properties of addition and subtraction of integers
•Download as PPTX, PDF•
3 likes•976 views
The document discusses properties of addition and subtraction of integers. It states that integers are closed under addition and subtraction, meaning the sum or difference of two integers is always an integer. For example, 2 + 3 = 5. It also explains that addition is commutative for integers, so changing the order of numbers being added does not change the result. However, subtraction is not commutative, so changing the order of numbers being subtracted can change the result. For example, 23 - (-30) = 53, but (-30) - 23 = -53.
Report
Share
Report
Share
Recommended
properties of multiplication of integers
1. Integers are closed under multiplication, meaning the product of two integers is always an integer. Multiplication is also commutative for integers, so changing the order of the integers does not change the result.
2. Multiplying an integer by zero always results in zero. Multiplying an integer by one results in the original integer.
3. Multiplication of integers is associative, meaning the grouping of factors does not change the result. It is also distributive over addition and subtraction, so a × (b + c) = (a × b) + (a × c) and a × (b - c) = (a × b) - (a × c).
Exponential Growth And Decay
The document discusses exponential growth and decay functions. Exponential growth functions have a base greater than 1, modeling an increasing pattern from small to big numbers over time. Exponential decay functions have a base between 0 and 1, modeling a decreasing pattern from big to small numbers. Examples are provided of functions modeling exponential growth and decay, along with explanations of how to determine which type of function based on the base.
Factorisation
This document provides an overview of factorizing algebraic expressions. It discusses several methods of factorizing including finding common factors, regrouping terms, using factorizing identities, and dividing polynomials. Examples are provided to illustrate each method. The document is divided into seven parts covering factors of natural numbers, common factors, regrouping terms, factorizing identities, factors in the form of (x+a)(x+b), and dividing algebraic expressions.
Linear Equation In one variable class 7
This document defines and provides examples of linear equations in one variable. It explains that a linear equation is an equation that can be written in the form ax + b = c or ax = b, where a, b, c are constants and a ≠ 0. Examples of linear equations given include 3x + 9 = 0 and 7x + 5 = 2x - 9. The document also discusses how to determine if a value is a solution to a linear equation by substitution and simplification. Steps for solving linear equations are provided, which include isolating the variable using inverse operations like addition/subtraction and multiplication/division.
Properties of addition
The document discusses properties of rational numbers. It defines a rational number as a number that can be expressed as p/q where p and q are integers and q is not equal to 0. It then lists properties of rational numbers under addition: 1) closure, 2) commutativity, 3) associativity, 4) existence of an additive identity element (0), and 5) existence of additive inverses. Examples are provided to illustrate each property.
Rational numbers
Rational numbers can be represented as fractions p/q where p and q are integers and q is not equal to 0. Rational numbers are closed under addition, subtraction, and multiplication, meaning these operations on rational numbers will always result in another rational number. Division of rational numbers is not closed as division by 0 is undefined. Rational numbers exhibit properties like commutativity for addition and multiplication but not for subtraction or division. They also demonstrate the associative and distributive properties. Every rational number has an additive inverse and a multiplicative inverse or reciprocal. Zero is the additive identity and one is the multiplicative identity.
Properties of Real Numbers
The document discusses properties of real numbers. It defines real numbers and distinguishes between rational and irrational numbers. Rational numbers can be expressed as ratios of integers, while irrational numbers have non-terminating, non-repeating decimals. It also outlines relationships between subsets of real numbers and properties that real number operations satisfy, such as commutativity, associativity, identities, inverses, and distribution.
Rationalnumbers
Rational numbers can be written as a ratio of two integers. They can be expressed as decimals that either terminate or repeat. Examples of rational numbers include integers like 16, 1/2, 8, and decimals like 1.33. Rational numbers have properties when added, subtracted, multiplied, and divided, such as closure, commutativity, associativity, and others. Every rational number has a reciprocal, and the reciprocal of a rational number a/b is b/a.
Recommended
properties of multiplication of integers
1. Integers are closed under multiplication, meaning the product of two integers is always an integer. Multiplication is also commutative for integers, so changing the order of the integers does not change the result.
2. Multiplying an integer by zero always results in zero. Multiplying an integer by one results in the original integer.
3. Multiplication of integers is associative, meaning the grouping of factors does not change the result. It is also distributive over addition and subtraction, so a × (b + c) = (a × b) + (a × c) and a × (b - c) = (a × b) - (a × c).
Exponential Growth And Decay
The document discusses exponential growth and decay functions. Exponential growth functions have a base greater than 1, modeling an increasing pattern from small to big numbers over time. Exponential decay functions have a base between 0 and 1, modeling a decreasing pattern from big to small numbers. Examples are provided of functions modeling exponential growth and decay, along with explanations of how to determine which type of function based on the base.
Factorisation
This document provides an overview of factorizing algebraic expressions. It discusses several methods of factorizing including finding common factors, regrouping terms, using factorizing identities, and dividing polynomials. Examples are provided to illustrate each method. The document is divided into seven parts covering factors of natural numbers, common factors, regrouping terms, factorizing identities, factors in the form of (x+a)(x+b), and dividing algebraic expressions.
Linear Equation In one variable class 7
This document defines and provides examples of linear equations in one variable. It explains that a linear equation is an equation that can be written in the form ax + b = c or ax = b, where a, b, c are constants and a ≠ 0. Examples of linear equations given include 3x + 9 = 0 and 7x + 5 = 2x - 9. The document also discusses how to determine if a value is a solution to a linear equation by substitution and simplification. Steps for solving linear equations are provided, which include isolating the variable using inverse operations like addition/subtraction and multiplication/division.
