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Integers
1) Rules for adding and subtracting integers include keeping the sign the same when adding like signs, and using the sign of the larger number when subtracting or adding opposite signs.
2) When multiplying integers, the sign of the product is determined by the number of negative factors. If even, the product is positive, and if odd, the product is negative.
3) Integers are closed under addition, subtraction, and multiplication, and follow properties like commutativity and associativity for these operations.
AFS7 Math 1
The document provides explanations and examples for adding, subtracting, multiplying, and dividing integers:
1) When adding integers with the same sign, add their absolute values and use the common sign. When adding integers with opposite signs, take the absolute difference and use the sign of the larger number.
2) To subtract an integer, add its opposite and then follow the addition rules.
3) When multiplying an even number of negatives, the result is positive. With an odd number of negatives, the result is negative.
AFS Math 3
The document provides explanations and examples for adding, subtracting, multiplying, and dividing integers.
It begins by explaining the rules for adding integers with the same sign and integers with different signs, providing examples such as -6 + -2 = -8. It then explains that subtracting integers uses the rule of "adding the opposite" and provides examples like 7 - (-6) = 13.
The document also covers multiplying and dividing integers, noting that an even number of negatives yields a positive result and an odd number yields a negative result. It provides examples such as -2(-2)(-2)= 16 and 2 (-5)= -10.
PRESENTATION ON THE PROPERTIES OF THE
The document discusses the six properties of operations on integers:
1. The closure property states that adding or multiplying integers results in an integer.
2. The commutative property does not change the result of addition or multiplication when the order of numbers is changed.
3. The associative property does not change the result when grouping numbers that are added or multiplied is changed.
4. The distributive property states that when two numbers are added/subtracted and multiplied by a factor, the result is the same as multiplying each number by the factor and then adding/subtracting.
5. The identity properties are: addition identity of 0, and multiplication identity of 1.
6. The inverse properties are: additive
Integers Class 7
Integers include both whole numbers and their negative counterparts. Addition and subtraction of integers follow predictable patterns based on the signs of the integers. Properties like closure, commutativity, and distributivity apply to integer addition and multiplication but not always to subtraction and division. Multiplication results in a positive integer if the number of negative factors is even, and a negative integer if the number of negative factors is odd. Division of a positive by a negative integer yields a negative result.
Class Presentation Math 1
This document contains notes from a 7th grade math class covering topics like subtracting integers, multiplying integers, and solving algebraic equations. Key points covered include: when subtracting integers, you add the opposite; when multiplying integers with the same sign the product is positive, and with different signs the product is negative; and to solve equations, you perform the same operation to both sides until the variable is isolated on one side.
Class Presentation Math 1
This document contains notes from a 7th grade math class covering topics like subtracting integers, multiplying integers, and solving algebraic equations. Key points covered include: when subtracting integers, you add the opposite; when multiplying integers with the same sign the product is positive, and with different signs the product is negative; and to solve equations, you perform the same operation to both sides until the variable is isolated on one side.
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Integers
1) Rules for adding and subtracting integers include keeping the sign the same when adding like signs, and using the sign of the larger number when subtracting or adding opposite signs.
2) When multiplying integers, the sign of the product is determined by the number of negative factors. If even, the product is positive, and if odd, the product is negative.
3) Integers are closed under addition, subtraction, and multiplication, and follow properties like commutativity and associativity for these operations.
AFS7 Math 1
The document provides explanations and examples for adding, subtracting, multiplying, and dividing integers:
1) When adding integers with the same sign, add their absolute values and use the common sign. When adding integers with opposite signs, take the absolute difference and use the sign of the larger number.
2) To subtract an integer, add its opposite and then follow the addition rules.
3) When multiplying an even number of negatives, the result is positive. With an odd number of negatives, the result is negative.
AFS Math 3
The document provides explanations and examples for adding, subtracting, multiplying, and dividing integers.
It begins by explaining the rules for adding integers with the same sign and integers with different signs, providing examples such as -6 + -2 = -8. It then explains that subtracting integers uses the rule of "adding the opposite" and provides examples like 7 - (-6) = 13.
The document also covers multiplying and dividing integers, noting that an even number of negatives yields a positive result and an odd number yields a negative result. It provides examples such as -2(-2)(-2)= 16 and 2 (-5)= -10.
