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01
A . R A T I P O R N C H O M R I T
NUMBER
Marketing Channel 1 Marketing Channel 2
Further elaborate on
the channel.
A number is an arithmetic value used for representing the quantity and used in making calculations. A written
symbol like “3” which represents a number is known as numerals. A number system is a writing system for
denoting numbers using digits or symbols in a logical manner.
Represents a useful set of numbers
Reflects the arithmetic and algebraic structure of a number
Provides standard representation
We use the digits from 0 to 9 to form all other numbers.
0 1 2 3 4 5 6 7 8 9
•Real Numbers: All the positive and negative integers, fractional and decimal numbers without imaginary
numbers are called real numbers. It is represented by the symbol “R”.
•Rational Numbers: Any number that can be written as a ratio of one number over another number is
written as rational numbers. This means that any number that can be written in the form of p/q. The
symbol “Q” represents the rational number.
•Irrational Numbers: The number that cannot be expressed as the ratio of one over another is known as
irrational numbers and it is represented by the symbol ”P”.
NUMBER
The different types of numbers in maths are:
•Natural Numbers: Natural numbers are known as counting numbers that contain the positive integers from 1 to infinity.
The set of natural numbers is denoted as “N” and it includes
•The natural number set is defined by: N = {1, 2, 3, 4, 5, ……….}
Examples: 35, 59, 110, etc.
Properties of Natural Numbers:
•Addition of natural numbers is closed, associative, and commutative.
•Natural Number multiplication is closed, associative, and commutative.
•The identity element of a natural number under addition is zero.
•The identity element of a natural number under Multiplication is one.
•Whole Numbers: Whole numbers are known as non-negative integers and it does not include any fractional or decimal
part. It is denoted as “W” and the set of whole numbers includes W = {0,1, 2, 3, 4, 5, ……….}
•Whole numbers are also known as natural numbers with zero. The set consists of non-negative integers where it does
not contain any decimal or fractional part.
The whole number set is represented by the letter “W”. The natural number set is defined by:
•W = {0,1, 2, 3, 4, 5, ……….}
•Examples: 67, 0, 49, 52, etc.
NUMBER
•Prime Numbers
A prime number is the one which has exactly two factors, which means, it can be divided by only “1” and itself.
But “1” is not a prime number.
•Example of Prime Number
•3 is a prime number because 3 can be divided by only two number’s i.e. 1 and 3 itself.
•3/1 = 3
•3/3 = 1
•In the same way, 2, 5, 7, 11, 13, 17 are prime numbers.
•Composite Numbers
A composite number has more than two factors, which means apart from getting divided by the number 1 and itself, it can
also be divided by at least one integer or number. We don’t consider ‘1’ as a composite number.
•Example of Composite Number
•12 is a composite number because it can be divided by 1, 2, 3, 4, 6 and 12. So, the number ‘12’ has 6 factors.
•12/1 = 12
•12/2 =6
•12/3 =4
•12/4 =3
•12/6 =2
•12/12 = 1
NUMBER
Number
Real Number
Irrational Number
Rational Number
Integer
Fraction
Zero
Negative Integer Positive Integer
Rational Numbers
Rational number is a type of real number, which is in the form of p/q where q is not equal to zero.
Some of the examples of rational numbers are 1/2, 1/5, 3/4, and so on. The number “0” is also a rational
number, as we can represent it in many forms such as 0/1, 0/2, 0/3, etc. But, 1/0, 2/0, 3/0, etc. are not
rational.
A rational number can be represented by the letter “Q”.
• Integer
• Fraction
• Repeating decimal
Irrational Numbers
The number that cannot be expressed in the form of p/q. It means a number that cannot be written as
the ratio of one over another is known as irrational numbers.
It is represented by the letter ”P”.
Examples: √2, π, 5.18118168473465 etc
NUMBER
•Integers: Integers are the set of all whole numbers but it includes a negative set of natural numbers also. “Z” represents
integers and the set of integers are Z = { -3, -2, -1, 0, 1, 2, 3}
•Integers are defined as the set of all whole numbers with a negative set of natural numbers. The integer set is represented by
the symbol “Z”. The set of integers is defined as:
•Z = {-3, -2, -1, 0, 1, 2, 3}
•Examples: -52, 0, -1, 16, 82, etc.
•Properties of Integers:
•Integers are closed under addition, subtraction, and multiplication.