Properties of addition
The document discusses properties of rational numbers. It defines a rational number as a number that can be expressed as p/q where p and q are integers and q is not equal to 0. It then lists properties of rational numbers under addition: 1) closure, 2) commutativity, 3) associativity, 4) existence of an additive identity element (0), and 5) existence of additive inverses. Examples are provided to illustrate each property.
Rational numbers
Rational numbers can be represented as fractions p/q where p and q are integers and q is not equal to 0. Rational numbers are closed under addition, subtraction, and multiplication, meaning these operations on rational numbers will always result in another rational number. Division of rational numbers is not closed as division by 0 is undefined. Rational numbers exhibit properties like commutativity for addition and multiplication but not for subtraction or division. They also demonstrate the associative and distributive properties. Every rational number has an additive inverse and a multiplicative inverse or reciprocal. Zero is the additive identity and one is the multiplicative identity.
Properties of Real Numbers
The document discusses properties of real numbers. It defines real numbers and distinguishes between rational and irrational numbers. Rational numbers can be expressed as ratios of integers, while irrational numbers have non-terminating, non-repeating decimals. It also outlines relationships between subsets of real numbers and properties that real number operations satisfy, such as commutativity, associativity, identities, inverses, and distribution.
Rationalnumbers
Rational numbers can be written as a ratio of two integers. They can be expressed as decimals that either terminate or repeat. Examples of rational numbers include integers like 16, 1/2, 8, and decimals like 1.33. Rational numbers have properties when added, subtracted, multiplied, and divided, such as closure, commutativity, associativity, and others. Every rational number has a reciprocal, and the reciprocal of a rational number a/b is b/a.
Properties Of Exponents
The document summarizes several key properties of exponents:
1) The Product Property states that when multiplying like bases, you add the exponents.
2) The Quotient of Powers Property allows you to divide one power by another by subtracting the exponents.
3) The Power of a Power Property allows you to raise a power to another exponent by multiplying the exponents.
Properties of real numbers
The document discusses several properties of real numbers including:
1) The closure properties of addition and multiplication - the sum and product of any two real numbers is a real number.
2) Commutative, associative, and distributive properties of addition and multiplication.
3) Identity properties of addition and multiplication with 0 and 1 being the identity elements.
4) Inverse properties of addition and multiplication where adding/multiplying a number and its inverse results in the identity element.
Algebraic fractions
Examples and explanation of adding fractions with a common denominator, different denominators, multiplying fractions and dividing fractions.
Ch 2 fractions and decimals
1. The document discusses fractions and decimals, including multiplication of fractions.
2. It introduces multiplying a whole number by a fraction by multiplying the whole number by the numerator and keeping the denominator the same.
3. To multiply two fractions, we multiply the numerators and multiply the denominators of the fractions being multiplied.
Proof by contradiction
This document discusses proof by contradiction, a form of indirect proof used to establish the truth of a statement. It does this by assuming the statement is false and then showing this assumption leads to a logical contradiction, proving the statement must be true. The document provides examples of using proof by contradiction to show a number is irrational, and to prove statements about even and odd integers. It also outlines the basic steps of proof by contradiction.
cubes and cube root
The document discusses cubes and cube roots. A cube is a three-dimensional solid shape where all sides are equal. The cube of a number is when that number is multiplied by itself three times. A perfect cube is a number that can be written as the cube of another number, such as 1, 8, 27, 64, 125 and so on. The cube root of a perfect cube will be an integer, while the cube root of a non-perfect cube may need to be simplified by factoring the number into terms with a cube root of an integer. Examples of simplifying cube roots and finding cube roots of perfect and non-perfect cubes are provided.
Cube numbers
The document discusses cubic numbers, which are numbers multiplied by themselves three times. It provides examples such as 3 cubed equaling 27 and 4 cubed equaling 64. The cubic root of a number is the value that when cubed produces the original number. Perfect cubes are the results of cubing whole numbers, with examples including 8 from 2 cubed and 1331 from 11 cubed. Cubic numbers can also have roots that are not whole numbers, such as the cubic root of 40 being between 3.4 and 3.5.
Integers
The document discusses the properties of integers. Integers include positive and negative whole numbers along with zero. Integers have closure properties under addition and subtraction. They also have commutative and associative properties for addition. Zero is the additive identity for integers. Integers are closed under multiplication and one is the multiplicative identity. Division of integers has closure and communality properties when the divisor is not equal to zero.
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.
Linear Equations Ppt
The document discusses linear functions and graphs. It explains that linear graphs form straight lines and linear expressions only contain one variable with no exponents. It also defines slope as the rate of incline or decline of a line and discusses how to find the slope and y-intercept of a linear equation in slope-intercept form. Finally, it provides an example of using a linear equation to generate a table of x-y points and graph those points on a line.
Factorials
Factorials are the product of all positive integers from a given number down to 1. This document provides examples of calculating factorials such as 4! = 24 and discusses the rules for operations involving factorials such as cancellation rules that apply when dividing factorials. Practice problems are included with explanations to reinforce understanding of how to calculate and operate with factorial expressions.