PRESENTATION ON THE PROPERTIES OF THE
The document discusses the six properties of operations on integers:
1. The closure property states that adding or multiplying integers results in an integer.
2. The commutative property does not change the result of addition or multiplication when the order of numbers is changed.
3. The associative property does not change the result when grouping numbers that are added or multiplied is changed.
4. The distributive property states that when two numbers are added/subtracted and multiplied by a factor, the result is the same as multiplying each number by the factor and then adding/subtracting.
5. The identity properties are: addition identity of 0, and multiplication identity of 1.
6. The inverse properties are: additive
Integers Class 7
Integers include both whole numbers and their negative counterparts. Addition and subtraction of integers follow predictable patterns based on the signs of the integers. Properties like closure, commutativity, and distributivity apply to integer addition and multiplication but not always to subtraction and division. Multiplication results in a positive integer if the number of negative factors is even, and a negative integer if the number of negative factors is odd. Division of a positive by a negative integer yields a negative result.
Class Presentation Math 1
This document contains notes from a 7th grade math class covering topics like subtracting integers, multiplying integers, and solving algebraic equations. Key points covered include: when subtracting integers, you add the opposite; when multiplying integers with the same sign the product is positive, and with different signs the product is negative; and to solve equations, you perform the same operation to both sides until the variable is isolated on one side.
Class Presentation Math 1
This document contains notes from a 7th grade math class covering topics like subtracting integers, multiplying integers, and solving algebraic equations. Key points covered include: when subtracting integers, you add the opposite; when multiplying integers with the same sign the product is positive, and with different signs the product is negative; and to solve equations, you perform the same operation to both sides until the variable is isolated on one side.
AFS Math 2
This document contains notes from a 7th grade math class covering topics like subtracting integers, multiplying integers, and solving algebraic equations. Key points covered include: when subtracting integers, you add the opposite; when multiplying integers with the same sign the product is positive, and with different signs the product is negative; and to solve equations, you perform the same operation to both sides until the variable is isolated on one side.
Vii ch 1 integers
The document contains examples of solving problems involving addition, subtraction, multiplication and division of integers. Some key examples include:
- Finding the balance in an account after a deposit and withdrawal.
- Calculating distance traveled east and west and the final position from a starting point.
- Verifying properties like commutativity, associativity and distributivity for operations on integers.
- Solving word problems involving gains, losses, temperature changes represented as positive and negative integers.
Integers
Integers are whole numbers and their negatives. On a number line, adding a positive integer moves right and adding a negative integer moves left.
Integers are closed under addition and subtraction. For any integers a and b, a + b and a - b are also integers. Addition is commutative but subtraction is not. Both operations are associative.
For integers a and b, a * (-b) = (-a) * b and a * (-b) = - (a * b). The product of two negative integers is positive. If the number of negative factors in a product is even, the product is positive, and if odd, the product is negative. Integers are closed
Real-Number-System.pptx
This document discusses key concepts in the real number system including:
- Rational numbers that can be expressed as ratios of integers, and irrational numbers that cannot.
- Integers, including positive, negative and whole numbers.
- Properties of addition like commutativity, associativity and closure.
- Properties of multiplication like commutativity, associativity and distributivity.
- Absolute value and rules for performing operations on signed numbers like addition, subtraction, multiplication and division.
Lesson 1 1 properties of real numbers
1) The document provides an overview of properties and operations of real numbers including identifying different types of real numbers like integers, rational numbers, and irrational numbers.
2) It discusses ordering real numbers and using symbols like <, >, ≤, ≥ to compare them. Properties of addition, multiplication and other operations are also covered.
3) Examples are provided to illustrate concepts like using properties of real numbers to evaluate expressions and convert between units like miles and kilometers.
DIPIN PPT OF INTEGER
The document defines integers and their properties under addition, subtraction, multiplication, and division. It states that integers include whole numbers and their negatives, and are closed under addition and subtraction. Properties discussed include commutativity, associativity, distributivity, and operations with zero. Integers are not closed under division and division is not always commutative.
Integers and Operation of Integers
This document provides information and examples about integer operations:
- Addition of integers follows the same rules as normal addition, such as 20 + 10 = 30 and -40 + -60 = -100.