•The commutative property is satisfied for addition and multiplication of integers.
•It obeys the associative property of addition and multiplication.
•It obeys the distributive property for addition and multiplication.
•Additive identity of integers is 0.
•Multiplicative identity of integers is 1.
NUMBER
Fractions
•Fractions, in Mathematics, are represented as a numerical value, which defines a part of a whole.
A fraction can be a portion or section of any quantity out of a whole, where the whole can be any number,
a specific value, or a thing.
Example. The following figure shows a pizza that is divided into 8 equal parts. Now, if we want to express
one selected part of the pizza, we can express it as 1/8 which shows that out of 8 equal parts, we are
referring to 1 part.
It means one in eight equal parts. It can also be read as:
•One-eighth, or
•1 by 8
NUMBER
Fractions and Decimals
Fractions and Decimals
Parts of a Fraction
All fractions consist of a numerator and a denominator and they are separated by a
horizontal bar known as the fraction bar.
• The numerator indicates how many sections of the fraction are represented or
selected. It is placed in the upper part of the fraction above the fractional bar.
• The denominator indicates the number of parts in which the whole has been
divided into. It is placed in the lower part of the fraction below the fractional bar.
numerator
denominator
Fractions and Decimals
Decimals are one of the types of numbers, which has a whole number and the fractional part
separated by a decimal point. The dot present between the whole number and fractions part is called
the decimal point. For example, 34.5 is a decimal number.
34 is a whole number part and 5 is the fractional part.
“.” is the decimal point.
Fractions and Decimals
Converting the Decimal Number into Decimal Fraction:
For the decimal point place “1” in the denominator and remove the decimal point.
“1” is followed by a number of zeros equal to the number of digits following the decimal point.
For Example:
8 1 . 7 5
↓ ↓ ↓
1 0 0
81.75 = 8175/100
8 represents the power of 101 that is the tenths position.
1 represents the power of 100 that is the units position.
7 represents the power of 10-1 that is the one-tenths position.
5 represents the power of 10-2 that is the one-hundredths position.
So that is how each digit is represented by a particular power of 10 in the decimal number.
Example 1:
Convert 8/10 in decimal form.
Solution:
To convert fraction to decimal, divide 8 by 10, we get the decimal form.
Thus, 8/10 = 0.8
Hence, the decimal form of 8/10 is 0.8
Fractions and Decimals
Example 1:
Convert 8/10 in decimal form.
Solution:
To convert fraction to decimal, divide 8 by 10, we get the decimal form.
Thus, 8/10 = 0.8
Hence, the decimal form of 8/10 is 0.8
What is Decimal Fraction?
A decimal fraction is defined as those fractions whose denominators are a power of 10, say 10, 100, 1000,
10000, and so on. Fraction is the relation between a part and a whole. So, in a decimal fraction, the whole is
always divided into parts equal to a power of 10 like 10, 100, 1000, and so on. For example, 7/10 implies that we
consider 7 parts out of a total of 10 parts. When we convert decimal to fraction, the first step is to write the
denominator as a power of 10 in which the number of zeros will be equal to the number of decimal places in the
given number. For example, 2.5 can be written as 25/10, so 25/10 is a decimal fraction. It is one of the types of
fractions which can be used for decimal fraction conversions. Look at the image below to understand what are
decimal fractions with the help of examples.
Factors And Multiples
Factors are the numbers which divide the given number exactly, whereas
the multiples are the numbers which are multiplied by the other number to get
specific numbers.
For example, 4 is a factor of 20, i.e. 4 divides 20 exactly giving 5 as quotient
and leaving zero as remainder.
HCF - Highest Common Factor
HCF or Highest Common Factor is the greatest number which divides each of the two or more
numbers. HCF is also called the Greatest Common Measure (GCM) and Greatest Common
Divisor(GCD).
The full form of HCF is Highest Common Factor. HCF of two numbers is the highest factor that can
divide the two numbers, evenly. HCF can be evaluated for two or more than two numbers. It is the greatest
divisor for any two or more numbers, that can equally or completely divide the given numbers.
The common factors of 18 and 27 are 1, 3, and 9. Among these numbers, 9 is the
highest (largest) number. So, the HCF of 18 and 27 is 9. This is written as: HCF (18, 27) = 9.
Observe the following figure to understand this concept.
