The document discusses the history and development of numbers and numerical systems. It begins with early counting methods using tallies and evolved to include the Egyptian, Babylonian, Roman, and Hindu-Arabic systems. The modern number system is then built up from the natural numbers to integers to rational numbers to real numbers, which include irrational numbers. Imaginary and complex numbers were later introduced to solve problems involving square roots of negative numbers. Place value systems and the ability to represent zero were important developments.
Mathematics for Primary School Teachers. Unit 2: NumerationSaide OER Africa
Mathematics for Primary School Teachers has been digitally published by Saide, with the Wits School of Education. It is a revised version of a course originally written for the Bureau for In-service Teacher Development (Bited) at the then Johannesburg College of Education (now Wits School of Education).
The course is for primary school teachers (Foundation and Intermediate Phase) and consists of six content units on the topics of geometry, numeration, operations, fractions, statistics and measurement. Though they do not cover the entire curriculum, the six units cover content from all five mathematics content areas represented in the curriculum.
This ppt slide is all about number system. here we learn-
To represent numbers
To know about different system
How number system works
The study of numbers is not only related to computers. We apply numbers everyday, and knowing how numbers work, will give us an insight of how computers manipulate and store numbers.
The document discusses different numeration systems used in various cultures and time periods around the world. It begins by explaining that the appropriate numbering system depends on the application. It then provides details on tally systems, the Hindu-Arabic numeral system, and various place-value systems including binary, decimal, and other bases. It also discusses the evolution of the concept of real numbers to include integers, rational numbers, and irrational numbers.
The document provides an overview of number systems used by different civilizations. It discusses ancient number systems including the Egyptian use of base-12 and Babylonian use of base-60. The decimal system is introduced as the most common modern system using base-10. Other number systems mentioned include binary, Mayan vigesimal, and fractions in ancient Egypt. Real numbers are defined as including integers, rational numbers like fractions, and irrational numbers like pi. Terminating, non-terminating recurring, and non-terminating non-recurring decimals are also briefly explained.
The document provides an overview of number systems throughout history. It discusses how ancient civilizations like the Egyptians and Babylonians experimented with different bases like base-12 and base-60 systems. It then covers the decimal system and describes number types like rational, irrational, integer, natural numbers and their properties. The document also discusses concepts like fractions in ancient Egypt, binary numbers and the expansion of numbers into terminating, non-terminating recurring and non-recurring decimals.
The document provides an overview of the history and types of number systems. It discusses how ancient civilizations like the Egyptians, Babylonians, and Mayans developed different base number systems based on counting fingers and toes. It then explains the modern decimal number system and provides examples of different types of numbers like rational, irrational, integer, natural numbers. The document also briefly touches on concepts like terminating and recurring decimals as well as scientists who contributed to the study of number systems.
This document contains a presentation on how mathematics is used in everyday life. It provides examples of how math is applied in art, business, cooking, music, and other domains. It also summarizes the contributions of important mathematicians like Pythagoras, Ramanujan, and properties of rational numbers, squares, cubes, and their roots. The presentation aims to demonstrate the pervasive and practical role of mathematics in various activities.
The document discusses operations on real numbers, including:
1) Addition and subtraction of real numbers follows rules based on sign, where numbers with the same sign are added and different signs are subtracted.
2) Multiplication and division of real numbers results in a positive number if the signs are the same, and negative if different.
3) Properties like commutativity, associativity, identity, inverse and distribution apply to real numbers as they do to other types of numbers.
Mathematics for Primary School Teachers. Unit 2: NumerationSaide OER Africa
Mathematics for Primary School Teachers has been digitally published by Saide, with the Wits School of Education. It is a revised version of a course originally written for the Bureau for In-service Teacher Development (Bited) at the then Johannesburg College of Education (now Wits School of Education).
The course is for primary school teachers (Foundation and Intermediate Phase) and consists of six content units on the topics of geometry, numeration, operations, fractions, statistics and measurement. Though they do not cover the entire curriculum, the six units cover content from all five mathematics content areas represented in the curriculum.
This ppt slide is all about number system. here we learn-
To represent numbers
To know about different system
How number system works
The study of numbers is not only related to computers. We apply numbers everyday, and knowing how numbers work, will give us an insight of how computers manipulate and store numbers.
The document discusses different numeration systems used in various cultures and time periods around the world. It begins by explaining that the appropriate numbering system depends on the application. It then provides details on tally systems, the Hindu-Arabic numeral system, and various place-value systems including binary, decimal, and other bases. It also discusses the evolution of the concept of real numbers to include integers, rational numbers, and irrational numbers.
The document provides an overview of number systems used by different civilizations. It discusses ancient number systems including the Egyptian use of base-12 and Babylonian use of base-60. The decimal system is introduced as the most common modern system using base-10. Other number systems mentioned include binary, Mayan vigesimal, and fractions in ancient Egypt. Real numbers are defined as including integers, rational numbers like fractions, and irrational numbers like pi. Terminating, non-terminating recurring, and non-terminating non-recurring decimals are also briefly explained.
The document provides an overview of number systems throughout history. It discusses how ancient civilizations like the Egyptians and Babylonians experimented with different bases like base-12 and base-60 systems. It then covers the decimal system and describes number types like rational, irrational, integer, natural numbers and their properties. The document also discusses concepts like fractions in ancient Egypt, binary numbers and the expansion of numbers into terminating, non-terminating recurring and non-recurring decimals.
The document provides an overview of the history and types of number systems. It discusses how ancient civilizations like the Egyptians, Babylonians, and Mayans developed different base number systems based on counting fingers and toes. It then explains the modern decimal number system and provides examples of different types of numbers like rational, irrational, integer, natural numbers. The document also briefly touches on concepts like terminating and recurring decimals as well as scientists who contributed to the study of number systems.
This document contains a presentation on how mathematics is used in everyday life. It provides examples of how math is applied in art, business, cooking, music, and other domains. It also summarizes the contributions of important mathematicians like Pythagoras, Ramanujan, and properties of rational numbers, squares, cubes, and their roots. The presentation aims to demonstrate the pervasive and practical role of mathematics in various activities.
The document discusses operations on real numbers, including:
1) Addition and subtraction of real numbers follows rules based on sign, where numbers with the same sign are added and different signs are subtracted.
2) Multiplication and division of real numbers results in a positive number if the signs are the same, and negative if different.
3) Properties like commutativity, associativity, identity, inverse and distribution apply to real numbers as they do to other types of numbers.
