This document provides information on indices, logarithms, and their applications. It defines indices and logarithms, outlines their basic properties and laws, and provides examples of using logarithms to perform calculations like multiplication, division, evaluating powers and roots. Logarithm tables are introduced as a tool to lookup logarithms and anti-logarithms before calculators. Worked examples demonstrate how to use logarithm tables to solve problems and determine unknown values.
This document defines a perfect number as a number where the sum of all its factors (excluding the number itself) equals the number. It provides 6 and 28 as examples of perfect numbers. It then explains a method for finding perfect numbers by taking the product of (2p-1) and 2p-1, where p is a prime number, and lists the first 3 perfect numbers found using this method: 6, 28, and 496. It challenges the reader to find the next perfect number and first 10 perfect numbers using this method.
The document discusses even, odd, and neither functions. It defines even functions as those where f(-x) = f(x), and odd functions as those where f(-x) = -f(x). Examples are provided of algebraic tests to determine if a function is even, odd, or neither. Students will use these algebraic methods to classify functions according to their symmetry.
Quantum mechanics 1st edition mc intyre solutions manualSelina333
Quantum Mechanics 1st Edition McIntyre Solutions Manual
Download at: https://goo.gl/SdC7Ef
quantum mechanics david mcintyre solutions pdf
quantum mechanics mcintyre pdf
quantum mechanics a paradigms approach solutions pdf
quantum mechanics mcintyre solutions pdf
quantum mechanics a paradigms approach solution manual
quantum mechanics mcintyre solutions manual pdf
hidden life of prayer
THE BINOMIAL THEOREM shows how to calculate a power of a binomial –
(x+ y)n -- without actually multiplying out.
For example, if we actually multiplied out the 4th power of (x + y) --
(x + y)4 = (x + y) (x + y) (x + y) (x + y)
-- then on collecting like terms we would find:
(x + y)4 = x4 + 4x3y + 6x2y2 + 4xy3 + y4 . . . . . (1)
The document discusses expanding binomial expressions like (x + y)n using Pascal's triangle and the binomial theorem. It explains that each term in the expansion has exponents of x and y that add up to n, with the x exponent decreasing by 1 and the y exponent increasing by 1 in subsequent terms. The coefficients of the terms form Pascal's triangle. It also presents the binomial theorem formula for finding the coefficients and discusses using factorials. It provides an example of finding a specific term in a binomial expansion by identifying which value of k corresponds to that term number.
This document discusses exponents and surds. It covers exponent or index notation, exponent or index laws, zero and negative indices, standard form, properties of surds, multiplication of surds, and division by surds. Examples are provided to illustrate exponent notation, evaluating exponents, writing numbers as products of prime factors, the laws of exponents, evaluating expressions with negative bases, and using a calculator to evaluate exponents.
FellowBuddy.com is an innovative platform that brings students together to share notes, exam papers, study guides, project reports and presentation for upcoming exams.
We connect Students who have an understanding of course material with Students who need help.
Benefits:-
# Students can catch up on notes they missed because of an absence.
# Underachievers can find peer developed notes that break down lecture and study material in a way that they can understand
# Students can earn better grades, save time and study effectively
Our Vision & Mission – Simplifying Students Life
Our Belief – “The great breakthrough in your life comes when you realize it, that you can learn anything you need to learn; to accomplish any goal that you have set for yourself. This means there are no limits on what you can be, have or do.”
Like Us - https://www.facebook.com/FellowBuddycom
Inter section of subspaces
Union of subspaces
Linear Sum of subspaces
Linear Span of a set
Linear Dependence of vector & Linearly Dependent set (LD set)
Linear Independence of vector & Linearly Independent set (LI set)
This document defines a perfect number as a number where the sum of all its factors (excluding the number itself) equals the number. It provides 6 and 28 as examples of perfect numbers. It then explains a method for finding perfect numbers by taking the product of (2p-1) and 2p-1, where p is a prime number, and lists the first 3 perfect numbers found using this method: 6, 28, and 496. It challenges the reader to find the next perfect number and first 10 perfect numbers using this method.
The document discusses even, odd, and neither functions. It defines even functions as those where f(-x) = f(x), and odd functions as those where f(-x) = -f(x). Examples are provided of algebraic tests to determine if a function is even, odd, or neither. Students will use these algebraic methods to classify functions according to their symmetry.