- Subtraction of integers is performed similarly to addition, such as -3 - 7 = -10 and 15 - 9 = 6.
- When multiplying integers, the product is positive if the signs are the same and negative if the signs are different, exemplified as -2 × 6 = -12 and 2 × -3 = -6.
- For division of integers, the quotient is positive if the signs are the same and negative if the signs are different, with examples like 12 ÷ -4 =
Complex numbers
The document appears to be discussing complex numbers in Urdu. It begins by stating that God is extremely merciful and compassionate. It then provides some key points about complex numbers, including:
- Complex numbers can be expressed in the form a + bi, where a and b are real numbers and i represents the imaginary unit.
- Operations like addition, subtraction, multiplication, and division can be performed with complex numbers by following specific rules.
- Complex numbers have properties like closure, commutativity, distributivity, identities, and inverses when performing operations.
- The conjugate of a complex number z = a + bi is a - bi. Conjugates have certain properties when performing operations
Intro adding integres
The document discusses rules for evaluating sums and products of positive and negative integers:
1) When adding positive integers, add the magnitudes and keep the positive sign. When adding negative integers, add the magnitudes and keep the negative sign.
2) When adding a positive and negative integer, subtract the magnitudes and keep the sign of the integer with the largest magnitude.
3) When multiplying integers, the sign of the product is positive if there is an even number of negative factors and negative if there is an odd number of negative factors.
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The lesson included in this file is Integers. This file can be used to introduced the concepts on integers such as its rules on the basic operations. Several examples are also provided in this file.
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Rational numbers can be used to solve equations that cannot be solved using only natural numbers, whole numbers, or integers. Rational numbers are numbers that can be expressed as fractions p/q where p and q are integers and q ≠ 0.
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Hemh101
cheeck this class 8 history ppt in class 8 students or below can refer this ppt and make their mind map for history. thank you
Números Reales - Genesis Sira
The document defines sets and set operations such as union, intersection, symmetric difference, and complement. It then discusses real numbers, defining them as any numeric expressions excluding imaginary and complex numbers, such as integers, fractions, irrational numbers, etc. It provides examples of different types of real numbers. The document also covers properties and operations of real numbers like commutativity, associativity, identity, and distribution. Finally, it defines inequalities and absolute value, providing properties and examples of solving inequalities with absolute value.
Real numbers
This document defines and explains key concepts related to real numbers and algebraic expressions. It introduces sets and subsets of real numbers like integers, rational numbers, and irrational numbers. It describes properties of real numbers including addition, multiplication, order, and absolute value. It also covers representing real numbers on a number line, algebraic expressions, and properties of negatives.
ch1.pdf
Rational numbers can be used to solve equations that cannot be solved using only natural numbers, whole numbers, or integers. Rational numbers are numbers that can be expressed as fractions p/q where p and q are integers and q ≠ 0. Rational numbers are closed under addition, subtraction, and multiplication but not division. Addition and multiplication of rational numbers are commutative, but subtraction and division are not. Addition is associative for rational numbers, but subtraction is not.
Integers
The document is about integers and their properties. Some key points:
- Integers include whole numbers and their negatives, but not fractions or imaginary numbers.
- The modulus or absolute value of a number gives its numerical value regardless of sign.
- Every integer has an additive inverse, such that when added the result is 0.
- Addition and subtraction follow rules based on sign: unlike signs subtract, like signs add.
- Multiplication and division of integers with unlike signs results in a negative product or quotient.
- Integers have properties for addition, subtraction, multiplication and division like commutativity, associativity and distribution.
Add and subtract pos and neg numbers 4 parts
This document provides an introduction to integers through five parts:
Part I defines key integer vocabulary like positive and negative numbers. It discusses integer properties like opposites and compares/orders integers on number lines. Real world applications like temperature, sea level, and money are explored.
Part II covers integer addition rules - signs the same means keep the sign, signs different means subtract the numbers and keep the larger absolute value sign. Number lines demonstrate adding integers visually.
Part III explains that subtracting a negative number is the same as adding a positive number through changing operation and number signs. More examples solidify this rule.