This document provides tips and tricks for analytical reasoning sections that commonly appear on aptitude tests. It discusses various question types like analogy, odd one out, relationships, series, coding/decoding, data sufficiency, statements and conclusions, visual reasoning, and logical reasoning. For each type, it provides examples of questions and explains conceptual approaches. It also recommends books for additional practice and skill development in analytical reasoning. The overall aim is to help students understand different reasoning concepts and scoring techniques to improve their performance on reasoning portions of employability tests.
The document contains 23 quantitative reasoning questions with numeric answers. It tests skills such as calculating percentages, solving equations, finding averages, and interpreting maps, diagrams, and word problems. The questions cover a wide range of math topics including rates, proportions, geometry, number properties, and algebra.
The document contains 15 multiple choice questions assessing verbal skills. The questions cover topics like sentence equivalence, distinguishing Latin and vernacular texts, evaluating tone and style in passages, and choosing words that complete sentences accurately. Each question is followed by 6 answer options in A-F format. The correct answers given are: Cf, Ab, Ac, Be, Cd, Df, Be, Be, Bc, Ac, Ce, Bf, Af, Cd, Ce.
1) A typist types 45 words per minute. Increasing their speed by 20%, they can now type 54 words per minute. In an hour (60 minutes) they can type 3,240 words.
2) Given the equations 2y - x = 8 and 3x - y = 1, the value of x is solved to be 2.
3) If the sum of four consecutive integers is 410, and consecutive integers increase by 1, the least of the integers is 101.
The document contains 30 questions from a verbal reasoning and sentence equivalence test. Each question has multiple answer choices and the correct answers are provided at the end. The questions cover a range of topics and test takers' ability to identify relationships between words and determine the meaning of sentences.
This document provides information and tips about the GRE text completion and sentence equivalence questions. It discusses the different types of blanks that may appear, the importance of identifying keywords and connectors, and strategies for choosing the best answer such as eliminating synonyms and having an answer in mind before looking at options. Positive and negative connectors are defined, as are cause-and-effect and implicit contrasting connectors. Practice is emphasized as crucial for improving performance on these question types.
Bloomberg Aptitude Test (BAT) Analytical Reasoning shivgan
1) Monica is at 25th place from top.
2) Total number of students is 50.
3) Then Monica must be at 26th place from bottom.
So, the total number of students = 50
The document contains 30 questions from a verbal reasoning test on topics like sentence equivalence, text completion, and logical reasoning. For each question, the correct answer is provided in a list at the end, with letters corresponding to answer choices. The test covers a range of vocabulary including words like ostentatious, laconic, vindicated, iconoclasm, and paradigm. Overall, the document provides answers to a verbal reasoning practice test on grammar, word meanings, and logical reasoning.
This document introduces analytical reasoning and provides guidance on how to solve analytical reasoning questions. It discusses that analytical reasoning questions present a complex situation and 3-7 related questions to test understanding. They can involve students in a row, committee members, or task scheduling. Solving requires understanding concepts and implications rather than formal logic. Understanding each detail is essential to answering all questions correctly. The document provides an overview of analytical reasoning and guidance on the skills needed to solve such questions.
GAT Preparation Book - www.NTSforums.comlahori_munda
The document discusses the benefits of meditation for reducing stress and anxiety. It states that regular meditation practice can calm the mind and body by lowering heart rate and easing muscle tension. Meditation may also make people feel more relaxed and in control of their thoughts.
This document provides information about a quantitative reasoning course. The course aims to help students gain a comprehensive understanding of mathematics and the ability to think critically and logically. It will cover topics like logical and quantitative thinking, arguments and reasoning, and the relationship between logic, science and mathematics. The course goals are to understand mathematics as a body of knowledge and a way of thinking, and to reason quantitatively on issues relevant to students and society. On completing the course, students should be able to analyze and evaluate arguments, understand mathematical concepts, and apply problem-solving skills to quantitative problems.
Teacher ::
a) Student: Class
b) Nurse: Doctor
c) Employee: Manager
d) Apprentice: Master
e) Secretary: President
The relationship between assistant and teacher is that of helper/aide to the person in charge. The parallel relationship here is apprentice to master, as an apprentice helps and learns from their master, similar to an assistant helping and learning from a teacher.
The answer is d) Apprentice: Master.
162 flashcards covering all of the formulas, concepts and strategies needed for the quantitative section of the GMAT. If, at any time, you need more information or instruction, each flashcard is linked to a video lesson (from GMAT Prep Now’s GMAT course)
This document discusses the history and development of numeral systems. It begins by explaining the key aspects of a numeral system and some of the earliest systems used, such as unary notation. It then describes the development of place-value systems, including the Hindu-Arabic decimal system. Various base systems are covered, such as base-2 (binary), base-5, base-8, base-10, base-12, base-20, and base-60. The document also discusses weighted and non-weighted binary coding systems, including excess-3 code and gray code. The history of binary numbers is outlined, from early concepts developed by ancient Indian and Chinese mathematicians to its modern implementation in digital circuits.
The document provides an overview of number systems used by different civilizations and an introduction to basic number concepts:
- It discusses ancient number systems including the Egyptian base-12 and Babylonian base-60 systems, as well as modern systems like binary and decimal.
- Basic number types are defined such as integers, rational numbers, irrational numbers, and real numbers. Fractions and decimal expansions are also introduced.
- Famous mathematicians who contributed to the study and development of number systems throughout history are acknowledged.
This document provides an overview of different number systems and concepts in mathematics related to numbers. It defines real numbers, rational numbers, integers, whole numbers, and natural numbers. It discusses that rational numbers can be divided into integers, whole numbers, and natural numbers. Irrational numbers are also introduced. Important mathematicians who contributed to the study and understanding of numbers are referenced, including Pythagoras, Archimedes, Aryabhatta, Dedekind, Cantor, Babylonians, and Euclid.
The Mesopotamian culture is often called Babylonian, after the lar.docxoreo10
The Mesopotamian culture is often called Babylonian, after the large metropolis of that name. We could “babble on”1 and on about their many fine achievements in architecture, irrigation, and commerce, but it is their mathematics that is truly remarkable, dwarfing that of other contemporary civilizations. One might not be impressed by their use of a vertical mark for “one” and a horizontal mark for “ten” – ten being a common unit in the mathematics of many societies, including Egypt, China, Rome, and our own society today. On the other hand, they were the first to employ a “positional” system which, except for minor changes, survives to this day!