Quantum mechanics 1st edition mc intyre solutions manualSelina333
Quantum Mechanics 1st Edition McIntyre Solutions Manual
Download at: https://goo.gl/SdC7Ef
quantum mechanics david mcintyre solutions pdf
quantum mechanics mcintyre pdf
quantum mechanics a paradigms approach solutions pdf
quantum mechanics mcintyre solutions pdf
quantum mechanics a paradigms approach solution manual
quantum mechanics mcintyre solutions manual pdf
hidden life of prayer
THE BINOMIAL THEOREM shows how to calculate a power of a binomial –
(x+ y)n -- without actually multiplying out.
For example, if we actually multiplied out the 4th power of (x + y) --
(x + y)4 = (x + y) (x + y) (x + y) (x + y)
-- then on collecting like terms we would find:
(x + y)4 = x4 + 4x3y + 6x2y2 + 4xy3 + y4 . . . . . (1)
The document discusses expanding binomial expressions like (x + y)n using Pascal's triangle and the binomial theorem. It explains that each term in the expansion has exponents of x and y that add up to n, with the x exponent decreasing by 1 and the y exponent increasing by 1 in subsequent terms. The coefficients of the terms form Pascal's triangle. It also presents the binomial theorem formula for finding the coefficients and discusses using factorials. It provides an example of finding a specific term in a binomial expansion by identifying which value of k corresponds to that term number.
This document discusses exponents and surds. It covers exponent or index notation, exponent or index laws, zero and negative indices, standard form, properties of surds, multiplication of surds, and division by surds. Examples are provided to illustrate exponent notation, evaluating exponents, writing numbers as products of prime factors, the laws of exponents, evaluating expressions with negative bases, and using a calculator to evaluate exponents.
FellowBuddy.com is an innovative platform that brings students together to share notes, exam papers, study guides, project reports and presentation for upcoming exams.
We connect Students who have an understanding of course material with Students who need help.
Benefits:-
# Students can catch up on notes they missed because of an absence.
# Underachievers can find peer developed notes that break down lecture and study material in a way that they can understand
# Students can earn better grades, save time and study effectively
Our Vision & Mission – Simplifying Students Life
Our Belief – “The great breakthrough in your life comes when you realize it, that you can learn anything you need to learn; to accomplish any goal that you have set for yourself. This means there are no limits on what you can be, have or do.”
Like Us - https://www.facebook.com/FellowBuddycom
Inter section of subspaces
Union of subspaces
Linear Sum of subspaces
Linear Span of a set
Linear Dependence of vector & Linearly Dependent set (LD set)
Linear Independence of vector & Linearly Independent set (LI set)
Pascal's triangle is a triangular array of the binomial coefficients that arises from the binomial formulas. It was studied extensively by the French mathematician Blaise Pascal in the 17th century. The binomial theorem states that the expansion of (a + b)^n can be written as the sum of terms involving the binomial coefficients, with the coefficient of each term found using the appropriate entry in Pascal's triangle. Examples are provided of using the binomial theorem to expand expressions like (x + y)^5 and determining coefficients of specific terms in the expansions.
The theory of sets was developed by German mathematician Georg Cantor in the late 19th century. Sets are collections of distinct objects, which can be used to represent mathematical concepts like numbers. There are different ways to represent sets, including listing elements within curly brackets or using set-builder notation to describe a property common to elements of the set. Basic set operations include union, intersection, and complement. Venn diagrams provide a visual representation of relationships between sets.
The document defines various sets of numbers and binary operations. It then provides examples of binary operations on sets of numbers, such as addition and multiplication on sets of natural numbers, integers, rational numbers, real numbers, and complex numbers. The document also defines properties of binary operations such as commutativity, associativity, identity elements, and inverse elements. It provides problems and solutions showing examples of binary operations and verifying their properties.
- Logarithmic functions are inverses of exponential functions and can be converted between logarithmic and exponential form.
- Logarithms allow working with very large numbers in a more manageable way by reducing them to an exponent.
- The logarithmic function loga(x) is defined as the exponent that the base a must be raised to in order to equal x.