Part IV proves this subtraction rule is true by using the same checking method as regular subtraction equations
CBSE-Class-8-NCERT-Maths-Book-Rational-Numbers-chapter-1.pdf
Rational numbers are numbers that can be written as fractions p/q, where p and q are integers and q ≠ 0. Some key properties of rational numbers are:
1) Rational numbers are closed under addition, subtraction, and multiplication but not division. For any two rational numbers a and b, a + b, a - b, and a × b are rational, but a ÷ b may not be rational if b = 0.
2) Addition and multiplication of rational numbers are commutative but subtraction and division are not.
3) Addition and multiplication of rational numbers are associative but subtraction and division are not.
1059331.pdf
Rational numbers are numbers that can be written as fractions p/q, where p and q are integers and q ≠ 0. Some key properties of rational numbers are:
1) Rational numbers are closed under addition, subtraction, and multiplication but not division. For any two rational numbers a and b, a + b, a - b, and a × b are rational, but a ÷ b may not be rational if b = 0.
2) Addition and multiplication of rational numbers are commutative but subtraction and division are not.
3) Addition and multiplication of rational numbers are associative but subtraction and division are not.
properties of whole numbers
The document discusses properties of operations on whole numbers. It states that addition and multiplication are associative for whole numbers, as the sum or product will be the same regardless of grouping order. As an example, it shows that 3 + (2 + 5) equals (3 + 2) + 5, both equalling 10. It also explains that the distributive property holds for whole numbers, where x(y + z) equals xy + xz. Finally, it notes that 0 is the additive identity and 1 is the multiplicative identity for whole numbers.
FRACTIONS
Learn about Fractions Basics.
For Lectures on fraction , visit my Udemy course. Link is
https://www.udemy.com/course/basics-of-math/?referralCode=6FF51E8FD0CE20F99D35
Basic logic gates
This document contains 6 examples of binary logic operations - AND, OR, NOT, AND-OR - represented through truth tables with variables A, B, and X. It shows how the output X changes based on the input values of A and B and whether the operation is AND, OR, NOT, AND-OR, or combinations of those operations.
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AFS Math 2
This document contains notes from a 7th grade math class covering topics like subtracting integers, multiplying integers, and solving algebraic equations. Key points covered include: when subtracting integers, you add the opposite; when multiplying integers with the same sign the product is positive, and with different signs the product is negative; and to solve equations, you perform the same operation to both sides until the variable is isolated on one side.
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The document contains examples of solving problems involving addition, subtraction, multiplication and division of integers. Some key examples include:
- Finding the balance in an account after a deposit and withdrawal.
- Calculating distance traveled east and west and the final position from a starting point.
- Verifying properties like commutativity, associativity and distributivity for operations on integers.
- Solving word problems involving gains, losses, temperature changes represented as positive and negative integers.
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Integers are closed under addition and subtraction. For any integers a and b, a + b and a - b are also integers. Addition is commutative but subtraction is not. Both operations are associative.
For integers a and b, a * (-b) = (-a) * b and a * (-b) = - (a * b). The product of two negative integers is positive. If the number of negative factors in a product is even, the product is positive, and if odd, the product is negative. Integers are closed
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This document discusses key concepts in the real number system including:
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1) The document provides an overview of properties and operations of real numbers including identifying different types of real numbers like integers, rational numbers, and irrational numbers.
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3) Examples are provided to illustrate concepts like using properties of real numbers to evaluate expressions and convert between units like miles and kilometers.
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The document defines integers and their properties under addition, subtraction, multiplication, and division. It states that integers include whole numbers and their negatives, and are closed under addition and subtraction. Properties discussed include commutativity, associativity, distributivity, and operations with zero. Integers are not closed under division and division is not always commutative.
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This document provides information and examples about integer operations:
- Addition of integers follows the same rules as normal addition, such as 20 + 10 = 30 and -40 + -60 = -100.
- Subtraction of integers is performed similarly to addition, such as -3 - 7 = -10 and 15 - 9 = 6.
- When multiplying integers, the product is positive if the signs are the same and negative if the signs are different, exemplified as -2 × 6 = -12 and 2 × -3 = -6.
- For division of integers, the quotient is positive if the signs are the same and negative if the signs are different, with examples like 12 ÷ -4 =
Complex numbers
The document appears to be discussing complex numbers in Urdu. It begins by stating that God is extremely merciful and compassionate. It then provides some key points about complex numbers, including:
- Complex numbers can be expressed in the form a + bi, where a and b are real numbers and i represents the imaginary unit.