1The authors would like to apologize for the easy pun, but we couldn’t resist.
Let’s remind ourselves how our current number system works. It does not suffice to say that it is based on grouping by tens. The Egyptians did this – yet we have left them in the dust by taking a giant step forward to the “position system.” We require only ten symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. Nevertheless, we can handle numbers of any size without the need to define a new symbol. This is because the value of a number is determined not just by the symbol. We must note the positionof the symbol as well. The two 3’s in the number 373 represent different quantities. You would rather have three hundred dollars than three dollars, right? To summarize, our number system employs a mere ten symbols, whose values depend on their position in the number. Moving one digit to the left multiplies its place value by ten, while moving to the right (not surprisingly) divides its place value by ten.
Observe, by the way, that this is true on both sides of the decimal point! In the number 3.1416, the 1 near the 6 is worth only one hundredth of the 1 near the 3. There is no number in the entire universe that is too large or too small for our clever (ten-digit!) number system (of Hindu-Arabic origin, by the way). We call our system the decimal system, because ten is the base.
The Babylonians used instead the sexagesimal system because they chose 60 as their base. While we are not sure why, we are fairly certain they did not have 60 fingers. One theory (which is very popular) is that 60 has a multitude of factors, that is, many numbers go into 60. Put another way, $60 can be divided without coin among 2, 3, 4, 5, 6, 10, 12, 15, 20, or 30 people. We shall follow the common practice of using commas to separate groups. Thus (3, 50)60 shall mean 3 sixties and 50 ones for a total of 230. What does (2, 3, 50)60 mean? Well in our position system, 357 means 3 hundreds, 5 tens, and 7 ones, right? Each column is ten times more valuable than its neighbor. In the same way, each column to the left in the Babylonian system is sixty times bigger! In the number (2, 3, 50)60, the 2 represents 2 3600’s – because 60 × 60 = 3600. The next column to the left would represent 60 × 3600 or 216000.
The Babylonians only used two symbols: a vertical mark for 1 and ...
This document provides an overview of the early development of number concepts and arithmetic. It discusses primitive counting methods using tally marks and how this led to more advanced systems like Roman numerals and the Hindu-Arabic number system. Key points covered include:
- Primitive counting involved using tally marks or objects to represent quantities, showing the earliest abstraction of numbers.
- Roman numerals and other early systems represented improvements but were still cumbersome for arithmetic.
- The Hindu-Arabic system introduced the place-value concept, allowing representation of any number using just 10 symbols.
- Early cultures also developed methods for addition, subtraction, multiplication and division, starting from concrete processes and advancing to more abstract algorithms and memorization of tables.
The document discusses several ancient numeration systems including the Egyptian, Babylonian, Roman, Mayan, and Hindu-Arabic systems. It provides examples of how each system represented numbers from 1 to 10. The Egyptian system used pictographs for the first nine numbers and logographs for higher numbers. The Babylonian system used a base-60 place value system with symbols for 1 and 10, requiring context to distinguish numbers. The Roman system used additive and subtractive principles with symbols for 1, 5, 10, 50, 100, 500, and 1000. The Mayan system was a base-20 place value system using dots, bars, and shells to represent numbers up to 19.
This document provides an overview of the number zero, including:
- Zero represents a count or amount of null size and plays a central role in mathematics as the additive identity.
- The concept of zero has evolved over time, from a placeholder used by ancient Babylonians to its formal definition in Brahmagupta's writings in the 7th century.
- Zero is now fundamental in fields like physics, where it represents the lowest possible value of some physical quantities, and computer science, where most modern programming languages use zero-based numbering.
This document provides tips and tricks for analytical reasoning sections that commonly appear on aptitude tests. It discusses various question types like analogy, odd one out, relationships, series, coding/decoding, data sufficiency, statements and conclusions, visual reasoning, and logical reasoning. For each type, it provides examples of questions and explains conceptual approaches. It also recommends books for additional practice and skill development in analytical reasoning. The overall aim is to help students understand different reasoning concepts and scoring techniques to improve their performance on reasoning portions of employability tests.
The document contains 23 quantitative reasoning questions with numeric answers. It tests skills such as calculating percentages, solving equations, finding averages, and interpreting maps, diagrams, and word problems. The questions cover a wide range of math topics including rates, proportions, geometry, number properties, and algebra.
The document contains 15 multiple choice questions assessing verbal skills. The questions cover topics like sentence equivalence, distinguishing Latin and vernacular texts, evaluating tone and style in passages, and choosing words that complete sentences accurately. Each question is followed by 6 answer options in A-F format. The correct answers given are: Cf, Ab, Ac, Be, Cd, Df, Be, Be, Bc, Ac, Ce, Bf, Af, Cd, Ce.
1) A typist types 45 words per minute. Increasing their speed by 20%, they can now type 54 words per minute. In an hour (60 minutes) they can type 3,240 words.
2) Given the equations 2y - x = 8 and 3x - y = 1, the value of x is solved to be 2.
3) If the sum of four consecutive integers is 410, and consecutive integers increase by 1, the least of the integers is 101.
The document contains 30 questions from a verbal reasoning and sentence equivalence test. Each question has multiple answer choices and the correct answers are provided at the end. The questions cover a range of topics and test takers' ability to identify relationships between words and determine the meaning of sentences.
This document provides information and tips about the GRE text completion and sentence equivalence questions. It discusses the different types of blanks that may appear, the importance of identifying keywords and connectors, and strategies for choosing the best answer such as eliminating synonyms and having an answer in mind before looking at options. Positive and negative connectors are defined, as are cause-and-effect and implicit contrasting connectors. Practice is emphasized as crucial for improving performance on these question types.
Bloomberg Aptitude Test (BAT) Analytical Reasoning shivgan
1) Monica is at 25th place from top.
2) Total number of students is 50.
3) Then Monica must be at 26th place from bottom.
So, the total number of students = 50
The document contains 30 questions from a verbal reasoning test on topics like sentence equivalence, text completion, and logical reasoning. For each question, the correct answer is provided in a list at the end, with letters corresponding to answer choices. The test covers a range of vocabulary including words like ostentatious, laconic, vindicated, iconoclasm, and paradigm. Overall, the document provides answers to a verbal reasoning practice test on grammar, word meanings, and logical reasoning.