- Graphs of logarithmic functions can be obtained by shifting the graph of f(x) = log(x) horizontally and vertically.
This document provides an overview of complex numbers. It defines complex numbers as numbers consisting of a real part and imaginary part written in the form a + bi. It discusses the subsets of complex numbers including real and imaginary numbers. It also covers topics such as the complex conjugate, modulus, addition, subtraction, multiplication, and division of complex numbers. Finally, it mentions applications of complex numbers in science, mathematics, engineering, and statistics.
This document contains information about a class on infinite series and sequences taught at Shantilal Shah Engineering College in Bhavnagar, Gujarat, India. It lists the names of 5 students in the Bachelor of Engineering Sem 1 Batch B1 instrumentation and control engineering class for the 2014-2015 academic year. It thanks the reader at the end.
The document announces a mathematics project competition open to students in forms 3 and 4 at Maria Regina College Boys' Junior Lyceum. Teams of two students can participate by creating one of the following: a statistics project, charts, or a PowerPoint presentation on a given theme related to mathematics history or concepts. The top five entries will represent the school in the national competition and prizes will be awarded to the top teams nationally. Proposals are due by November 30th and completed projects by January 18th.
This document discusses approaches to teaching complex numbers. It describes an axiomatic approach, utilitarian approach, and historical approach. The historical approach builds on prior knowledge of quadratic equations and introduces complex numbers to solve problems like finding the roots of quadratic and cubic equations. The document also covers definitions of complex numbers, addition, subtraction, multiplication, and division of complex numbers. It discusses pedagogical considerations like using multiple representations and building on students' prior knowledge.
This document discusses binomial expansion, which is the process of expanding expressions with two terms like (x + a) to higher powers without lengthy multiplication. It introduces Pascal's triangle as a way to determine the coefficients in the expanded terms. It then defines the factorial operation and provides a general formula for determining the coefficients of any term when expanding a binomial to a given power.
The document provides an introduction to the binomial theorem. It begins by discussing binomial coefficients through the Pascal's triangle. It then derives an explicit formula for binomial coefficients using factorials. Finally, it states the binomial theorem and provides examples of using it to expand algebraic expressions and estimate numerical values.
This chapter discusses the theory of angular momentum in quantum mechanics and its applications. Eigenvectors of the angular momentum operator J satisfy certain eigenvalue equations involving the quantum numbers j and m. Specific cases of spin-1/2 and spin-1 systems are then derived. The chapter covers topics like coupling of angular momentum systems and angular momentum matrix elements.
The document discusses interpolation, which involves using a function to approximate values between known data points. It provides examples of Lagrange interpolation, which finds a polynomial passing through all data points, and Newton's interpolation, which uses divided differences to determine coefficients for approximating between points. The examples demonstrate constructing Lagrange and Newton interpolation polynomials using given data sets.
This MATLAB code uses finite difference methods to calculate the derivative of a function at discrete points. It takes in input vectors x and y, representing the independent and dependent variables. It then calculates the derivative at each point using forward, backward, and central difference formulas, storing the results in an output vector dx.
This document discusses Gaussian quadrature, a method for numerical integration. It begins by comparing Gaussian quadrature to Newton-Cotes formulae, noting that Gaussian quadrature selects both weights and locations of integration points to exactly integrate higher order polynomials. The document then provides examples of 2-point and 3-point Gaussian quadrature on the interval [-1,1], showing how to determine the points and weights to integrate polynomials up to a certain order exactly. It also discusses extending Gaussian quadrature to other intervals via a coordinate transformation, and provides an example integration problem.
This document discusses exponential and logarithmic functions. It defines exponential functions as functions of the form f(x) = bx where b is the base, and defines logarithmic functions as the inverses of exponential functions. Properties of exponential and logarithmic functions are presented, including their domains, ranges, and asymptotes. Examples of graphing common exponential and logarithmic functions are shown. Methods for solving exponential and logarithmic equations are also provided.
Liner algebra-vector space-1 introduction to vector space and subspace Manikanta satyala
This document discusses the key differences between scalar and vector quantities. Scalars only have magnitude, while vectors have both magnitude and direction. It then defines vector spaces as sets of vectors that are closed under vector addition and scalar multiplication. Examples of vector spaces include n-dimensional spaces, matrix spaces, polynomial spaces, and function spaces. Subspaces are also introduced as vector spaces that are subsets of a larger vector space and satisfy the same properties.