- Operations like addition, subtraction, multiplication, and division can be performed with complex numbers by following specific rules.
- Complex numbers have properties like closure, commutativity, distributivity, identities, and inverses when performing operations.
- The conjugate of a complex number z = a + bi is a - bi. Conjugates have certain properties when performing operations
Intro adding integres
The document discusses rules for evaluating sums and products of positive and negative integers:
1) When adding positive integers, add the magnitudes and keep the positive sign. When adding negative integers, add the magnitudes and keep the negative sign.
2) When adding a positive and negative integer, subtract the magnitudes and keep the sign of the integer with the largest magnitude.
3) When multiplying integers, the sign of the product is positive if there is an even number of negative factors and negative if there is an odd number of negative factors.
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Rational numbers can be used to solve equations that cannot be solved using only natural numbers, whole numbers, or integers. Rational numbers are numbers that can be expressed as fractions p/q where p and q are integers and q ≠ 0.
Rational numbers are closed under addition, subtraction, and multiplication, but not division. Addition, subtraction and multiplication of rational numbers are commutative, but division is not. Addition of rational numbers is associative, but subtraction is not.
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cheeck this class 8 history ppt in class 8 students or below can refer this ppt and make their mind map for history. thank you
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The document defines sets and set operations such as union, intersection, symmetric difference, and complement. It then discusses real numbers, defining them as any numeric expressions excluding imaginary and complex numbers, such as integers, fractions, irrational numbers, etc. It provides examples of different types of real numbers. The document also covers properties and operations of real numbers like commutativity, associativity, identity, and distribution. Finally, it defines inequalities and absolute value, providing properties and examples of solving inequalities with absolute value.
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This document defines and explains key concepts related to real numbers and algebraic expressions. It introduces sets and subsets of real numbers like integers, rational numbers, and irrational numbers. It describes properties of real numbers including addition, multiplication, order, and absolute value. It also covers representing real numbers on a number line, algebraic expressions, and properties of negatives.
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Rational numbers can be used to solve equations that cannot be solved using only natural numbers, whole numbers, or integers. Rational numbers are numbers that can be expressed as fractions p/q where p and q are integers and q ≠ 0. Rational numbers are closed under addition, subtraction, and multiplication but not division. Addition and multiplication of rational numbers are commutative, but subtraction and division are not. Addition is associative for rational numbers, but subtraction is not.
Integers
The document is about integers and their properties. Some key points:
- Integers include whole numbers and their negatives, but not fractions or imaginary numbers.
- The modulus or absolute value of a number gives its numerical value regardless of sign.
- Every integer has an additive inverse, such that when added the result is 0.
- Addition and subtraction follow rules based on sign: unlike signs subtract, like signs add.
- Multiplication and division of integers with unlike signs results in a negative product or quotient.
- Integers have properties for addition, subtraction, multiplication and division like commutativity, associativity and distribution.
Add and subtract pos and neg numbers 4 parts
This document provides an introduction to integers through five parts:
Part I defines key integer vocabulary like positive and negative numbers. It discusses integer properties like opposites and compares/orders integers on number lines. Real world applications like temperature, sea level, and money are explored.
Part II covers integer addition rules - signs the same means keep the sign, signs different means subtract the numbers and keep the larger absolute value sign. Number lines demonstrate adding integers visually.
Part III explains that subtracting a negative number is the same as adding a positive number through changing operation and number signs. More examples solidify this rule.
Part IV proves this subtraction rule is true by using the same checking method as regular subtraction equations
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Rational numbers are numbers that can be written as fractions p/q, where p and q are integers and q ≠ 0. Some key properties of rational numbers are:
1) Rational numbers are closed under addition, subtraction, and multiplication but not division. For any two rational numbers a and b, a + b, a - b, and a × b are rational, but a ÷ b may not be rational if b = 0.
2) Addition and multiplication of rational numbers are commutative but subtraction and division are not.
3) Addition and multiplication of rational numbers are associative but subtraction and division are not.