Powers and Exponents
The document discusses powers and exponents. It explains that multiplication is a shortcut for repeated addition, and exponents are a shortcut for repeated multiplication. An exponent written as a base number with a little number on top, where the base is the number being multiplied and the exponent tells how many times to multiply the base by itself. Common mistakes in working with exponents are also described.
Quadratic functions my maths presentation
1. The document discusses key concepts related to quadratic equations including factorizing, solving, graphing, finding roots and axis of symmetry, locating maximum/minimum values, and using graphs to solve related equations and inequalities.
2. The key steps provided to factorize a quadratic equation are to find the product and factors of the coefficients that add up to the middle term, and then pair the factors.
3. To solve a quadratic equation, take all terms to one side, factorize, set each factor equal to 0 and solve for x to find the two roots.
Distributive Property
The document discusses three methods for solving the numeric expression 2(4+3):
1) Using the order of operations (BODMAS/PEMDAS) to evaluate the expression inside the parentheses first before multiplying.
2) Changing multiplication to addition by thinking of it as having "lots of" something.
3) Using the distributive property, where the number outside the parentheses is distributed across each term inside the parentheses.
alternate angles
The document presents a proof that alternate angles are equal. It defines two types of alternate angles: alternate interior angles and alternate exterior angles. It gives an example of two lines intersected by a transversal line to represent the angles. The proof uses vertically opposite angles and corresponding angles properties to show that alternate interior angles are congruent. It also asks to identify angle relationships and find values of variables in examples of alternate exterior and interior angles.
INTRODUCTION TO ALGEBRA
This document provides an overview of the key concepts in the language of algebra. It defines algebra as the branch of mathematics involving expressions with variables. Algebraic expressions combine numbers and variables using letters, symbols, and operation symbols. The language of algebra consists of numerals, variables to represent unknown numbers, constants, and operational symbols to express addition, subtraction, multiplication, and division of expressions. It also defines important terminology like terms, expressions, coefficients and evaluates sample expressions.
Classifying numbers
This document provides an overview of different types of numbers: whole numbers, integers, rational numbers, and irrational numbers. It defines each type and provides examples. Whole numbers are the counting numbers including 0. Integers include both positive and negative whole numbers. Rational numbers can be written as fractions, while irrational numbers cannot be written as fractions and have non-repeating decimals. The document uses examples and diagrams to illustrate how the different number types are related and how to identify each one.
Transformation Geometry
This document discusses translations in geometry. It begins by defining a translation as a horizontal or vertical slide that moves a figure without changing its shape or size. It then provides examples of translating points and figures by:
1) Translating individual points on a coordinate plane by moving them horizontally and vertically.
2) Translating triangles by moving their vertices horizontally and vertically, which demonstrates the triangle retains its shape and size.
3) Describing translations algebraically using notation like (x+p; y+q) to indicate a figure is translated p units horizontally and q units vertically.
Fractions & decimals class vii
This document provides an overview of fractions, decimals, and operations involving them. It discusses types of fractions like proper, mixed, and improper fractions. It also covers equivalent fractions, addition and subtraction of like and unlike fractions, and multiplication and division of fractions. Regarding decimals, it defines them as numbers with a decimal point separating the whole number part from the fractional part. It discusses comparing and performing operations like addition, subtraction, multiplication, and division on decimals. Finally, it briefly mentions conversion of metric units.
Properties of rational number explained
The document discusses properties of rational numbers. It states that multiplication and addition of rational numbers are commutative, but division and subtraction are not. It provides examples to show that multiplication, addition, and subtraction are associative for rational numbers, but division is not. It also discusses the distributive property of multiplication over addition and subtraction for rational numbers. Finally, it notes that zero is the additive identity and one is the multiplicative identity for rational numbers.
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.
Rational numbers ppt
Rational numbers are numbers that can be written as fractions p/q, where p and q are integers and q is not equal to 0. Rational numbers have important properties:
1) They are closed under addition and multiplication, meaning the sum or product of two rational numbers is also rational.
2) Operations like addition and multiplication are commutative and associative, following standard order of operations rules.
3) They follow the distributive property, where multiplying a number times the sum of two other numbers equals the sum of multiplying each number individually.
4) Each rational number has an additive inverse (its negative) and a multiplicative inverse (its reciprocal), such that adding/multiplying a number by its inverse results
More Related Content
What's hot
Properties Of Exponents
The document summarizes several key properties of exponents:
1) The Product Property states that when multiplying like bases, you add the exponents.
2) The Quotient of Powers Property allows you to divide one power by another by subtracting the exponents.
3) The Power of a Power Property allows you to raise a power to another exponent by multiplying the exponents.
Properties of real numbers
The document discusses several properties of real numbers including:
1) The closure properties of addition and multiplication - the sum and product of any two real numbers is a real number.
2) Commutative, associative, and distributive properties of addition and multiplication.
3) Identity properties of addition and multiplication with 0 and 1 being the identity elements.
4) Inverse properties of addition and multiplication where adding/multiplying a number and its inverse results in the identity element.
Algebraic fractions
Examples and explanation of adding fractions with a common denominator, different denominators, multiplying fractions and dividing fractions.
Ch 2 fractions and decimals
1. The document discusses fractions and decimals, including multiplication of fractions.
2. It introduces multiplying a whole number by a fraction by multiplying the whole number by the numerator and keeping the denominator the same.
3. To multiply two fractions, we multiply the numerators and multiply the denominators of the fractions being multiplied.