The document discusses the method of mathematical induction. It is used to verify infinitely many related statements without checking each one individually. As an example, it examines the statement that the sum of the first n odd numbers equals n^2 for all natural numbers n. It shows the base case of this statement is true, and if the statement is true for an arbitrary n, it must also be true for n+1. Therefore, by the principle of mathematical induction, the statement is true for all natural numbers n.
Gauss solved his teacher's challenge of summing the integers from 1 to 100 immediately by realizing that the normal sum (1 + 2 + ... + 100) and the reverse sum (100 + 99 + ... + 1) would be equal. Adding these sums together results in 101 multiplied by the number of terms (100), giving the solution of 5050. This insight showed that mechanical solving is not as useful as understanding the underlying patterns in mathematical problems.
The Gauss-Elemination method is used to solve systems of linear equations by reducing the system to upper triangular form using elementary row operations. It works by first making the coefficients of the variables above the main diagonal equal to zero one by one, then back-substituting the solutions. The method is illustrated using a 3x3 system that is reduced to upper triangular form by subtracting appropriate multiples of rows from each other. The unique solution can then be found by back-substituting the values of z, y, and x.
This document discusses indices, logarithms, and their properties and relationships. It includes:
- Five laws of indices for manipulating expressions with exponents.
- Examples of simplifying expressions using index laws.
- A definition of logarithms as the power to which a base must be raised to equal the value.
- Three logarithm rules for manipulating logarithmic expressions.
- Examples of calculating logarithms and using logarithm rules and properties.
- An overview of using logarithm tables to look up logarithm values.
Ib grade 11 physics lesson 1 measurements and uncertainitiesMESUT MIZRAK
This document discusses measurement and uncertainties in physics. It covers:
1) Expressing quantities using orders of magnitude to easily understand scale.
2) The ranges of distances, masses, and times that occur in the universe from smallest to greatest.
3) Calculating ratios as differences in orders of magnitude to compare quantities.
Pascal's triangle is a triangular array of the binomial coefficients that arises from the binomial formulas. It was studied extensively by the French mathematician Blaise Pascal in the 17th century. The binomial theorem states that the expansion of (a + b)^n can be written as the sum of terms involving the binomial coefficients, with the coefficient of each term found using the appropriate entry in Pascal's triangle. Examples are provided of using the binomial theorem to expand expressions like (x + y)^5 and determining coefficients of specific terms in the expansions.
The theory of sets was developed by German mathematician Georg Cantor in the late 19th century. Sets are collections of distinct objects, which can be used to represent mathematical concepts like numbers. There are different ways to represent sets, including listing elements within curly brackets or using set-builder notation to describe a property common to elements of the set. Basic set operations include union, intersection, and complement. Venn diagrams provide a visual representation of relationships between sets.
The document defines various sets of numbers and binary operations. It then provides examples of binary operations on sets of numbers, such as addition and multiplication on sets of natural numbers, integers, rational numbers, real numbers, and complex numbers. The document also defines properties of binary operations such as commutativity, associativity, identity elements, and inverse elements. It provides problems and solutions showing examples of binary operations and verifying their properties.
- Logarithmic functions are inverses of exponential functions and can be converted between logarithmic and exponential form.
- Logarithms allow working with very large numbers in a more manageable way by reducing them to an exponent.
- The logarithmic function loga(x) is defined as the exponent that the base a must be raised to in order to equal x.
- Graphs of logarithmic functions can be obtained by shifting the graph of f(x) = log(x) horizontally and vertically.
This document provides an overview of complex numbers. It defines complex numbers as numbers consisting of a real part and imaginary part written in the form a + bi. It discusses the subsets of complex numbers including real and imaginary numbers. It also covers topics such as the complex conjugate, modulus, addition, subtraction, multiplication, and division of complex numbers. Finally, it mentions applications of complex numbers in science, mathematics, engineering, and statistics.
This document contains information about a class on infinite series and sequences taught at Shantilal Shah Engineering College in Bhavnagar, Gujarat, India. It lists the names of 5 students in the Bachelor of Engineering Sem 1 Batch B1 instrumentation and control engineering class for the 2014-2015 academic year. It thanks the reader at the end.