This document introduces analytical reasoning and provides guidance on how to solve analytical reasoning questions. It discusses that analytical reasoning questions present a complex situation and 3-7 related questions to test understanding. They can involve students in a row, committee members, or task scheduling. Solving requires understanding concepts and implications rather than formal logic. Understanding each detail is essential to answering all questions correctly. The document provides an overview of analytical reasoning and guidance on the skills needed to solve such questions.
GAT Preparation Book - www.NTSforums.comlahori_munda
The document discusses the benefits of meditation for reducing stress and anxiety. It states that regular meditation practice can calm the mind and body by lowering heart rate and easing muscle tension. Meditation may also make people feel more relaxed and in control of their thoughts.
This document provides information about a quantitative reasoning course. The course aims to help students gain a comprehensive understanding of mathematics and the ability to think critically and logically. It will cover topics like logical and quantitative thinking, arguments and reasoning, and the relationship between logic, science and mathematics. The course goals are to understand mathematics as a body of knowledge and a way of thinking, and to reason quantitatively on issues relevant to students and society. On completing the course, students should be able to analyze and evaluate arguments, understand mathematical concepts, and apply problem-solving skills to quantitative problems.
Teacher ::
a) Student: Class
b) Nurse: Doctor
c) Employee: Manager
d) Apprentice: Master
e) Secretary: President
The relationship between assistant and teacher is that of helper/aide to the person in charge. The parallel relationship here is apprentice to master, as an apprentice helps and learns from their master, similar to an assistant helping and learning from a teacher.
The answer is d) Apprentice: Master.
162 flashcards covering all of the formulas, concepts and strategies needed for the quantitative section of the GMAT. If, at any time, you need more information or instruction, each flashcard is linked to a video lesson (from GMAT Prep Now’s GMAT course)
This document discusses the history and development of numeral systems. It begins by explaining the key aspects of a numeral system and some of the earliest systems used, such as unary notation. It then describes the development of place-value systems, including the Hindu-Arabic decimal system. Various base systems are covered, such as base-2 (binary), base-5, base-8, base-10, base-12, base-20, and base-60. The document also discusses weighted and non-weighted binary coding systems, including excess-3 code and gray code. The history of binary numbers is outlined, from early concepts developed by ancient Indian and Chinese mathematicians to its modern implementation in digital circuits.
The document provides an overview of number systems used by different civilizations and an introduction to basic number concepts:
- It discusses ancient number systems including the Egyptian base-12 and Babylonian base-60 systems, as well as modern systems like binary and decimal.
- Basic number types are defined such as integers, rational numbers, irrational numbers, and real numbers. Fractions and decimal expansions are also introduced.
- Famous mathematicians who contributed to the study and development of number systems throughout history are acknowledged.
This document provides an overview of different number systems and concepts in mathematics related to numbers. It defines real numbers, rational numbers, integers, whole numbers, and natural numbers. It discusses that rational numbers can be divided into integers, whole numbers, and natural numbers. Irrational numbers are also introduced. Important mathematicians who contributed to the study and understanding of numbers are referenced, including Pythagoras, Archimedes, Aryabhatta, Dedekind, Cantor, Babylonians, and Euclid.
The Mesopotamian culture is often called Babylonian, after the lar.docxoreo10
The Mesopotamian culture is often called Babylonian, after the large metropolis of that name. We could “babble on”1 and on about their many fine achievements in architecture, irrigation, and commerce, but it is their mathematics that is truly remarkable, dwarfing that of other contemporary civilizations. One might not be impressed by their use of a vertical mark for “one” and a horizontal mark for “ten” – ten being a common unit in the mathematics of many societies, including Egypt, China, Rome, and our own society today. On the other hand, they were the first to employ a “positional” system which, except for minor changes, survives to this day!
1The authors would like to apologize for the easy pun, but we couldn’t resist.
Let’s remind ourselves how our current number system works. It does not suffice to say that it is based on grouping by tens. The Egyptians did this – yet we have left them in the dust by taking a giant step forward to the “position system.” We require only ten symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. Nevertheless, we can handle numbers of any size without the need to define a new symbol. This is because the value of a number is determined not just by the symbol. We must note the positionof the symbol as well. The two 3’s in the number 373 represent different quantities. You would rather have three hundred dollars than three dollars, right? To summarize, our number system employs a mere ten symbols, whose values depend on their position in the number. Moving one digit to the left multiplies its place value by ten, while moving to the right (not surprisingly) divides its place value by ten.
Observe, by the way, that this is true on both sides of the decimal point! In the number 3.1416, the 1 near the 6 is worth only one hundredth of the 1 near the 3. There is no number in the entire universe that is too large or too small for our clever (ten-digit!) number system (of Hindu-Arabic origin, by the way). We call our system the decimal system, because ten is the base.
The Babylonians used instead the sexagesimal system because they chose 60 as their base. While we are not sure why, we are fairly certain they did not have 60 fingers. One theory (which is very popular) is that 60 has a multitude of factors, that is, many numbers go into 60. Put another way, $60 can be divided without coin among 2, 3, 4, 5, 6, 10, 12, 15, 20, or 30 people. We shall follow the common practice of using commas to separate groups. Thus (3, 50)60 shall mean 3 sixties and 50 ones for a total of 230. What does (2, 3, 50)60 mean? Well in our position system, 357 means 3 hundreds, 5 tens, and 7 ones, right? Each column is ten times more valuable than its neighbor. In the same way, each column to the left in the Babylonian system is sixty times bigger! In the number (2, 3, 50)60, the 2 represents 2 3600’s – because 60 × 60 = 3600. The next column to the left would represent 60 × 3600 or 216000.
The Babylonians only used two symbols: a vertical mark for 1 and ...
This document provides an overview of the early development of number concepts and arithmetic. It discusses primitive counting methods using tally marks and how this led to more advanced systems like Roman numerals and the Hindu-Arabic number system. Key points covered include:
- Primitive counting involved using tally marks or objects to represent quantities, showing the earliest abstraction of numbers.
- Roman numerals and other early systems represented improvements but were still cumbersome for arithmetic.
- The Hindu-Arabic system introduced the place-value concept, allowing representation of any number using just 10 symbols.
- Early cultures also developed methods for addition, subtraction, multiplication and division, starting from concrete processes and advancing to more abstract algorithms and memorization of tables.