Proof by contradiction
This document discusses proof by contradiction, a form of indirect proof used to establish the truth of a statement. It does this by assuming the statement is false and then showing this assumption leads to a logical contradiction, proving the statement must be true. The document provides examples of using proof by contradiction to show a number is irrational, and to prove statements about even and odd integers. It also outlines the basic steps of proof by contradiction.
cubes and cube root
The document discusses cubes and cube roots. A cube is a three-dimensional solid shape where all sides are equal. The cube of a number is when that number is multiplied by itself three times. A perfect cube is a number that can be written as the cube of another number, such as 1, 8, 27, 64, 125 and so on. The cube root of a perfect cube will be an integer, while the cube root of a non-perfect cube may need to be simplified by factoring the number into terms with a cube root of an integer. Examples of simplifying cube roots and finding cube roots of perfect and non-perfect cubes are provided.
Cube numbers
The document discusses cubic numbers, which are numbers multiplied by themselves three times. It provides examples such as 3 cubed equaling 27 and 4 cubed equaling 64. The cubic root of a number is the value that when cubed produces the original number. Perfect cubes are the results of cubing whole numbers, with examples including 8 from 2 cubed and 1331 from 11 cubed. Cubic numbers can also have roots that are not whole numbers, such as the cubic root of 40 being between 3.4 and 3.5.
Integers
The document discusses the properties of integers. Integers include positive and negative whole numbers along with zero. Integers have closure properties under addition and subtraction. They also have commutative and associative properties for addition. Zero is the additive identity for integers. Integers are closed under multiplication and one is the multiplicative identity. Division of integers has closure and communality properties when the divisor is not equal to zero.
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.
Linear Equations Ppt
The document discusses linear functions and graphs. It explains that linear graphs form straight lines and linear expressions only contain one variable with no exponents. It also defines slope as the rate of incline or decline of a line and discusses how to find the slope and y-intercept of a linear equation in slope-intercept form. Finally, it provides an example of using a linear equation to generate a table of x-y points and graph those points on a line.
Factorials
Factorials are the product of all positive integers from a given number down to 1. This document provides examples of calculating factorials such as 4! = 24 and discusses the rules for operations involving factorials such as cancellation rules that apply when dividing factorials. Practice problems are included with explanations to reinforce understanding of how to calculate and operate with factorial expressions.
Powers and Exponents
The document discusses powers and exponents. It explains that multiplication is a shortcut for repeated addition, and exponents are a shortcut for repeated multiplication. An exponent written as a base number with a little number on top, where the base is the number being multiplied and the exponent tells how many times to multiply the base by itself. Common mistakes in working with exponents are also described.
Quadratic functions my maths presentation
1. The document discusses key concepts related to quadratic equations including factorizing, solving, graphing, finding roots and axis of symmetry, locating maximum/minimum values, and using graphs to solve related equations and inequalities.
2. The key steps provided to factorize a quadratic equation are to find the product and factors of the coefficients that add up to the middle term, and then pair the factors.
3. To solve a quadratic equation, take all terms to one side, factorize, set each factor equal to 0 and solve for x to find the two roots.
Distributive Property
The document discusses three methods for solving the numeric expression 2(4+3):
1) Using the order of operations (BODMAS/PEMDAS) to evaluate the expression inside the parentheses first before multiplying.
2) Changing multiplication to addition by thinking of it as having "lots of" something.
3) Using the distributive property, where the number outside the parentheses is distributed across each term inside the parentheses.
alternate angles
The document presents a proof that alternate angles are equal. It defines two types of alternate angles: alternate interior angles and alternate exterior angles. It gives an example of two lines intersected by a transversal line to represent the angles. The proof uses vertically opposite angles and corresponding angles properties to show that alternate interior angles are congruent. It also asks to identify angle relationships and find values of variables in examples of alternate exterior and interior angles.
INTRODUCTION TO ALGEBRA
This document provides an overview of the key concepts in the language of algebra. It defines algebra as the branch of mathematics involving expressions with variables. Algebraic expressions combine numbers and variables using letters, symbols, and operation symbols. The language of algebra consists of numerals, variables to represent unknown numbers, constants, and operational symbols to express addition, subtraction, multiplication, and division of expressions. It also defines important terminology like terms, expressions, coefficients and evaluates sample expressions.
Classifying numbers
This document provides an overview of different types of numbers: whole numbers, integers, rational numbers, and irrational numbers. It defines each type and provides examples. Whole numbers are the counting numbers including 0. Integers include both positive and negative whole numbers. Rational numbers can be written as fractions, while irrational numbers cannot be written as fractions and have non-repeating decimals. The document uses examples and diagrams to illustrate how the different number types are related and how to identify each one.
Transformation Geometry
This document discusses translations in geometry. It begins by defining a translation as a horizontal or vertical slide that moves a figure without changing its shape or size. It then provides examples of translating points and figures by:
1) Translating individual points on a coordinate plane by moving them horizontally and vertically.
2) Translating triangles by moving their vertices horizontally and vertically, which demonstrates the triangle retains its shape and size.
3) Describing translations algebraically using notation like (x+p; y+q) to indicate a figure is translated p units horizontally and q units vertically.