The document announces a mathematics project competition open to students in forms 3 and 4 at Maria Regina College Boys' Junior Lyceum. Teams of two students can participate by creating one of the following: a statistics project, charts, or a PowerPoint presentation on a given theme related to mathematics history or concepts. The top five entries will represent the school in the national competition and prizes will be awarded to the top teams nationally. Proposals are due by November 30th and completed projects by January 18th.
This document discusses approaches to teaching complex numbers. It describes an axiomatic approach, utilitarian approach, and historical approach. The historical approach builds on prior knowledge of quadratic equations and introduces complex numbers to solve problems like finding the roots of quadratic and cubic equations. The document also covers definitions of complex numbers, addition, subtraction, multiplication, and division of complex numbers. It discusses pedagogical considerations like using multiple representations and building on students' prior knowledge.
This document discusses binomial expansion, which is the process of expanding expressions with two terms like (x + a) to higher powers without lengthy multiplication. It introduces Pascal's triangle as a way to determine the coefficients in the expanded terms. It then defines the factorial operation and provides a general formula for determining the coefficients of any term when expanding a binomial to a given power.
The document provides an introduction to the binomial theorem. It begins by discussing binomial coefficients through the Pascal's triangle. It then derives an explicit formula for binomial coefficients using factorials. Finally, it states the binomial theorem and provides examples of using it to expand algebraic expressions and estimate numerical values.
This chapter discusses the theory of angular momentum in quantum mechanics and its applications. Eigenvectors of the angular momentum operator J satisfy certain eigenvalue equations involving the quantum numbers j and m. Specific cases of spin-1/2 and spin-1 systems are then derived. The chapter covers topics like coupling of angular momentum systems and angular momentum matrix elements.
The document discusses interpolation, which involves using a function to approximate values between known data points. It provides examples of Lagrange interpolation, which finds a polynomial passing through all data points, and Newton's interpolation, which uses divided differences to determine coefficients for approximating between points. The examples demonstrate constructing Lagrange and Newton interpolation polynomials using given data sets.
This MATLAB code uses finite difference methods to calculate the derivative of a function at discrete points. It takes in input vectors x and y, representing the independent and dependent variables. It then calculates the derivative at each point using forward, backward, and central difference formulas, storing the results in an output vector dx.
This document discusses Gaussian quadrature, a method for numerical integration. It begins by comparing Gaussian quadrature to Newton-Cotes formulae, noting that Gaussian quadrature selects both weights and locations of integration points to exactly integrate higher order polynomials. The document then provides examples of 2-point and 3-point Gaussian quadrature on the interval [-1,1], showing how to determine the points and weights to integrate polynomials up to a certain order exactly. It also discusses extending Gaussian quadrature to other intervals via a coordinate transformation, and provides an example integration problem.
This document discusses exponential and logarithmic functions. It defines exponential functions as functions of the form f(x) = bx where b is the base, and defines logarithmic functions as the inverses of exponential functions. Properties of exponential and logarithmic functions are presented, including their domains, ranges, and asymptotes. Examples of graphing common exponential and logarithmic functions are shown. Methods for solving exponential and logarithmic equations are also provided.
Liner algebra-vector space-1 introduction to vector space and subspace Manikanta satyala
This document discusses the key differences between scalar and vector quantities. Scalars only have magnitude, while vectors have both magnitude and direction. It then defines vector spaces as sets of vectors that are closed under vector addition and scalar multiplication. Examples of vector spaces include n-dimensional spaces, matrix spaces, polynomial spaces, and function spaces. Subspaces are also introduced as vector spaces that are subsets of a larger vector space and satisfy the same properties.
The document discusses the method of mathematical induction. It is used to verify infinitely many related statements without checking each one individually. As an example, it examines the statement that the sum of the first n odd numbers equals n^2 for all natural numbers n. It shows the base case of this statement is true, and if the statement is true for an arbitrary n, it must also be true for n+1. Therefore, by the principle of mathematical induction, the statement is true for all natural numbers n.