The document discusses several ancient numeration systems including the Egyptian, Babylonian, Roman, Mayan, and Hindu-Arabic systems. It provides examples of how each system represented numbers from 1 to 10. The Egyptian system used pictographs for the first nine numbers and logographs for higher numbers. The Babylonian system used a base-60 place value system with symbols for 1 and 10, requiring context to distinguish numbers. The Roman system used additive and subtractive principles with symbols for 1, 5, 10, 50, 100, 500, and 1000. The Mayan system was a base-20 place value system using dots, bars, and shells to represent numbers up to 19.
This document provides an overview of the number zero, including:
- Zero represents a count or amount of null size and plays a central role in mathematics as the additive identity.
- The concept of zero has evolved over time, from a placeholder used by ancient Babylonians to its formal definition in Brahmagupta's writings in the 7th century.
- Zero is now fundamental in fields like physics, where it represents the lowest possible value of some physical quantities, and computer science, where most modern programming languages use zero-based numbering.
This document discusses number systems and different types of numbers. It begins by introducing rational numbers as numbers that can be written as fractions with integer numerators and denominators. Irrational numbers are then defined as numbers that cannot be expressed as rational numbers. Some examples of irrational numbers like square roots and pi are provided. Finally, it is explained that the set of all rational and irrational numbers together make up the real numbers, and every real number can be represented by a unique point on the number line.
The document summarizes the history and development of the concept of zero. It discusses how zero was conceptualized and used in different ancient civilizations like the Maya, Babylonians, Indians, and Chinese. Key developments include the Maya using zero as a placeholder in their calendar system, the Babylonians using a placeholder in their place value system without treating it as a number, Indians developing the concept of zero as a number in the 9th century, and Chinese using empty space in counting rods to represent zero. The document also outlines the importance of zero in developing the place value number system and its role in mathematics and measuring physical quantities.
This document introduces different number systems, including natural numbers, whole numbers, integers, rational numbers, and irrational numbers. It defines each type of number and provides examples. Rational numbers can be written as fractions using integers, while irrational numbers cannot be expressed as ratios of integers. Together, rational and irrational numbers make up the set of real numbers, which correspond one-to-one with points on the real number line. Famous mathematicians like Pythagoras, Cantor, and Dedekind contributed to the understanding and classification of these different number systems.
2.1 numbers and their practical applications(part 2)Raechel Lim
This document discusses the development of the modern number system, including natural numbers, integers, rational numbers, real numbers, imaginary and complex numbers. It provides examples and definitions of these different types of numbers. The document also focuses on prime numbers, describing properties like twin primes and Mersenne primes. Methods for finding primes like the Sieve of Eratosthenes are explained. Applications of prime numbers in cryptography and factoring are also mentioned.
Zero is a number that represents nothing or empty space. It was invented in ancient India and was an important development that allowed more advanced mathematics. Zero holds a central role and is the identity element in addition. It allows place-value systems like binary to function and is essential for computer science. Zero's existence was initially debated but is now widely used and crucial for science, mathematics, banking, and many other aspects of modern life.
This is meant for age group 11 to 14 years.
For Class VIII CBSE.
Some viewers have requested me to send the file through mail.
So I allowed everybody to download.My request is whenever you are using plz acknowledge me.
Pratima Nayak ,Teacher,Kendriya Vidyalaya,Fort William,Kolkata
pnpratima@gmail.com
Based on Text book
The document discusses the history and importance of zero. It describes how zero emerged over thousands of years, starting with early cultures like the Egyptians, Greeks, Romans, and Babylonians making early uses of placeholders or empty values without a true numerical concept of zero. Zero was formally developed in India and spread through Arabic mathematicians. It was resisted in Europe but became widely used by the 1500s. Zero is now recognized as a crucial concept in mathematics and other fields as a placeholder, separator of positive and negative numbers, and allowing calculations and systems like computers.
Roman numeral symbols have values that correspond to numbers, such as I=1 and X=10. While Roman numerals are not well-suited for complex math, basic operations like addition and subtraction can be performed by canceling out or converting matching symbols between the numerals. The document demonstrates adding and subtracting Roman numerals through examples such as XXXXVIII + XXIIII = LXXXXVI and CLXXXXVI - XXXXVIII = CXXXXVIII. It also notes that some typical Roman numeral representations, such as XLVIII for 48, are incorrect and exceed the true value.
The document provides a history of mathematics from ancient civilizations to modern times. It discusses how mathematics originated in ancient Babylon and Egypt around 3000 BC, with the Egyptians having an advanced decimal number system and knowledge of geometry. It also describes the Babylonian sexagesimal system of representing numbers. Key figures discussed include Pythagoras, whose theorem is summarized, and Carl Gauss, a mathematical genius whose contributions transformed number theory and other fields in the 19th century. The document concludes by outlining the fundamental purposes and objectives of teaching mathematics.
The document provides a timeline of key developments in mathematics from 6000 BCE to the present. Some of the highlights include:
- The earliest written Egyptian numbers dating back to 2700 BCE which used symbols for units, tens, hundreds, and thousands.
- Babylonian mathematics from 1800 BCE which had multiplication tables and worked on solving quadratic and cubic equations.
- Early Chinese mathematics from 1600 BC which included the use of an efficient decimal place value system using bamboo rods.
- Indian mathematics from 1000 BCE which developed concepts like zero, negative numbers, and trigonometry that were later transmitted worldwide.
- Classical Greek mathematics from 624 BC which included theorems attributed to Thales and Euclid's Elements textbook.
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2. 2.1 Numbers and Their
Practical Applications
Goals:
1.To integrate quantitative reasoning with logical
reasoning by investigating the concept of
numbers; and
2.To develop methods for interpreting large and
small numbers.
3. 2.1.1 The Concept of Number
and the Language of
Nature
This section includes a discussion of:
The history of numbers
How numbers are used
The modern system of numbers
4. 4
Mathematics is used to model andMathematics is used to model and
describe natural phenomena.describe natural phenomena.
It is also used to model phenomenaIt is also used to model phenomena
of human nature, including manyof human nature, including many
economic and social interactions.economic and social interactions.
Mathematics is said to beMathematics is said to be the languagethe language
of natureof nature..
5. (REVENUE)(REVENUE)
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6. It has its own vocabulary and its own grammatical
rules.
It uses many symbols, equations and other
terminology.
It expresses ideas through numbers (just as
spoken languages through words) which may be
written and used in many different ways.
As other languages, it uses abstraction.
Mathematical statements always can be
translated into English (or other languages).
7. 2.1.2 A Brief History of Numbers
No single, formal definition of the concept
of numbers exists. The term numbers is used
to describe many different ideas.