Fractions & decimals class vii
This document provides an overview of fractions, decimals, and operations involving them. It discusses types of fractions like proper, mixed, and improper fractions. It also covers equivalent fractions, addition and subtraction of like and unlike fractions, and multiplication and division of fractions. Regarding decimals, it defines them as numbers with a decimal point separating the whole number part from the fractional part. It discusses comparing and performing operations like addition, subtraction, multiplication, and division on decimals. Finally, it briefly mentions conversion of metric units.
Properties of rational number explained
The document discusses properties of rational numbers. It states that multiplication and addition of rational numbers are commutative, but division and subtraction are not. It provides examples to show that multiplication, addition, and subtraction are associative for rational numbers, but division is not. It also discusses the distributive property of multiplication over addition and subtraction for rational numbers. Finally, it notes that zero is the additive identity and one is the multiplicative identity for rational numbers.
What's hot (20)
Similar to properties of addition and subtraction of integers
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.
Rational numbers ppt
Rational numbers are numbers that can be written as fractions p/q, where p and q are integers and q is not equal to 0. Rational numbers have important properties:
1) They are closed under addition and multiplication, meaning the sum or product of two rational numbers is also rational.
2) Operations like addition and multiplication are commutative and associative, following standard order of operations rules.
3) They follow the distributive property, where multiplying a number times the sum of two other numbers equals the sum of multiplying each number individually.
4) Each rational number has an additive inverse (its negative) and a multiplicative inverse (its reciprocal), such that adding/multiplying a number by its inverse results
Algebra 2 Section 1-1
The document provides examples and explanations for solving linear equations. It begins by defining key vocabulary like open sentence, equation, and solution. It then shows how to translate between verbal and algebraic expressions. Various properties of equality like reflexive, symmetric, and transitive properties are explained. Finally, it demonstrates solving linear equations by isolating the variable using the inverse operations property of equality. Examples include solving equations with variables on both sides and checking solutions.
Determinants
This document provides information about determinants of square matrices:
- It defines the determinant of a matrix as a scalar value associated with the matrix. Determinants are computed using minors and cofactors.
- Properties of determinants are described, such as how determinants change with row/column operations or identical rows/columns.
- Examples are provided to demonstrate computing determinants by expanding along rows or columns and using cofactors and minors.
- Applications of determinants include finding the area of triangles and solving systems of linear equations.
Solving Linear Equations Over p-Adic Integers
The document discusses solving linear equations over the p-adic integers Zp. It begins by introducing p-adic integers as infinite series representations and describing how basic arithmetic operations of addition, subtraction, and multiplication can be performed on p-adic integers using techniques from modular arithmetic. It then shows that every linear equation over the p-adic integers has a unique solution, unlike equations over the usual integers Z.
multiplication of integers
This document discusses the multiplication of integers. It states that multiplying two integers of the same sign results in a positive integer, while multiplying an integer of one sign with an integer of the opposite sign results in a negative integer. It also explains that the product of an even number of negative integers is positive, while the product of an odd number of negative integers is negative.
Abstract Algebra Beamer Lesson1
This document provides an abstract for a course on Abstract Algebra taught by Gemma P. Salasalan. It includes the following key points:
- The course will cover topics like sets, arithmetic in the integers, congruence and congruence classes, and relations and operations. It will use the textbooks "Algebra" by Hungerford and "Fundamental Abstract Algebra" by Malik.
- Definitions and examples are provided for fundamental concepts in abstract algebra like sets, the division algorithm, divisibility, greatest common divisors, and congruence modulo n.
- Properties of binary relations like reflexivity, symmetry, and transitivity are defined.
Números repunits de lehmer
This document presents two theorems about repunit Lehmer numbers. Theorem 1 states that for any fixed base g > 1, there are only finitely many positive integers n such that the repunit number un = (gn - 1)/(g - 1) is a Lehmer number, and these can all be effectively computed. Theorem 2 states that there are no Lehmer numbers of the form un when 2 ≤ g ≤ 1000. The document provides background on Lehmer numbers and repunit numbers, establishes some preliminary results, and gives the proof of Theorem 1 by considering primitive divisors of the repunit numbers.
Geometry unit 2.5.ppt
Algebraic properties of equality, such as the additive, multiplicative, and substitution properties, can be used when solving problems in geometry. These properties are accepted as true in geometry, just as certain postulates and properties are. Properties of congruence between geometric objects follow similar rules to the properties of equality between numbers. Two-column proofs in geometry are structured with statements in one column and reasons or justifications in the other column, citing given information, definitions, theorems, or postulates.
HEC 222 LECTURE 1.pptx
The document provides an introduction to quantitative economics and basic numeracy and arithmetic skills needed to analyze economic theory. It explains why learning math is important for economics, such as being able to critically analyze assumptions, data, and isolate factors. The document outlines how to properly learn and apply mathematics, including understanding implications, equivalencies, deductions from assumptions rather than facts. It provides examples of logical implications, linear equations, and arithmetic rules. It demonstrates solving systems of linear equations using substitution and elimination methods. It also gives examples of using linear equations and functions in economic models of supply/demand equilibrium, national income, and a firm's demand function.
Stochastic Processes Homework Help
I am Geoffrey J. I am a Stochastic Processes Homework Expert at excelhomeworkhelp.com. I hold a Ph.D. in Statistics, from Edinburgh, UK. I have been helping students with their homework for the past 6 years. I solve homework related to Stochastic Processes. Visit excelhomeworkhelp.com or email info@excelhomeworkhelp.com. You can also call on +1 678 648 4277 for any assistance with Stochastic Processes Homework.