Gauss solved his teacher's challenge of summing the integers from 1 to 100 immediately by realizing that the normal sum (1 + 2 + ... + 100) and the reverse sum (100 + 99 + ... + 1) would be equal. Adding these sums together results in 101 multiplied by the number of terms (100), giving the solution of 5050. This insight showed that mechanical solving is not as useful as understanding the underlying patterns in mathematical problems.
The Gauss-Elemination method is used to solve systems of linear equations by reducing the system to upper triangular form using elementary row operations. It works by first making the coefficients of the variables above the main diagonal equal to zero one by one, then back-substituting the solutions. The method is illustrated using a 3x3 system that is reduced to upper triangular form by subtracting appropriate multiples of rows from each other. The unique solution can then be found by back-substituting the values of z, y, and x.
This document discusses indices, logarithms, and their properties and relationships. It includes:
- Five laws of indices for manipulating expressions with exponents.
- Examples of simplifying expressions using index laws.
- A definition of logarithms as the power to which a base must be raised to equal the value.
- Three logarithm rules for manipulating logarithmic expressions.
- Examples of calculating logarithms and using logarithm rules and properties.
- An overview of using logarithm tables to look up logarithm values.
Ib grade 11 physics lesson 1 measurements and uncertainitiesMESUT MIZRAK
This document discusses measurement and uncertainties in physics. It covers:
1) Expressing quantities using orders of magnitude to easily understand scale.
2) The ranges of distances, masses, and times that occur in the universe from smallest to greatest.
3) Calculating ratios as differences in orders of magnitude to compare quantities.
Logarithms are used to replace multiplication with addition and division with subtraction. They allow calculations with large numbers to be simplified. There are three main laws of logarithms: 1) the logarithm of a product is equal to the sum of the logarithms of the factors, 2) the logarithm of a power is the power multiplied by the logarithm of the number, and 3) the logarithm of a quotient is equal to the logarithm of the dividend minus the logarithm of the divisor. Logarithms are commonly written to bases 10 and e, known as common logarithms and natural logarithms.
Logarithms are used to replace multiplication with addition and division with subtraction. They allow calculations with large numbers to be simplified. There are three main laws of logarithms: 1) the logarithm of a product is equal to the sum of the logarithms of the factors, 2) the logarithm of a power is the power multiplied by the logarithm of the number, and 3) the logarithm of a quotient is equal to the logarithm of the dividend minus the logarithm of the divisor. Logarithms are commonly written to bases 10 and e, known as common logarithms and natural logarithms.
1) Logarithms provide an alternative way to express exponential expressions like 16=24 by writing log216=4, where the logarithm is equivalent to the power or index in the original expression.
2) The three laws of logarithms describe relationships between logarithms and exponents: the first law states that loga(xy)=logax + logay, the second law states that loga(xm)= mlogax, and the third law states that loga(x/y)=logax - logay.
3) Logarithms can be used to solve equations where the unknown is in the power by taking logarithms of both sides and using the laws of logarithms to isolate the unknown.
The document discusses different types of numbers and operations involving positive and negative numbers. It explains rules for addition, subtraction, multiplication, and division of positive and negative numbers. It also covers order of operations using PEMDAS and provides examples of solving expressions using proper order. Finally, it discusses properties and rules for exponents, including adding, subtracting, multiplying, and dividing terms with the same base and combining exponents.
Logarithms allow multiplication and division operations to be converted to addition and subtraction. They were invented in the early 1600s by John Napier and Henry Briggs as a way to simplify calculations. Logarithms are closely related to exponents - if x = an then equivalently loga x = n. Studying logarithms allows complex multiplications and divisions to be performed using simpler addition and subtraction operations. They remain important in many fields such as science, engineering, economics and acoustics.
The document discusses different types of numbers including whole numbers, natural numbers, integers, rational numbers, and real numbers. It then covers the order of operations using PEMDAS and provides examples. Finally, it discusses properties of exponents such as multiplying, dividing, and raising exponents to other exponents.
Here are the answers to the exercises:
1. The 2007th digit after the period in the decimal expansion of 1/7 is 7, since the expansion repeats with a period of 7 digits (142857...).
2. a) and b) have finite decimal expansions, while c) does not.