HCF - Highest Common Factor
HCF Examples
Using the above HCF definition, the HCF of a few sets of numbers can be listed as follows:
•HCF of 60 and 40 is 20, i.e., HCF (60, 40) = 20
•HCF of 100 and 150 is 50, i.e., HCF (150, 50) = 50
•HCF of 144 and 24 is 24, i.e., HCF (144, 24) = 24
•HCF of 17 and 89 is 1, i.e., HCF (17, 89) = 1
HCF - Highest Common Factor
How to Find HCF?
There are many ways to find the highest common factor of the given numbers.
Irrespective of the method, the answer to the HCF of the numbers is always the same.
There are 3 methods to calculate the HCF of two numbers:
•HCF by listing factors method
•HCF by prime factorization
•HCF by division method
Let us discuss each method in detail with the help of examples.
https://www.cuemath.com/numbers/hcf-highest-common-factor/
HCF - Highest Common Factor
Least Common Multiple-LCM
'Least Common Multiple' or the Lowest Common Multiple. The least common
multiple (LCM) of two numbers is the lowest possible number that can be divisible by
both numbers. It can be calculated for two or more numbers as well.
The least common multiple is also known as LCM (or) the lowest common multiple in
math. The least common multiple of two or more numbers is the smallest number
among all common multiples of the given numbers. Let us take two numbers, 2 and 5.
Each will have its own set of multiples.
•Multiples of 2 are 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, and so on.
•Multiples of 5 are 5, 10, 15, 20, and so on.
Now, let us represent these multiples on the number line and circle the common
multiples
Thus, the common multiples of 2 and 5 are 10, 20, and so on. The smallest number
among 10, 20, and so on is 10. So the least common multiple of 2 and 5 is 10. It can be
written as LCM (2, 5) = 10.
MARKETING CHANNELS
Marketing Channel 1 Marketing Channel 2
Further elaborate on
the channel.
System Base Symbols
Used by
humans?
Used in computers?
Decimal 10 0, 1, … 9 Yes No
Binary 2 0, 1 No Yes
Octal 8 0, 1, … 7 No No
Hexa-
decimal
16 0, 1, … 9,
A, B, … F
No No
O V E R V I E W
Chap1_Number and Number Sydtem.pptx

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Chap1_Number and Number Sydtem.pptx

  • 1. 01 A . R A T I P O R N C H O M R I T
  • 2. NUMBER Marketing Channel 1 Marketing Channel 2 Further elaborate on the channel. A number is an arithmetic value used for representing the quantity and used in making calculations. A written symbol like “3” which represents a number is known as numerals. A number system is a writing system for denoting numbers using digits or symbols in a logical manner. Represents a useful set of numbers Reflects the arithmetic and algebraic structure of a number Provides standard representation We use the digits from 0 to 9 to form all other numbers. 0 1 2 3 4 5 6 7 8 9
  • 3. •Real Numbers: All the positive and negative integers, fractional and decimal numbers without imaginary numbers are called real numbers. It is represented by the symbol “R”. •Rational Numbers: Any number that can be written as a ratio of one number over another number is written as rational numbers. This means that any number that can be written in the form of p/q. The symbol “Q” represents the rational number. •Irrational Numbers: The number that cannot be expressed as the ratio of one over another is known as irrational numbers and it is represented by the symbol ”P”. NUMBER
  • 4. The different types of numbers in maths are: •Natural Numbers: Natural numbers are known as counting numbers that contain the positive integers from 1 to infinity. The set of natural numbers is denoted as “N” and it includes •The natural number set is defined by: N = {1, 2, 3, 4, 5, ……….} Examples: 35, 59, 110, etc. Properties of Natural Numbers: •Addition of natural numbers is closed, associative, and commutative. •Natural Number multiplication is closed, associative, and commutative. •The identity element of a natural number under addition is zero. •The identity element of a natural number under Multiplication is one. •Whole Numbers: Whole numbers are known as non-negative integers and it does not include any fractional or decimal part. It is denoted as “W” and the set of whole numbers includes W = {0,1, 2, 3, 4, 5, ……….} •Whole numbers are also known as natural numbers with zero. The set consists of non-negative integers where it does not contain any decimal or fractional part. The whole number set is represented by the letter “W”. The natural number set is defined by: •W = {0,1, 2, 3, 4, 5, ……….} •Examples: 67, 0, 49, 52, etc. NUMBER
