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Chapter I
Algebra Review
MATH-020
Dr. Farhana Shaheen
3/3/2017
1
Chapter I
Algebra Review
1. The Real Number System
2. Sets
3. Inequality & Interval Notation
4. Integer Exponents
5. Ratios, Proportions, and Percentages
6. Simple and Compound Interest
1.1 Real Number System
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3
STORY OF NUMBERS
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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
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
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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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12
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”
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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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15
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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17
• 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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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
‘
and
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
6
1
.
3/3/2017
25
Rational Number
Rational 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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26
Rational Number
• Examples of Rational Numbers
1) The number 0.75 is a rational number because it is written as
fraction
3
4
.
2) The integer 8 is a rational number because it can be written as
8
1
.
3) The number 0.3333333... =
1
3
, so 0.333333.... is a rational
number. This number is repeated but not terminated.
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27
Irrational Numbers
Irrational Numbers are decimals which are
Never ending and Never Repeating.
•
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28
‘
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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29
‘
Irrational Number
• The value of pi to 100 significant figures is
3.141592653589793238462643383279502884197169399375
10582097494459230781640628620899862803482534211706
7...
• Note:
Rational and Irrational numbers both exist on the number line.
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30
‘
Irrational Number
Examples of Irrational Numbers
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31
Irrational Numbers
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32
Activity
• Tell whether the following are rational or irrational numbers:
1.
1
2
=
2.
3
4
=
3. 0.2345234… =
4. 3=
5. 2=
6. 0. 315315315..... =
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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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35
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
3/3/2017
36
• 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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37
3/3/2017
38
Complex Numbers
• The largest existing numbers, comprising
of Real and Imaginary numbers
(a+i b, where , a, b are real).
3/3/2017
39
40
3/3/2017

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1.1 real number system dfs

  • 1. Chapter I Algebra Review MATH-020 Dr. Farhana Shaheen 3/3/2017 1
  • 2. Chapter I Algebra Review 1. The Real Number System 2. Sets 3. Inequality & Interval Notation 4. Integer Exponents 5. Ratios, Proportions, and Percentages 6. Simple and Compound Interest
  • 3. 1.1 Real Number System 3/3/2017 3
  • 5. Who invented number systems? • The Mayans according to historians are first who invented the number systems 3400 BC. 5
  • 6. Tally Marks: Numerals used for counting 6
  • 7. • After them independently Egyptians around 3100 BC invented their numeral system. 7
  • 9. 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
  • 10. How were numbers invented? 10
  • 11. 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 3/3/2017
  • 12. 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. 3/3/2017 12
  • 13. 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” 3/3/2017 13
  • 14. 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. 3/3/2017 14
  • 15. 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 3/3/2017 15
  • 16. 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 3/3/2017 16
  • 17. 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. 3/3/2017 17
  • 18. • 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 3/3/2017 18
  • 19. Adding and subtracting Integers • 3 - 4 = ? 3/3/2017 19
  • 21. • Examples: • 7 + 5 = 12 • -7 -5 = -12 • -7 + 5 = -2 • 7 – 5 = 2 3/3/2017 21
  • 22. Multiplying Integers • + . + = + • - . - = + • + . - = - • - . + = - Examples: 7 x 5 = 35 (-7)(-5) = 35 (-7)(5) = -35 (7)(– 5) =- 35 3/3/2017 22
  • 24. 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… 3/3/2017 24
  • 25. 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 6 1 . 3/3/2017 25
  • 26. Rational Number Rational numbers are numbers which are either repeated, or terminated. Like, 0.25 0.7645 0.232323..... 0.333333…. 0.714285714285…. are all rational numbers. 3/3/2017 26
  • 27. Rational Number • Examples of Rational Numbers 1) The number 0.75 is a rational number because it is written as fraction 3 4 . 2) The integer 8 is a rational number because it can be written as 8 1 . 3) The number 0.3333333... = 1 3 , so 0.333333.... is a rational number. This number is repeated but not terminated. 3/3/2017 27
  • 28. Irrational Numbers Irrational Numbers are decimals which are Never ending and Never Repeating. • 3/3/2017 28 ‘
  • 29. 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. 3/3/2017 29 ‘
  • 30. Irrational Number • The value of pi to 100 significant figures is 3.141592653589793238462643383279502884197169399375 10582097494459230781640628620899862803482534211706 7... • Note: Rational and Irrational numbers both exist on the number line. 3/3/2017 30 ‘
  • 31. Irrational Number Examples of Irrational Numbers 3/3/2017 31
  • 33. Activity • Tell whether the following are rational or irrational numbers: 1. 1 2 = 2. 3 4 = 3. 0.2345234… = 4. 3= 5. 2= 6. 0. 315315315..... = 3/3/2017 33
  • 34. Rational and Irrational Numbers • Rational Numbers: Either repeat, or terminate or both. • Irrational Numbers: Neither repeat, nor terminate. 3/3/2017 34
  • 36. 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 3/3/2017 36
  • 37. • 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 3/3/2017 37
  • 39. Complex Numbers • The largest existing numbers, comprising of Real and Imaginary numbers (a+i b, where , a, b are real). 3/3/2017 39