3. A = [-1, 2], B = (-∞, -1] ∪ (2, ∞). C = (-∞, 1) ∪ (2, ∞). D = (-∞, 1) ∪ (3, ∞). E = [-1, 2].
The finite sets are A and E.
3. Functions
3
Ratio and Proportion, Indices and Logarithm Part 4FellowBuddy.com
FellowBuddy.com is an innovative platform that brings students together to share notes, exam papers, study guides, project reports and presentation for upcoming exams.
We connect Students who have an understanding of course material with Students who need help.
Benefits:-
# Students can catch up on notes they missed because of an absence.
# Underachievers can find peer developed notes that break down lecture and study material in a way that they can understand
# Students can earn better grades, save time and study effectively
Our Vision & Mission – Simplifying Students Life
Our Belief – “The great breakthrough in your life comes when you realize it, that you can learn anything you need to learn; to accomplish any goal that you have set for yourself. This means there are no limits on what you can be, have or do.”
Like Us - https://www.facebook.com/FellowBuddycom
This document contains lecture notes for a first semester calculus course. It begins by discussing different types of numbers like integers, rational numbers, and real numbers which are represented by possibly infinite decimal expansions. It then introduces functions and their properties like inverse functions and implicit functions. The notes provide examples and exercises to accompany the explanations.
Tenth class state syllabus-text book-em-ap-ts-mathematicsNaukriTuts
This document discusses rational and irrational numbers. It begins by recalling rational numbers as numbers that can be written as p/q where p and q are integers and q is not equal to 0. Irrational numbers cannot be written in this form and include numbers like √2, √3, and π. Together, rational and irrational numbers make up the set of real numbers. The document then explores properties of rational numbers, such as when their decimal expansions terminate or repeat periodically. It introduces the Fundamental Theorem of Arithmetic and uses it to prove results about rational numbers and their representations as decimals.
The document provides an overview of topics in number theory including:
- Number systems such as natural numbers, integers, and real numbers
- Properties of real numbers like closure, commutativity, associativity, identity, and inverse properties
- Rational and irrational numbers
- Order of operations
- Absolute value
- Intervals on the number line
- Finite and repeating decimals
- Converting between fractions and decimals
This chapter reviews real numbers including:
[1] Classifying numbers as natural numbers, integers, rational numbers, irrational numbers, and real numbers. Rational numbers can be written as fractions while irrational numbers cannot.
[2] Approximating irrational numbers like π as decimals to a given number of decimal places by rounding or truncating.
[3] How calculators handle decimals by either truncating or rounding values based on their display capabilities. A scientific or graphing calculator is recommended for this course.
This document discusses different types of number systems, including natural numbers, integers, rational numbers, real numbers, and complex numbers. It provides details on key concepts for each system. Natural numbers are the counting numbers starting from 1. Integers add negative whole numbers. Rational numbers are fractions with integer numerators and denominators. Real numbers include rational and irrational numbers, which can be represented by decimals. Complex numbers consist of real numbers combined with imaginary numbers using the imaginary unit i, where i^2 = -1. Each number system forms a proper subset of the next largest system.
This document discusses properties of logarithms, including:
1) Logarithms with the same base "undo" each other according to the inverse function relationship between logarithms and exponents.
2) Logarithmic expressions can be expanded using properties to write them as sums or differences of individual logarithmic terms, or condensed into a single logarithm.
3) The change of base formula allows converting between logarithms with different bases, with common uses being to change to base 10 or the base of natural logarithms.
The document provides definitions and explanations of key concepts in algebra including:
1. Types of numbers such as complex, rational, irrational, and integer numbers.
2. Properties of real numbers like commutative, associative, and distributive properties.
3. Exponents, radicals, logarithms, progressions, the binomial theorem, and word problems.
This document provides an overview of real numbers including:
1) Natural numbers, whole numbers, integers, decimals, rational numbers and irrational numbers are introduced.
2) Properties of real numbers such as being ordered and closed under addition and multiplication are described.
3) Converting fractions to decimals and vice versa is explained through examples.
4) Terminating, non-terminating, and recurring decimals are defined.
This document discusses notable products, which are algebraic multiplication expressions that can be written directly from inspection without verifying the multiplication, according to fixed rules. It notes that the most common notable products are the sum-difference identity and the square of a binomial. It also covers synthetic division, which is a method for easily dividing polynomials where the divisor is linear and of the form ax + b. Rational expressions and their domains are defined, and different types of factorization are described.