Integers
The document discusses integers and operations on integers. It defines integers as the set of whole numbers and their opposites, including negative numbers. It introduces the concepts of the additive inverse and absolute value of integers. The four basic operations on integers - addition, subtraction, multiplication and division - are then explained through their properties and rules. Addition and subtraction of integers follows rules about keeping or changing signs based on the signs of the integers. Multiplication and division may result in changing the sign of the product or quotient depending on the signs of the integers.
Integers
The document discusses integers and operations on integers. It defines integers as the set of whole numbers and their opposites, including negative numbers. It introduces the concepts of the additive inverse and absolute value of integers. The four basic operations on integers - addition, subtraction, multiplication and division - are then explained through their properties and rules. Key rules include taking the additive inverse when subtracting and giving the quotient a positive or negative sign depending on whether the integers have like or unlike signs.
04
This document summarizes key concepts in commutative ring theory, beginning with divisibility and ideals. It defines what it means for an element to divide another in a ring, as well as irreducible, prime, and unit elements. Ideals are introduced as additive subgroups that are closed under multiplication. Polynomial rings over a field are then discussed, proving they have unique factorizations. Irreducible polynomials are shown to be prime. The document provides numerous examples and proofs of fundamental results about divisibility, ideals, and factorization in commutative rings.
Vectors2
The document discusses various topics related to vectors and planes:
1. It explains the vector product in three forms - mathematical calculation, 3D picture, and in terms of sine. It provides an example to calculate the vector product of two vectors.
2. It discusses the different forms of the equation of a plane - parametric, scalar product, and Cartesian forms. It provides examples to write the equation of a plane in these different forms.
3. It explains how to find the foot of the perpendicular from a point to a plane. It provides examples to find the foot and shortest distance.
4. It discusses how to find acute angles between lines, planes, and a line and plane. Examples
Asymtotic Appoach.ppt
This document discusses asymptotic notation and its use in analyzing algorithm running times. It begins with an introduction to asymptotic notation and defines common notations like Big-O, Big-Theta, and Big-Omega. Examples are provided to demonstrate how to determine if a function is a member of a notation class. The document also reviews properties of asymptotic notation like transitivity, reflexivity, and symmetry. Common functions like polynomials, logarithms, and exponentials are examined from an asymptotic perspective. In the end, the document provides a brief review of summations and their application in analyzing loop-based algorithms.
Real number by G R Ahmed of KVK
Here are the step-by-step workings:
1) 224-1 can be factorized as (212+1)(212-1)
2) 212-1 can be further factorized as (26+1)(26-1)
3) Therefore, 224-1 = (212+1)(26+1)(26-1)
4) The numbers between 60 and 70 that divide (26+1)(26-1) are 65 and 63.
So the two numbers that exactly divide 224-1 and lie between 60 and 70 are 65 and 63.
Think Like Scilab and Become a Numerical Programming Expert- Notes for Beginn...
Notes for Scilab Programming. This notes includes the mathematics used behind scilab numerical programming. Illustrated with suitable graphics and examples. Each function is explained well with complete example. Helpful to beginners. GUI programming is also explained.
Scilab help book 2 of 2
This document provides information about various matrix operations and functions in Scilab, including:
1. flipdim, which flips a matrix with respect to a specified dimension.
2. permute, which permutes matrix elements into new array dimensions.
3. repmat, which replicates a given matrix to make a new matrix with specified row and column dimensions.
4. Various functions for cumulative operations on matrices like cumprod (cumulative product), cumsum (cumulative summation), and their use with different orientation arguments.
5. Other matrix functions covered include kron (Kronecker product), prod (product), sum (summation), and their use with different orientation arguments.
Pigeonhole Principle,Cardinality,Countability
The document discusses the pigeonhole principle, countability, and cardinality. The pigeonhole principle states that if n items are placed into m pigeonholes with n > m, then at least one pigeonhole must contain more than one item. Countability refers to sets being either finite or denumerable (having the same cardinality as the natural numbers). Cardinality compares the sizes of sets based on whether a bijection exists between them. The document provides examples and proofs of these concepts.
Similar to properties of addition and subtraction of integers (20)
Think Like Scilab and Become a Numerical Programming Expert- Notes for Beginn...
Think Like Scilab and Become a Numerical Programming Expert- Notes for Beginn...
More from sufiyafatima
playing with numbers
The document defines factors, multiples, and perfect numbers. Factors are numbers that divide evenly into a given number. Multiples are numbers that the given number divides into evenly. A perfect number is one where the sum of its factors is equal to twice the number. Examples given are that 6 is a perfect number because the factors of 1, 2, 3, and 6 sum to 12, which is twice 6.
integers word problems grade 7
The document contains examples of math word problems involving subtraction of negative numbers.
The first example asks for the current bank balance after an overdraft fee and deposit. It shows subtracting a negative overdraft amount and fee from a positive deposit amount.
The second example asks for the temperature difference between the optimum temperatures for two types of bacteria. It shows subtracting one negative temperature from an even lower negative temperature to get a positive difference.
integers word problems grade 7
The document contains two math word problems. The first problem asks for the room temperature after 10 hours if the room temperature is lowered by 5 degrees Celsius every hour, starting at 40 degrees Celsius. The answer is that the room temperature will be -10 degrees Celsius after 10 hours.
