1.1 real number system dfs
TRANSCRIPT
![Page 1: 1.1 real number system dfs](https://reader035.vdocuments.site/reader035/viewer/2022062522/58ceb81c1a28abb2218b6253/html5/thumbnails/1.jpg)
05/0
2/20
23
Chapter I
Algebra ReviewMATH-020
Dr. Farhana Shaheen1
![Page 2: 1.1 real number system dfs](https://reader035.vdocuments.site/reader035/viewer/2022062522/58ceb81c1a28abb2218b6253/html5/thumbnails/2.jpg)
Chapter I
Algebra Review1. The Real Number System2. Sets3. Inequality & Interval Notation4. Integer Exponents5. Ratios, Proportions, and Percentages6. Simple and Compound Interest
![Page 3: 1.1 real number system dfs](https://reader035.vdocuments.site/reader035/viewer/2022062522/58ceb81c1a28abb2218b6253/html5/thumbnails/3.jpg)
05/0
2/20
23
1.1 Real Number System
3
![Page 4: 1.1 real number system dfs](https://reader035.vdocuments.site/reader035/viewer/2022062522/58ceb81c1a28abb2218b6253/html5/thumbnails/4.jpg)
05/0
2/20
23
STORY OF NUMBERS
4
![Page 5: 1.1 real number system dfs](https://reader035.vdocuments.site/reader035/viewer/2022062522/58ceb81c1a28abb2218b6253/html5/thumbnails/5.jpg)
Who invented number systems?• The Mayans according to historians are first who invented the
number systems 3400 BC.
5
![Page 6: 1.1 real number system dfs](https://reader035.vdocuments.site/reader035/viewer/2022062522/58ceb81c1a28abb2218b6253/html5/thumbnails/6.jpg)
Tally Marks: Numerals used for counting
6
![Page 7: 1.1 real number system dfs](https://reader035.vdocuments.site/reader035/viewer/2022062522/58ceb81c1a28abb2218b6253/html5/thumbnails/7.jpg)
• After them independently Egyptians around 3100 BC invented their numeral system.
7
![Page 8: 1.1 real number system dfs](https://reader035.vdocuments.site/reader035/viewer/2022062522/58ceb81c1a28abb2218b6253/html5/thumbnails/8.jpg)
ROMAN NUMERALS
8
![Page 9: 1.1 real number system dfs](https://reader035.vdocuments.site/reader035/viewer/2022062522/58ceb81c1a28abb2218b6253/html5/thumbnails/9.jpg)
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
![Page 10: 1.1 real number system dfs](https://reader035.vdocuments.site/reader035/viewer/2022062522/58ceb81c1a28abb2218b6253/html5/thumbnails/10.jpg)
How were numbers invented?
10
![Page 11: 1.1 real number system dfs](https://reader035.vdocuments.site/reader035/viewer/2022062522/58ceb81c1a28abb2218b6253/html5/thumbnails/11.jpg)
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
![Page 12: 1.1 real number system dfs](https://reader035.vdocuments.site/reader035/viewer/2022062522/58ceb81c1a28abb2218b6253/html5/thumbnails/12.jpg)
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
![Page 13: 1.1 real number system dfs](https://reader035.vdocuments.site/reader035/viewer/2022062522/58ceb81c1a28abb2218b6253/html5/thumbnails/13.jpg)
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
![Page 14: 1.1 real number system dfs](https://reader035.vdocuments.site/reader035/viewer/2022062522/58ceb81c1a28abb2218b6253/html5/thumbnails/14.jpg)
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
![Page 15: 1.1 real number system dfs](https://reader035.vdocuments.site/reader035/viewer/2022062522/58ceb81c1a28abb2218b6253/html5/thumbnails/15.jpg)
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
![Page 16: 1.1 real number system dfs](https://reader035.vdocuments.site/reader035/viewer/2022062522/58ceb81c1a28abb2218b6253/html5/thumbnails/16.jpg)
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
![Page 17: 1.1 real number system dfs](https://reader035.vdocuments.site/reader035/viewer/2022062522/58ceb81c1a28abb2218b6253/html5/thumbnails/17.jpg)
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
![Page 18: 1.1 real number system dfs](https://reader035.vdocuments.site/reader035/viewer/2022062522/58ceb81c1a28abb2218b6253/html5/thumbnails/18.jpg)
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
![Page 19: 1.1 real number system dfs](https://reader035.vdocuments.site/reader035/viewer/2022062522/58ceb81c1a28abb2218b6253/html5/thumbnails/19.jpg)
05/0
2/20
23
Adding and subtracting Integers
• 3 - 4 = ?