1059331.pdf
Rational numbers are numbers that can be written as fractions p/q, where p and q are integers and q ≠ 0. Some key properties of rational numbers are:
1) Rational numbers are closed under addition, subtraction, and multiplication but not division. For any two rational numbers a and b, a + b, a - b, and a × b are rational, but a ÷ b may not be rational if b = 0.
2) Addition and multiplication of rational numbers are commutative but subtraction and division are not.
3) Addition and multiplication of rational numbers are associative but subtraction and division are not.
properties of whole numbers
The document discusses properties of operations on whole numbers. It states that addition and multiplication are associative for whole numbers, as the sum or product will be the same regardless of grouping order. As an example, it shows that 3 + (2 + 5) equals (3 + 2) + 5, both equalling 10. It also explains that the distributive property holds for whole numbers, where x(y + z) equals xy + xz. Finally, it notes that 0 is the additive identity and 1 is the multiplicative identity for whole numbers.
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INTEGERS
- 1. INTEGERS
- 2. What are INTEGERS ? Integers is a bigger collection of numbers which includes Whole numbers and Negative numbers. OR We can say also that Integers is a bigger collection of number which includes zero , positive numbers and negative numbers.
- 3. Properties of Addition/Subtraction of Integers CLOSURE PROPERTY UNDER ADDITION 20 + 10 = 30 50 + 60 = 110 - 100 + 200 = 100 57 + ( -13 ) = 44 - 42 + ( -11 ) = - 53 - 78 + ( - 40 ) = - 118 In above examples , we are getting integer after addition of two integers.
- 4. Since addition of integers , gives integers , we can say integers are closed under addition. Therefore, we can say in general For any two integers a and b , a + b is also an integer. https://www.udemy.com/course/basics-of- math/?referralCode=6FF51E8FD0CE20F9 9D35
- 5. COMMUTATIVE PROPERTY (ADDITION) 7 + ( - 2 ) = 5 AND - 2 + 7 = 5 - 3 + ( - 9 ) = - 12 AND - 9 + ( - 3 ) = - 12 - 6 + 21 = 15 AND 21 + ( - 6 ) = 15 23 + 50 = 73 AND 50 + 23 = 73 From above examples we can say addition is commutative for integers
- 6. We can say in general For any two integers a and b , a + b = b + a
- 7. Multiplication of Integers Multiplication of a positive and a negative integer 4 x ( - 3 ) = - 12 - 10 x 33 = - 330 - 5 x 7 = - 35 101 x ( - 8 ) = - 808 From the above examples it is clear that while multiplying a positive integer and a negative integer, we simply multiply them and put a negative sign before them . We thus get a negative integer.
- 8. Multiplication of two negative integers - 4 x ( - 3 ) = 12 - 100 x ( - 20 ) = 2000 - 5 x - 7 = 35 - 7 x ( - 3 ) = 21 Product ( multiplication ) of two negative integers is a positive integer. We multiply two negative integers by ignoring the negative sign of both integers and put the positive sign before the product.
- 9. Multiplication properties of Integers CLOSURE PROPERTY UNDER MULTIPLICATION 1000 x ( - 37 ) = - 37000 ( - 50 ) x ( - 10 ) = 500 5621 x 20 = 11240 ( - 15 ) x 6 = - 90 The product of two integers is again an integer. Therefore integers are closed under multiplication.
- 10. - 12 ÷ ( - 2 ) = 6 - 35 ÷ ( - 7 ) = 5 When we divide a negative integer by a negative integer, we first divide them and then put a positive sign ( + ) before the quotient . https://www.udemy.com/course/basics-of- math/?referralCode=6FF51E8FD0CE20F99 D35
- 11. Properties of Division of integers - 8 ÷ ( - 4 ) = 2 - 4 ÷ ( - 8 ) = 0.5 80 ÷ 20 = 4 20 ÷ 80 = 0.25 Division is not commutative for integers Integers are also not closed under division. Since division of two integers is not always an integer.
- 12. Any integer divided by zero is not defined a ÷ 0 is not defined , a is an integer Zero divided by an integer other than zero is equal to zero. 0 ÷ a = 0 where a is an integer and is not equal to zero
- 13. FOR COMPLETE TOPICS/LECTURES ON INTEGERS Visit my Udemy lectures. Link is given below https://www.udemy.com/course/basics-of- math/?referralCode=6FF51E8FD0CE20F99D35