The concept of numbers has evolved over
time. It developed in parallel with methods
for writing numerals which are symbols that
represent numbers.
8. The Origin of Modern Numerals
Numbers were originally used for simple
counting.
Numeral systems relied on tallies with fingers or
toes, piles of stones, or notches cut on a bone or a
piece of wood. But these systems are inadequate
for large numbers.
To simplify the process of counting, counts are
grouped by 2’s, 3’s, then eventually, by 5’s, 10’s,
and 20’s.
In 3000 B.C., the Egyptians and Babylonians
independently introduced the first numeral
system to go beyond tallying.
12. Modern numerals trace directly to the work of
Hindu mathematicians in India in the first few
centuries A.D.
In A.D. 800, Hindu numerals became part of
the Arab culture when major Hindu works on
astronomy were translated into Arabic.
The Arabs then led the development of
mathematics during the next several centuries.
The written shapes of the Hindu numerals
slowly changed over time. As a result, modern
numerals are called Hindu-Arabic numerals.
These numerals took their current form when
great works of Arab mathematicians were
translated into Latin in about A.D. 1200.
13. Thinking About:
The Uses of Numbers
1. Counting. Cardinal numbers
answer the question “how
many?”.
3. Labeling. Nominal numbers are
used as labels or names.
2. Ordering. Ordinal numbers
indicate the order of members in
a set.
14. Additive Numeral Systems
The Roman Numerals
Many other systems of numerals came into
use. The Roman numerals, developed in 500
B.C., were used in ancient Greece and Rome,
and became dominant in Europe for more
than 1,000 years.
It is an additive numeral system, in which
values are determined by adding the values of
individual symbols. The position of the symbol
does not affect its value.
It does not have a symbol for zero.
15. Subtraction was introduced only in the 16th or
17th century, but by then Hindu-Arabic numerals
were far more common.
They were less useful because: (1) writing large
numbers is extremely difficult, and (2) they offer
no convenient way to represent fractions.
They are still used for decorative or artistic
purposes.
16. Place-Value Numeral Systems
The Decimal System
The Hindu-Arabic system is a decimal, or base-10, place-
value system. Decimal (from Lat. decimus meaning “tenth”).
The value of a numeral depends on its place or position.
The symbols 0,1,2,…,9 are called digits (from Lat. digitus
which means “finger”)
Place-value systems require a symbol for zero, which is
crucial in the development of the modern number system.
Zero became a meaningful number only in A.D. 600 when
Hindu mathematicians introduced it.
The Mayan civilization in America however independently
developed the idea of zero 500 years earlier.
18. Decimal Fractions
The Babylonians in 2000 B.C. invented the
method of writing fractions, which was nearly
identical to the modern method of writing
decimal fractions.
However, the Babylonian system was based on
powers of 60 instead of powers of 10.
The method of writing fractions with a numerator
and denominator was probably developed by
Hindu mathematicians.
19. Numbers in other bases
The binary system is a place-value system
that uses only two symbols, 0 and 1, called
bits or binary digits.
It is easy to convert base-2 numeral to
base-10 numeral.
A numeral in any base can be represented
in any other base.
20. 20
From base 2 to base 10
112 = 1*21
+ 1*20
= 2 + 1 = 3
1012 = 1*22
+ 0*21
+ 1*20
= 4 + 1 = 5
From base 10 to base 2
21 =16 + 4 + 1
= 24
+ 22
+ 20
= 1*24
+ 0*23
+ 1*22
+ 0*21
+ 1*20
= 101012
21. 21
Base 10Base 10 As a sum of powers ofAs a sum of powers of
22
Base 2Base 2
11 11
22 1010
33 1111
44 100100
55 101101
66 110110
77 111111
88 10001000
99 10011001
1010 10101010
1
2 + 0
2 + 1
22
22
+ 1
22
+ 2
22
+ 2 + 1
23
23
+ 1
23
+ 2
22. The Babylonian system is a base-60 system.
Vestiges of this remain in time keeping (1 hour
= 60 minutes, 1 minute = 60 seconds), and in
angle measurement.
The Mayans used base-20 system.
The base system used for computers is base-2.
23. 2.1.3 Building the Modern Number System
The Natural Numbers
We build the modern number system beginning with
numbers used for counting.
Counting numbers, or natural numbers, comprise the set
{1,2,3,4,…}.
Natural numbers are further categorized according to their
factors (or divisors). Natural numbers are either prime or
composite.
The Fundamental Theorem of Arithmetic: every composite
number can be uniquely expressed as a product of prime
numbers.
25. The Integers
Negative numbers came out of subtracting natural
numbers.
Uses of negative numbers:
1. In commerce, where debts and losses are
represented by negative numbers.
2. In temperature and elevation measurements
The set of all numbers that we can make by
adding or subtracting natural numbers is called
the set of integers.
26. The integers include the natural numbers,
also called positive integers, zero, and the
negatives of all natural numbers, or negative
integers.
The set of whole numbers comprise zero and
the positive integers.
Properties of Integers:
1. Every integer, except 0, has a sign
which indicates whether it is positive
(+) or negative (-). 0 is neither positive nor
negative.
2. Every integer has a magnitude
(or absolute value) which indicates
how far it lies from 0 on the number
line.
27. The Rational Numbers
The set of all possible outcomes of dividing
integers (except dividing by 0) is called the set
of rational numbers.
Rational (from the word ratio which refers to
the division of two numbers)
The set of rational numbers is the set of all
numbers that can be expressed in the form x/y
where both x and y are integers and y ≠ 0.
The set of integers is a subset of the set of
rational numbers.
28. At one time in ancient Greece, all numbers were
believed to be rational numbers.
A secret society of followers of Pythagoras (500
B.C.) believed that numbers had special and
mystical meanings. Examples:
1 was considered divine.
Even numbers were considered feminine.
Odd numbers besides 1 were considered masculine.
The number 5, sum of the first feminine and masculine numbers,
represented marriage.
7 represented the seven “planets” known to the Greeks; the belief
that 7 was a “lucky number” probably came from them.
29. The motto of the Pythagoreans: “All is number.”
Their sacred belief was that all numbers were either “whole”,
by which they meant the natural numbers (they did not
recognize zero or negative numbers), or fractions made by
the division of “whole” numbers.
But using the Pythagorean Theorem, they realized that a
right triangle with two sides of length 1 has a third side of
length equal to the square root of 2 , which they could not
express as a fraction.
