Story of Numbers, how they emerged, subsets of Real numbers. Dr. Farhana Shaheen

1. Chapter I
Algebra Review
MATH-020
Dr. Farhana Shaheen
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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
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4. STORY OF NUMBERS
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5. Who invented number
systems?
• The Mayans according to historians are first who invented the
number systems 3400 BC.
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6. Tally Marks: Numerals used
for counting
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7. • After them independently Egyptians around 3100 BC invented
their numeral system.
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8. ROMAN NUMERALS
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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.
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10. How were numbers invented?
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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’
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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.
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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”
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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.
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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
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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
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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.
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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
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19. Adding and subtracting Integers
• 3 - 4 = ?
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20. Adding and subtracting
Integers
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21. • Examples:
• 7 + 5 = 12
• -7 -5 = -12
• -7 + 5 = -2
• 7 – 5 = 2
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22. Multiplying Integers
• + . + = +
• - . - = +
• + . - = -
• - . + = -
Examples:
7 x 5 = 35
(-7)(-5) = 35
(-7)(5) = -35
(7)(– 5) =- 35
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23. Rational and Irrational
Numbers
‘
and

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…
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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
.
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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.
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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.
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28. Irrational Numbers
Irrational Numbers are decimals which are
Never ending and Never Repeating.
•
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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.
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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.
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31. Irrational Number
Examples of Irrational Numbers
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32. Irrational Numbers
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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..... =
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34. Rational and Irrational Numbers
• Rational Numbers: Either repeat, or terminate or both.
• Irrational Numbers: Neither repeat, nor terminate.
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35. Real Number System
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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
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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
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39. Complex Numbers
• The largest existing numbers, comprising
of Real and Imaginary numbers
(a+i b, where , a, b are real).
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40. 40
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