The second problem contains three sub-problems about students' scores on a 10 question test where 5 marks are given for each correct answer and -2 marks for each incorrect answer. The document calculates the scores for (i) Mohan who got 4 correct and 6 incorrect answers (score of 8), (ii) Reshma who got 5 correct and 5 incorrect answers (score of 15), and (iii) Heena who got 2 correct and 5
properties of addition and subtraction of integers
Addition of integers is associative, meaning the grouping of numbers does not change the result. For any three integers p, q, and r, p + (q + r) equals (p + q) + r. Addition is also associative because adding zero to any integer p yields p, demonstrating zero is the additive identity for integers.
patterns grade 6
This document discusses using patterns to understand numbers and do verbal calculations more easily. It presents examples of patterns where numbers are arranged in elementary shapes like triangles, squares and lines. The patterns show that multiplying a number by itself increases the result by one repeated digit, and that multiplying a number by 9 and adding a digit follows a pattern of increasing the result by one repeated digit as well. These patterns help with verbal calculations and understanding relationships between numbers.
whole numbers sum by suitable arrangement
The document contains examples of solving math problems step-by-step and answering word problems about quantities of liquids purchased and their total costs. It first shows additions and multiplications of numbers, then works through word problems about a taxi driver filling his tank with petrol over two days totaling 90 liters at Rs. 44 per liter, costing Rs. 3960. Another problem is about a milk vendor supplying 32 liters in the morning and 68 liters in the evening to a hotel at Rs. 15 per liter, totaling Rs. 1500 due per day.
Ppt 7 integers
An integer is a positive or negative whole number including zero. There are four integer operations: addition, subtraction, multiplication, and division. Integers have specific rules for addition - the sum of two positives is positive, the sum of two negatives is negative, and the sum of a positive and negative can be positive, negative, or zero.
7th integers class 7 ppt
Integers include all whole numbers from negative infinity to positive infinity, including zero, and are denoted by the letter Z. On a number line, positive integers are to the right of zero and negative integers are to the left. The additive inverse of a number is its opposite - for example, the additive inverse of 5 is -5. To subtract integers, the subtraction sign is changed to addition and the number after the sign is changed to its opposite. This allows subtraction problems to be solved as addition problems.
6th class ppt whole numbers
Whole numbers include the natural numbers (1, 2, 3, etc.) and zero. They can be represented on a number line and have important properties - zero is the smallest whole number, there are an infinite number of whole numbers, and each whole number has a unique successor and predecessor obtained by adding or subtracting 1.
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.
More from sufiyafatima (10)
properties of addition and subtraction of integers
properties of addition and subtraction of integers
Recently uploaded
Hindi varnamala | hindi alphabet PPT.pdf
हिंदी वर्णमाला पीपीटी, hindi alphabet PPT presentation, hindi varnamala PPT, Hindi Varnamala pdf, हिंदी स्वर, हिंदी व्यंजन, sikhiye hindi varnmala, dr. mulla adam ali, hindi language and literature, hindi alphabet with drawing, hindi alphabet pdf, hindi varnamala for childrens, hindi language, hindi varnamala practice for kids, https://www.drmullaadamali.com
Your Skill Boost Masterclass: Strategies for Effective Upskilling
Your Skill Boost Masterclass: Strategies for Effective UpskillingExcellence Foundation for South Sudan
Strategies for Effective Upskilling is a presentation by Chinwendu Peace in a Your Skill Boost Masterclass organisation by the Excellence Foundation for South Sudan on 08th and 09th June 2024 from 1 PM to 3 PM on each day.ANATOMY AND BIOMECHANICS OF HIP JOINT.pdf
it describes the bony anatomy including the femoral head , acetabulum, labrum . also discusses the capsule , ligaments . muscle that act on the hip joint and the range of motion are outlined. factors affecting hip joint stability and weight transmission through the joint are summarized.
Constructing Your Course Container for Effective Communication
Communicating effectively and consistently with students can help them feel at ease during their learning experience and provide the instructor with a communication trail to track the course's progress. This workshop will take you through constructing an engaging course container to facilitate effective communication.
BBR 2024 Summer Sessions Interview Training
Qualitative research interview training by Professor Katrina Pritchard and Dr Helen Williams
BÀI TẬP BỔ TRỢ TIẾNG ANH 8 CẢ NĂM - GLOBAL SUCCESS - NĂM HỌC 2023-2024 (CÓ FI...
BÀI TẬP BỔ TRỢ TIẾNG ANH 8 CẢ NĂM - GLOBAL SUCCESS - NĂM HỌC 2023-2024 (CÓ FI...Nguyen Thanh Tu Collection
https://app.box.com/s/y977uz6bpd3af4qsebv7r9b7s21935vdHow to Setup Warehouse & Location in Odoo 17 Inventory
In this slide, we'll explore how to set up warehouses and locations in Odoo 17 Inventory. This will help us manage our stock effectively, track inventory levels, and streamline warehouse operations.
BÀI TẬP DẠY THÊM TIẾNG ANH LỚP 7 CẢ NĂM FRIENDS PLUS SÁCH CHÂN TRỜI SÁNG TẠO ...
