + 5 = 12• -7 -5 = -12• -7 + 5 = -2• 7 – 5 = 2

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Multiplying Integers• + . + = +• - . - = +• + . - = -• - . + = -

Examples: 7 x 5 = 35(-7)(-5) = 35 (-7)(5) = -35 (7)(– 5) =- 35

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Rational and Irrational Numbers

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Rational Numbers

• A Rational number is a number that can be written as a ratio a/b, for any two integers a and b.

• The notation is also called a fraction. • For example, 3/4, 5/7, 9/4 etc. are all fractions.• 1/2= 0.5• 3/4 = 0. 75• 5/7 = 0.714285714285….• 9/4 = 2.25• 1/3 = 0.333333333…

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Rational Number• Note-1:

The numerator (the number on top) and the denominator (the number at the bottom) must be integers. • Note-2:

Every integer is a rational number simply because it can be written as a fraction. For example, 6 is a rational number because it can be written as .

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Rational NumberRational numbers are numbers which are either repeated, or terminated. Like, 0.25 0.7645 0.232323..... 0.333333….0.714285714285….are all rational numbers.

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Rational Number• Examples of Rational Numbers

1) The number 0.75 is a rational number because it is written as fraction . 2) The integer 8 is a rational number because it can be written as .3) The number 0.3333333... = , so 0.333333.... is a rational number. This number is repeated but not terminated.

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Irrational Numbers Irrational Numbers are decimals which are Never ending and Never Repeating.•

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Irrational Number• An Irrational Number is basically a non-rational number; it

consists of numbers that are not whole numbers. Irrational numbers can be written as decimals, but not as fractions.

• Irrational Numbers are non-repeating and non-ending.

• For example, the mathematical constant Pi = π = 3.14159… has a decimal representation which consists of an infinite number of non-repeating digits.

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Irrational Number• The value of pi to 100 significant figures is

3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067...

• Note: Rational and Irrational numbers both exist on the number line.

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Irrational NumberExamples of Irrational Numbers

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Irrational Numbers

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Activity • Tell whether the following are rational or irrational numbers:

1. =

2. =

3. 0.2345234… =

4. =

5. =

6. 0. 315315315..... = 33

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Rational and Irrational Numbers

• Rational Numbers: Either repeat, or terminate or both.

• Irrational Numbers: Neither repeat, nor terminate.

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Real Number System

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Real Number System • Real Number System:

The collection of all rational and irrational numbers form the set of real numbers, usually denoted by R. • The real number system has many subsets:1. Natural Numbers 2. Whole Numbers 3. Integers

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• Natural numbers are the set of counting numbers.{1, 2, 3,4,5,6,…}

• Whole numbers are the set of numbers that include 0 plus the set of natural numbers.

{0, 1, 2, 3, 4, 5,…}

• Integers are the set of whole numbers and their opposites.{…,-3, -2, -1, 0, 1, 2, 3,…}

Real Number System

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Complex Numbers• The largest existing numbers, comprising of Real and Imaginary numbers (a+i b, where , a, b are real).

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