  • 5. •Prime Numbers A prime number is the one which has exactly two factors, which means, it can be divided by only “1” and itself. But “1” is not a prime number. •Example of Prime Number •3 is a prime number because 3 can be divided by only two number’s i.e. 1 and 3 itself. •3/1 = 3 •3/3 = 1 •In the same way, 2, 5, 7, 11, 13, 17 are prime numbers. •Composite Numbers A composite number has more than two factors, which means apart from getting divided by the number 1 and itself, it can also be divided by at least one integer or number. We don’t consider ‘1’ as a composite number. •Example of Composite Number •12 is a composite number because it can be divided by 1, 2, 3, 4, 6 and 12. So, the number ‘12’ has 6 factors. •12/1 = 12 •12/2 =6 •12/3 =4 •12/4 =3 •12/6 =2 •12/12 = 1 NUMBER
  • 6. Number Real Number Irrational Number Rational Number Integer Fraction Zero Negative Integer Positive Integer
  • 7. Rational Numbers Rational number is a type of real number, which is in the form of p/q where q is not equal to zero. Some of the examples of rational numbers are 1/2, 1/5, 3/4, and so on. The number “0” is also a rational number, as we can represent it in many forms such as 0/1, 0/2, 0/3, etc. But, 1/0, 2/0, 3/0, etc. are not rational. A rational number can be represented by the letter “Q”. • Integer • Fraction • Repeating decimal Irrational Numbers The number that cannot be expressed in the form of p/q. It means a number that cannot be written as the ratio of one over another is known as irrational numbers. It is represented by the letter ”P”. Examples: √2, π, 5.18118168473465 etc NUMBER
  • 8. •Integers: Integers are the set of all whole numbers but it includes a negative set of natural numbers also. “Z” represents integers and the set of integers are Z = { -3, -2, -1, 0, 1, 2, 3} •Integers are defined as the set of all whole numbers with a negative set of natural numbers. The integer set is represented by the symbol “Z”. The set of integers is defined as: •Z = {-3, -2, -1, 0, 1, 2, 3} •Examples: -52, 0, -1, 16, 82, etc. •Properties of Integers: •Integers are closed under addition, subtraction, and multiplication. •The commutative property is satisfied for addition and multiplication of integers. •It obeys the associative property of addition and multiplication. •It obeys the distributive property for addition and multiplication. •Additive identity of integers is 0. •Multiplicative identity of integers is 1. NUMBER
  • 9. Fractions •Fractions, in Mathematics, are represented as a numerical value, which defines a part of a whole. A fraction can be a portion or section of any quantity out of a whole, where the whole can be any number, a specific value, or a thing. Example. The following figure shows a pizza that is divided into 8 equal parts. Now, if we want to express one selected part of the pizza, we can express it as 1/8 which shows that out of 8 equal parts, we are referring to 1 part. It means one in eight equal parts. It can also be read as: •One-eighth, or •1 by 8 NUMBER Fractions and Decimals
  • 10. Fractions and Decimals Parts of a Fraction All fractions consist of a numerator and a denominator and they are separated by a horizontal bar known as the fraction bar. • The numerator indicates how many sections of the fraction are represented or selected. It is placed in the upper part of the fraction above the fractional bar. • The denominator indicates the number of parts in which the whole has been divided into. It is placed in the lower part of the fraction below the fractional bar. numerator denominator
  • 11. Fractions and Decimals Decimals are one of the types of numbers, which has a whole number and the fractional part separated by a decimal point. The dot present between the whole number and fractions part is called the decimal point. For example, 34.5 is a decimal number. 34 is a whole number part and 5 is the fractional part. “.” is the decimal point.