19
![Page 20: 1.1 real number system dfs](https://reader035.vdocuments.site/reader035/viewer/2022062522/58ceb81c1a28abb2218b6253/html5/thumbnails/20.jpg)
05/0
2/20
23
Adding and subtracting Integers
20
![Page 21: 1.1 real number system dfs](https://reader035.vdocuments.site/reader035/viewer/2022062522/58ceb81c1a28abb2218b6253/html5/thumbnails/21.jpg)
05/0
2/20
23
• Examples:• 7 + 5 = 12• -7 -5 = -12• -7 + 5 = -2• 7 – 5 = 2
21
![Page 22: 1.1 real number system dfs](https://reader035.vdocuments.site/reader035/viewer/2022062522/58ceb81c1a28abb2218b6253/html5/thumbnails/22.jpg)
05/0
2/20
23
Multiplying Integers• + . + = +• - . - = +• + . - = -• - . + = -
Examples: 7 x 5 = 35(-7)(-5) = 35 (-7)(5) = -35 (7)(– 5) =- 35
22
![Page 23: 1.1 real number system dfs](https://reader035.vdocuments.site/reader035/viewer/2022062522/58ceb81c1a28abb2218b6253/html5/thumbnails/23.jpg)
Rational and Irrational Numbers
‘ and
![Page 24: 1.1 real number system dfs](https://reader035.vdocuments.site/reader035/viewer/2022062522/58ceb81c1a28abb2218b6253/html5/thumbnails/24.jpg)
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
![Page 25: 1.1 real number system dfs](https://reader035.vdocuments.site/reader035/viewer/2022062522/58ceb81c1a28abb2218b6253/html5/thumbnails/25.jpg)
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
![Page 26: 1.1 real number system dfs](https://reader035.vdocuments.site/reader035/viewer/2022062522/58ceb81c1a28abb2218b6253/html5/thumbnails/26.jpg)
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
![Page 27: 1.1 real number system dfs](https://reader035.vdocuments.site/reader035/viewer/2022062522/58ceb81c1a28abb2218b6253/html5/thumbnails/27.jpg)
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
![Page 28: 1.1 real number system dfs](https://reader035.vdocuments.site/reader035/viewer/2022062522/58ceb81c1a28abb2218b6253/html5/thumbnails/28.jpg)
05/0
2/20
23
Irrational Numbers Irrational Numbers are decimals which are Never ending and Never Repeating.•
28
‘
![Page 29: 1.1 real number system dfs](https://reader035.vdocuments.site/reader035/viewer/2022062522/58ceb81c1a28abb2218b6253/html5/thumbnails/29.jpg)
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
‘
![Page 30: 1.1 real number system dfs](https://reader035.vdocuments.site/reader035/viewer/2022062522/58ceb81c1a28abb2218b6253/html5/thumbnails/30.jpg)
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
‘
![Page 31: 1.1 real number system dfs](https://reader035.vdocuments.site/reader035/viewer/2022062522/58ceb81c1a28abb2218b6253/html5/thumbnails/31.jpg)
05/0
2/20
23
Irrational NumberExamples of Irrational Numbers
31
![Page 32: 1.1 real number system dfs](https://reader035.vdocuments.site/reader035/viewer/2022062522/58ceb81c1a28abb2218b6253/html5/thumbnails/32.jpg)
05/0
2/20
23
Irrational Numbers
32
![Page 33: 1.1 real number system dfs](https://reader035.vdocuments.site/reader035/viewer/2022062522/58ceb81c1a28abb2218b6253/html5/thumbnails/33.jpg)
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
![Page 34: 1.1 real number system dfs](https://reader035.vdocuments.site/reader035/viewer/2022062522/58ceb81c1a28abb2218b6253/html5/thumbnails/34.jpg)
05/0
2/20
23
Rational and Irrational Numbers
• Rational Numbers: Either repeat, or terminate or both.
• Irrational Numbers: Neither repeat, nor terminate.
34
![Page 35: 1.1 real number system dfs](https://reader035.vdocuments.site/reader035/viewer/2022062522/58ceb81c1a28abb2218b6253/html5/thumbnails/35.jpg)
05/0
2/20
23
Real Number System
35
![Page 36: 1.1 real number system dfs](https://reader035.vdocuments.site/reader035/viewer/2022062522/58ceb81c1a28abb2218b6253/html5/thumbnails/36.jpg)
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
![Page 37: 1.1 real number system dfs](https://reader035.vdocuments.site/reader035/viewer/2022062522/58ceb81c1a28abb2218b6253/html5/thumbnails/37.jpg)
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
![Page 38: 1.1 real number system dfs](https://reader035.vdocuments.site/reader035/viewer/2022062522/58ceb81c1a28abb2218b6253/html5/thumbnails/38.jpg)
05/0
2/20
23
38
![Page 39: 1.1 real number system dfs](https://reader035.vdocuments.site/reader035/viewer/2022062522/58ceb81c1a28abb2218b6253/html5/thumbnails/39.jpg)
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
![Page 40: 1.1 real number system dfs](https://reader035.vdocuments.site/reader035/viewer/2022062522/58ceb81c1a28abb2218b6253/html5/thumbnails/40.jpg)
05/0
2/20
23
40