Eventually they proved that the square root of 2 cannot be
expressed by dividing two whole numbers, that is, it is an
irrational number.
They attempted to keep this as a secret because their
fundamental beliefs may be challenged, and even killed one
of their members, Hippasus, for telling others of their
discovery.
30. The Real Numbers
The combination of the rational and irrational
numbers is called the real numbers.
Another way to describe real numbers is as the
rational numbers and “everything in between”.
Each point on the number line has a
corresponding real number, and vice versa.
31. Imaginary and Complex Numbers
Finding a real number that is a square root of a
negative number is impossible. Thus another type of
“non-real” numbers, called imaginary numbers, was
invented to solve this problem.
Imaginary numbers are numbers that represent the
square root of negative numbers.
A special number called i (for “imaginary”) is defined
to be the square root of negative 1.
Imaginary numbers cannot be shown on a real number
line because they are not real numbers.
The complex numbers are numbers that include all the
real numbers and all the imaginary numbers.
32. Complex Numbers
Real Numbers Imaginary Numbers
Irrational Rational
Integers Other fractions
Negative 0 Positive
Prime 1 Composite
33. Numbers are Beautiful
If you don’t see why,
No one can tell you.
If they aren’t beautiful,
Nothing is.
-Paul Erdös
33
34. 34
NameName U.S. meaningU.S. meaning
MillionMillion 1,000,000 (6 zeros)1,000,000 (6 zeros)
BillionBillion 1,000,000,000 (9 zeros)1,000,000,000 (9 zeros)
TrillionTrillion 1,000,000,000,000 (12 zeros)1,000,000,000,000 (12 zeros)
QuadrillionQuadrillion 1 followed by 15 zeros1 followed by 15 zeros
QuintillionQuintillion 1 followed by 18 zeros1 followed by 18 zeros
SextillionSextillion 1 followed by 21 zeros1 followed by 21 zeros
SeptillionSeptillion 1 followed by 24 zeros1 followed by 24 zeros
OctillionOctillion 1 followed by 27 zeros1 followed by 27 zeros
Names and values of numbers
35. 35
Metric Prefixes for small values
prefixprefix Abbrev.Abbrev. valuevalue
decideci dd 1010-1-1
centicenti cc 1010-2-2
millimilli mm 1010-3-3
micromicro µµ 1010-6-6
nanonano nn 1010-9-9
picopico pp 1010-12-12
femtofemto ff 1010-15-15
attoatto aa 1010-18-18
zeptozepto zz 1010-21-21
yoctoyocto yy 1010-24-24
=
ns
s
1
1µ
s
s
9
6
10
10
−
−
.,000110 3
== −
36. 36
Metric Prefixes for large values
prefixprefix Abbrev.Abbrev. valuevalue
decadeca dada 101011
hectohecto hh 101022
kilokilo kk 101033
megamega MM 101066
gigagiga GG 101099
teratera TT 10101212
petapeta PP 10101515
exaexa EE 10101818
zettazetta ZZ 10102121
yottayotta YY 10102424
37. 2.1.4 Prime Numbers: Mysteries and
Applications
It is difficult to generate the sequence of prime
numbers.
Basic question: How many primes are there?
Euclid (c. 300 B.C.) proved that there are infinitely
many primes.
Erathosthenes, Greek mathematician who lived in
the third century B.C., devised a systematic
method for generating primes, called the Sieve of
Erathosthenes.
38. The Search for the Largest Prime
As of 2008, the largest prime is
with 12978189 digits. This is also the
largest prime so far.
38
1243112609
−
12 −p
• This is called a Mersenne prime, named
after F. Marin Mersenne. For a prime p, a
Mersenne prime is any prime of the form
39. Other Interesting Primes
Twin primes are two primes of the form p and p+2. The
largest as of 2009 is
with 100355 digits.
39
1256551646835 333333
±⋅
122
+
n
• A Fermat prime is a prime expressible in the
form
where .n 0≥
40. The Sieve of Erathosthenes (An
Algorithm)
Given a list of natural numbers from 1 to n.
Cross out 1 because it is neither prime nor composite.
The next number 2 is prime; cross out all subsequent multiples of
2 because they are composite. (We call 2 a sieve number because
it helps us “sift through” or remove other numbers in the list.)
The next number 3 is prime; then cross out all subsequent
multiples of 3.
Move to the next number that has not been crossed out. Use 5 as
a sieve number and cross out all multiples of 5 that have not been
crossed out yet.
Continue this process until we reach the end of the list.
The numbers that remain after all the “crossings out” are the
primes on the list.
42. In principle, the sieve of Erathosthenes could be used
on a list of numbers of any length.
The method, however, is extremely tedious.
Kulik, a 19th century Austrian astronomer, spent 20
years using this method to find all primes between 1
and 100 million!
Moreover, the library to which he gave his manuscripts
lost the sections containing the primes between
12,642,000 and 22,852,800.
43. A formula to produce primes could generate
lists with much less effort.
Mathematicians have searched in vain for such
a formula for more than 2,000 years!
A few formulas work over a limited range of
numbers before failing.
Example: the expression n2
- n + 41 successfully
produces primes for small values of n. This
formula fails after n = 41, that is, it generates
only 41 primes before it produces a composite
number. In addition, the formula misses all
primes less then 41.
Other formulas also fail. Mathematicians
believe that a suitable formula does not exist.
44. The Search is On!
Great Internet Mersenne Prime Search
A project pioneered by George Woltman in 1997
A free software
So far has discovered 13 Mersenne primes
The 47th
Mersenne prime is the largest prime, which has been
discovered before the 45th
and 46th
.
44
45. Applications of prime numbers
In cryptography, wherein messages are written in code
to protect privacy and maintain security.
A security system can use large composite numbers as
a lock. The two primes multiplied to make the
composite represent the keys.
Because there is no efficient way to find the prime
factorization, the lock can be opened only by people
who hold the keys.
Research seeks efficient methods of factoring large
numbers, and computers are getting faster. But as
larger and larger primes are found, more inviolable
locks can be designed.
46. 46
Divisibility rules for 2,3,4,5,9,10
All even numbers are divisible by 2.
A number is divisible by 3 if the sum of its
digits is divisible by 3.
A number is divisible by 4 if the last 2 digits is
divisible by 4.
Example: 2451 is divisible by 3 since
(2+4+5+1) = 12 is divisible by 3.