  • 12. Fractions and Decimals Converting the Decimal Number into Decimal Fraction: For the decimal point place “1” in the denominator and remove the decimal point. “1” is followed by a number of zeros equal to the number of digits following the decimal point. For Example: 8 1 . 7 5 ↓ ↓ ↓ 1 0 0 81.75 = 8175/100 8 represents the power of 101 that is the tenths position. 1 represents the power of 100 that is the units position. 7 represents the power of 10-1 that is the one-tenths position. 5 represents the power of 10-2 that is the one-hundredths position. So that is how each digit is represented by a particular power of 10 in the decimal number. Example 1: Convert 8/10 in decimal form. Solution: To convert fraction to decimal, divide 8 by 10, we get the decimal form. Thus, 8/10 = 0.8 Hence, the decimal form of 8/10 is 0.8
  • 13. Fractions and Decimals Example 1: Convert 8/10 in decimal form. Solution: To convert fraction to decimal, divide 8 by 10, we get the decimal form. Thus, 8/10 = 0.8 Hence, the decimal form of 8/10 is 0.8 What is Decimal Fraction? A decimal fraction is defined as those fractions whose denominators are a power of 10, say 10, 100, 1000, 10000, and so on. Fraction is the relation between a part and a whole. So, in a decimal fraction, the whole is always divided into parts equal to a power of 10 like 10, 100, 1000, and so on. For example, 7/10 implies that we consider 7 parts out of a total of 10 parts. When we convert decimal to fraction, the first step is to write the denominator as a power of 10 in which the number of zeros will be equal to the number of decimal places in the given number. For example, 2.5 can be written as 25/10, so 25/10 is a decimal fraction. It is one of the types of fractions which can be used for decimal fraction conversions. Look at the image below to understand what are decimal fractions with the help of examples.
  • 14. Factors And Multiples Factors are the numbers which divide the given number exactly, whereas the multiples are the numbers which are multiplied by the other number to get specific numbers. For example, 4 is a factor of 20, i.e. 4 divides 20 exactly giving 5 as quotient and leaving zero as remainder.
  • 15. HCF - Highest Common Factor HCF or Highest Common Factor is the greatest number which divides each of the two or more numbers. HCF is also called the Greatest Common Measure (GCM) and Greatest Common Divisor(GCD). The full form of HCF is Highest Common Factor. HCF of two numbers is the highest factor that can divide the two numbers, evenly. HCF can be evaluated for two or more than two numbers. It is the greatest divisor for any two or more numbers, that can equally or completely divide the given numbers.
  • 16. The common factors of 18 and 27 are 1, 3, and 9. Among these numbers, 9 is the highest (largest) number. So, the HCF of 18 and 27 is 9. This is written as: HCF (18, 27) = 9. Observe the following figure to understand this concept. HCF - Highest Common Factor
  • 17. HCF Examples Using the above HCF definition, the HCF of a few sets of numbers can be listed as follows: •HCF of 60 and 40 is 20, i.e., HCF (60, 40) = 20 •HCF of 100 and 150 is 50, i.e., HCF (150, 50) = 50 •HCF of 144 and 24 is 24, i.e., HCF (144, 24) = 24 •HCF of 17 and 89 is 1, i.e., HCF (17, 89) = 1 HCF - Highest Common Factor
  • 18. How to Find HCF? There are many ways to find the highest common factor of the given numbers. Irrespective of the method, the answer to the HCF of the numbers is always the same. There are 3 methods to calculate the HCF of two numbers: •HCF by listing factors method •HCF by prime factorization •HCF by division method Let us discuss each method in detail with the help of examples. https://www.cuemath.com/numbers/hcf-highest-common-factor/ HCF - Highest Common Factor
  • 19. Least Common Multiple-LCM 'Least Common Multiple' or the Lowest Common Multiple. The least common multiple (LCM) of two numbers is the lowest possible number that can be divisible by both numbers. It can be calculated for two or more numbers as well. The least common multiple is also known as LCM (or) the lowest common multiple in math. The least common multiple of two or more numbers is the smallest number among all common multiples of the given numbers. Let us take two numbers, 2 and 5. Each will have its own set of multiples. •Multiples of 2 are 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, and so on. •Multiples of 5 are 5, 10, 15, 20, and so on. Now, let us represent these multiples on the number line and circle the common multiples Thus, the common multiples of 2 and 5 are 10, 20, and so on. The smallest number among 10, 20, and so on is 10. So the least common multiple of 2 and 5 is 10. It can be written as LCM (2, 5) = 10.
  • 20.
  • 21. MARKETING CHANNELS Marketing Channel 1 Marketing Channel 2 Further elaborate on the channel. System Base Symbols Used by humans? Used in computers? Decimal 10 0, 1, … 9 Yes No Binary 2 0, 1 No Yes Octal 8 0, 1, … 7 No No Hexa- decimal 16 0, 1, … 9, A, B, … F No No
  • 22. O V E R V I E W

Editor's Notes

  1. Prime Numbers จนเฉพาะ Composite Numbers จนประกอบ