1.1 real number system dfs

05/0

2/20

23

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

05/0

2/20

23

1.1 Real Number System

3

05/0

2/20

23

STORY OF NUMBERS

4

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

number systems 3400 BC.

5

Tally Marks: Numerals used for counting

6

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

7

ROMAN NUMERALS

8

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.

9

How were numbers invented?

10

05/0

2/20

23

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’

11

05/0

2/20

23

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.

12

SYMBOLS FOR NUMBERS

05/0

2/20

23

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

05/0

2/20

23

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.

14

05/0

2/20

23

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

15

05/0

2/20

23

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

16

05/0

2/20

23

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.

17

05/0

2/20

23

• 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

18

SET OF POSITIVE AND NEGATIVE INTEGERS

05/0

2/20

23

Adding and subtracting Integers

• 3 - 4 = ?

19

05/0

2/20

23

Adding and subtracting Integers

20

05/0

2/20

23

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

21

05/0

2/20

23

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

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

22

Rational and Irrational Numbers

‘ and

05/0

2/20

23

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…

24

05/0

2/20

23

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 .

25

05/0

2/20

23

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.

26

05/0

2/20

23

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.

27

05/0

2/20

23

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

28

‘

05/0

2/20

23

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.

29

‘

05/0

2/20

23

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

3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067...

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

30

‘

05/0

2/20

23

Irrational NumberExamples of Irrational Numbers

31

05/0

2/20

23

Irrational Numbers

32

05/0

2/20

23

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

1. =

2. =

3. 0.2345234… =

4. =

5. =

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

05/0

2/20

23

Rational and Irrational Numbers

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

• Irrational Numbers: Neither repeat, nor terminate.

34

05/0

2/20

23

Real Number System

35

05/0

2/20

23

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

36

05/0

2/20

23

• 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

37

05/0

2/20

23

38

05/0

2/20

23

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

39

05/0

2/20

23

40