A number is divisible by 5 if its last digit is 0 or
5.
47. 47
A number is divisible by 9 if the sum of its
digits is divisible by 9.
Example: 23454 is divisible by 9 since
2 + 3 +4 + 5 + 4 = 18 is divisible by 9.
A number is divisible by 10 if its last digit is 0.
48. 2.1.5 Infinity
Infinity may be the most astonishing aspect of the
concept of numbers.
Georg Cantor (1845-1918) began a serious study of
infinity over a century ago. His results shocked
the mathematical world at that time.
Cardinality is another term for the number of
elements of a set.
49. How can you determine whether two sets have the
same cardinality?
One way is by counting the elements of each set
and see whether the count is the same for both
sets.
Another way is to determine if there is a one-to-
one correspondence between the members of the
two sets.
50. The Paradox of Infinite Sets
The set of natural numbers and the set of even integers
have the same cardinality.
1 2 3 4 5 6 …
2 4 6 8 10 12 ...
Similarly, there are as many natural numbers as odd
numbers, and as many natural numbers as multiples of
3, etc.
Galileo in 1638 considered these as unexplainable
paradoxes and chose not to work with infinity further.
51. The Arithmetic of Infinity
After 250 years, Cantor took these paradoxes as
starting points for further work and invented a new
arithmetic, called transfinite arithmetic, that applies to
infinity.
The cardinality of the natural numbers is symbolized
by ℵo (pronounced “aleph naught” or aleph null”).
We have the following results:
ℵo+ 1= ℵo and ℵo+ ℵo= ℵo
52. Consider the set of positive
rational numbers. What is the
cardinality of this set?
The infinite array above contains all positive rational numbers. We haveThe infinite array above contains all positive rational numbers. We have
the one-to-one correspondence:the one-to-one correspondence:
11 22 33 44 55 66 77 88 ......
1/11/1 1/21/2 2/12/1 3/13/1 2/22/2 1/31/3 1/41/4 2/32/3 ......
1 2 3 4 5 …
1 1/1 1/2 1/3 1/4 1/5
2 2/1 2/2 2/3 2/4 2/5
3 3/1 3/2 3/3 3/4 3/5
4 4/1 4/2 4/3 4/4 4/5
5 5/1 5/2 5/3 5/4 5/5
.
53. Thus, there are as many natural numbers as
there are rational numbers!
And we have the result:
ℵox ℵo= ℵo and (ℵo)2
= ℵo
Similarly, (ℵo)3
= ℵo and so on.
54. The Bane of Pythagoreans
Cantor showed that some infinite sets have cardinality
greater than ℵo.
He showed that the real numbers cannot be put into
one-to-one correspondence with the natural numbers,
and that there are more irrationals than rationals.
In fact, between any two points in the number line, the
number of irrationals is greater than the number of all
rational numbers.
The cardinality of this new, higher infinity is
designated ℵ1.
55. Thus the symbol ∞ cannot be used because
infinity has more than one “level”.
Outline of Cantor’s argument that there are more
irrationals than rationals:
Irrational numbers cannot be written exactly in decimal form
because they are non-terminating decimals. Suppose now that
there is a scheme for matching the irrationals to the natural
numbers, as in the list:
1 → 0.142678435…
2 → 0.383902892…
3 → 0.293758778…
4 → 0.563856365…
:
56. Regardless of the method used for matching,
we will always be able to write another
irrational number that is not already on the
list.
To do so, for the first digit of the new number,
we choose something other than the first digit
of the first number on the list, that is, anything
other than 1. For the second digit, we choose
something other than the second digit of the
second number on the list, or something other
than 8. And so on to infinity.
57. The resulting irrational number will differ in at
least one digit from every number on the list.
In other words, we will have found a number
that was “missed” by the matching scheme.
Thus, the natural numbers cannot be put in
one-to-one correspondence with the
irrationals.
Our conclusion: the cardinality of the
irrationals is greater than that of either the
natural or rational numbers.
58. Higher Orders of Infinity
Does a level of infinity exist between
ℵo and ℵ1?
The answer is unknown, but a set with such
cardinality has never been found.
The “continuum hypothesis” says that no set with
such cardinality exists.
59. Does a set with higher cardinality than the
reals exist?
Yes, as Cantor proved.
In fact, he showed that an infinite number of
higher levels of infinity exist, and their cardinality
might be designated
ℵo,ℵ1, ℵ2, ℵ3, ℵ4,...
But no one has ever been able to describe a set
with an infinity higher than ℵ2.
60. 2.1.6 Putting Numbers in Perspective
In ancient times, there was no way to express
extremely large or small numbers; in fact it was
unnecessary.
Today, these seemingly incomprehensible numbers
are dealt with in the real world.
Goal: learn to think quantitatively by developing
methods for interpreting such numbers.
61. tens or hundreds of billions of pesos of
spending and taxation
the collective impact of six billion people on
the environment
a nuclear weapon with one megaton of
explosive power
a computer with gigabytes of memory and
processing times measured in nanoseconds,
microseconds or milliseconds
Can you assess the values of these numbers?Can you assess the values of these numbers?
62. Survival and prosperity in the modern world
depend on decisions that involve numbers that
may, at first, seem incomprehensibly large or
small.
To make wise decisions, you must find ways of
putting such numbers into perspective.
Our task: to learn how to make extremely large or
small numbers comprehensible by relating them
to numbers which we are already familiar with.
63. 2.1.7 Writing Large or Small Numbers
Consider the following numbers:
The diameter of the Galaxy is about
1,000,000,000,000,000,000 kilometers
The nucleus of a hydrogen atom has a
diameter of about 0.000000000000001 meters
These numbers are difficult to read and most
people will just skip right over them. There is a
better way of expressing such numbers.
64. The Scientific Notation
Dealing with large and small numbers is much
easier with a special notation.
Numbers written with a number between
1 and 10 multiplied by a power of 10 are said to
be in scientific notation.
A number written in scientific notation can be
quickly converted to ordinary notation.
There is no shortcut for adding or subtracting
numbers in scientific notation.
65. Advantages of the Scientific Notation
The scientific notation simplifies writing extremely
large or small numbers.
Rounding and expressing numbers in scientific
notation allow quick approximations of the exact
answers.
Example: Estimate the product of 5795
and 326.
66. The danger of scientific notation
The scientific notation makes extremely large or
small numbers deceptively easy to write.
Example:
1026
does not look much different from 1020
, when
written, but is a million times larger