BÀI TẬP DẠY THÊM TIẾNG ANH LỚP 7 CẢ NĂM FRIENDS PLUS SÁCH CHÂN TRỜI SÁNG TẠO ...Nguyen Thanh Tu Collection
https://app.box.com/s/qhtvq32h4ybf9t49ku85x0n3xl4jhr15How to Create a More Engaging and Human Online Learning Experience
How to Create a More Engaging and Human Online Learning Experience Wahiba Chair Training & Consulting
Wahiba Chair's Talk at the 2024 Learning Ideas Conference. Walmart Business+ and Spark Good for Nonprofits.pdf
"Learn about all the ways Walmart supports nonprofit organizations.
You will hear from Liz Willett, the Head of Nonprofits, and hear about what Walmart is doing to help nonprofits, including Walmart Business and Spark Good. Walmart Business+ is a new offer for nonprofits that offers discounts and also streamlines nonprofits order and expense tracking, saving time and money.
The webinar may also give some examples on how nonprofits can best leverage Walmart Business+.
The event will cover the following::
Walmart Business + (https://business.walmart.com/plus) is a new shopping experience for nonprofits, schools, and local business customers that connects an exclusive online shopping experience to stores. Benefits include free delivery and shipping, a 'Spend Analytics” feature, special discounts, deals and tax-exempt shopping.
Special TechSoup offer for a free 180 days membership, and up to $150 in discounts on eligible orders.
Spark Good (walmart.com/sparkgood) is a charitable platform that enables nonprofits to receive donations directly from customers and associates.
Answers about how you can do more with Walmart!"
RHEOLOGY Physical pharmaceutics-II notes for B.pharm 4th sem students
Physical pharmaceutics notes for B.pharm students
Gender and Mental Health - Counselling and Family Therapy Applications and In...
A proprietary approach developed by bringing together the best of learning theories from Psychology, design principles from the world of visualization, and pedagogical methods from over a decade of training experience, that enables you to: Learn better, faster!
Temple of Asclepius in Thrace. Excavation results
The temple and the sanctuary around were dedicated to Asklepios Zmidrenus. This name has been known since 1875 when an inscription dedicated to him was discovered in Rome. The inscription is dated in 227 AD and was left by soldiers originating from the city of Philippopolis (modern Plovdiv).
Recently uploaded (20)
Your Skill Boost Masterclass: Strategies for Effective Upskilling
Your Skill Boost Masterclass: Strategies for Effective Upskilling
Constructing Your Course Container for Effective Communication
Constructing Your Course Container for Effective Communication
BÀI TẬP BỔ TRỢ TIẾNG ANH 8 CẢ NĂM - GLOBAL SUCCESS - NĂM HỌC 2023-2024 (CÓ FI...
BÀI TẬP BỔ TRỢ TIẾNG ANH 8 CẢ NĂM - GLOBAL SUCCESS - NĂM HỌC 2023-2024 (CÓ FI...
B. Ed Syllabus for babasaheb ambedkar education university.pdf
B. Ed Syllabus for babasaheb ambedkar education university.pdf
How to Setup Warehouse & Location in Odoo 17 Inventory
How to Setup Warehouse & Location in Odoo 17 Inventory
BÀI TẬP DẠY THÊM TIẾNG ANH LỚP 7 CẢ NĂM FRIENDS PLUS SÁCH CHÂN TRỜI SÁNG TẠO ...
BÀI TẬP DẠY THÊM TIẾNG ANH LỚP 7 CẢ NĂM FRIENDS PLUS SÁCH CHÂN TRỜI SÁNG TẠO ...
How to Create a More Engaging and Human Online Learning Experience
How to Create a More Engaging and Human Online Learning Experience
Walmart Business+ and Spark Good for Nonprofits.pdf
Walmart Business+ and Spark Good for Nonprofits.pdf
RHEOLOGY Physical pharmaceutics-II notes for B.pharm 4th sem students
RHEOLOGY Physical pharmaceutics-II notes for B.pharm 4th sem students
Gender and Mental Health - Counselling and Family Therapy Applications and In...
Gender and Mental Health - Counselling and Family Therapy Applications and In...
Film vocab for eal 3 students: Australia the movie
Film vocab for eal 3 students: Australia the movie
properties of addition and subtraction of integers
- 1. Properties of Addition and Subtraction of Integers
- 2. 1. Closure property a. Closure under Addition 1. Closure under Addition For the closure property the sum of two integers must be an integer then it will be closed under addition. Example 2 + 3 = 5 2+ (-3) = -1 (-2) + 3 = 1 (-2) + (-3) = -5 As you can see that the addition of two integers will always be an integer, hence integers are closed under addition. If we have two integers p and q, p + q is an integer.
- 3. b. Closure under Subtraction If the difference between two integers is also an integer then it is said to be closed under subtraction. Example 7 – 2 = 5 7 – (- 2) = 9 - 7 – 2 = – 9 - 7 – (- 2) = – 5 As you can see that the subtraction of two integers will always be an integer, hence integers are closed under subtraction. For any two integers p and q, p - q is an integer.
- 4. 3. Commutative Property a. Commutative Property under addition If we change the order of the integers while adding then also the result is the same then it is said that addition is commutative for integers. For any two integers p and q p + q = q + p Example 23 + (-30) = – 7 (-30) + 23 = – 7 There is no difference in answer after changing the order of the numbers.
- 5. b. Commutative Property under subtraction If we change the order of the integers while subtracting then the result is not the same so subtraction is not commutative for integers. For any two integers p and q p – q ≠ q – p will not always equal. Example 23 - (-30) = 53 (-30) - 23 = -53 The answer is different after changing the order of the numbers