Classification of Living & Non Living ThingsPuna Ripiye
The document discusses the key differences between living and non-living things, and between plants and animals. Living things can grow, move, respire, and respond to their environment, while non-living things cannot. Plants and animals also differ in that plants can produce their own food, have cell walls, and lack advanced sensory and nervous systems, whereas animals consume other organisms for food and have more complex cellular structures and sensory abilities.
Revision cards on financial mahts [autosaved]Puna Ripiye
This document provides information on various financial mathematics concepts:
1) Simple interest, compound interest, linear depreciation, and reducing-balance depreciation formulas are defined.
2) Examples of simple interest, compound interest, and percentage gain/loss word problems are worked out in 3 steps or less.
3) Additional word problems involving topics like installment payments, savings deposits, cost price and selling price are solved concisely.
- The document discusses differentiation and integration of algebraic functions.
- It provides rules for finding the derivatives of functions such as y = xn, y = axn, and the sum or difference of functions.
- It also discusses that the derivative of a constant is 0, and provides examples such as dy/dx = 0 for y = 1.
- Integration is discussed as the reverse process of differentiation, with rules provided for indefinite integrals of functions like xn and definite integrals over an interval.
This document provides information on indices, logarithms, and their applications. It defines indices and logarithms, outlines their basic properties and laws, and provides examples of using logarithms to perform calculations like multiplication, division, evaluating powers and roots. Logarithm tables are also introduced as a tool to lookup logarithm and anti-logarithm values when direct calculation is difficult. Examples demonstrate how to use logarithm and anti-logarithm tables to solve problems.
1. Rationalizing surds means removing the radical sign from the denominator by multiplying the numerator and denominator by the conjugate.
2. The conjugate of a surd term keeps the radicand the same but changes the sign of any terms outside the radical.
3. Rationalizing terms of the form a - b involves multiplying the numerator and denominator by the conjugate a + b.
Direct and indirect variation problems can be solved by writing the appropriate variation equation based on whether the quantities vary directly, inversely, or jointly. The variation equation introduces a constant of proportionality that can be solved for by substituting known values. Common variation equations include: y = kx for direct variation, y = k/x for inverse variation, and z = kxy or z = kx^2 for joint variation, where k is the constant of proportionality.
This document provides formulas and examples for calculating probabilities of events. It defines mutually exclusive events as events that cannot occur at the same time. It gives the formulas for calculating the probability of the union or intersection of events. Examples include calculating probabilities of rolling dice, picking beads from a bag, and selecting fruits or students at random.
Modular arithmetic involves finding the remainder when one number is divided by another. It deals with operations like addition, subtraction, and multiplication performed on numbers modulo a given value. Some key points about modulo are that it always yields a non-negative remainder and can be used to solve congruences and expressions involving remainders of division.
- The document discusses differentiation and integration of algebraic functions.
- It provides rules for finding the derivatives of functions such as y = xn, y = axn, and the sum or difference of functions.
- It also discusses that the derivative of a constant function is equal to 0.
- The document concludes by discussing integration as the reverse process of differentiation and provides rules for indefinite and definite integrals of simple algebraic functions.
The document describes how to create and interpret a pie chart. It explains that to make a pie chart, you first find the total quantity being represented and divide a 360 degree circle proportionally based on each quantity's value out of the total. It then gives examples of solving problems using information from pie charts, such as calculating the sector angle that corresponds to a specific value or the total represented by the entire pie chart.
The document defines mode as the data value that occurs most frequently in a data set. It defines mean as the average of the values, calculated by summing all values and dividing by the total number of values. It provides examples of calculating mean from raw data sets and frequency tables. It also provides word problems calculating mean, median, and mode from data sets and using relationships between variable values.
Histogram,frequency distribution revision cardPuna Ripiye
The document contains a series of histograms showing frequency distributions of test scores. It provides solutions to questions about analyzing the distributions, including identifying the modal class, range of scores, class boundaries, frequency within classes, total number of students, and number of students within certain score ranges. Key information that can be extracted from the histograms includes the class boundaries, frequencies, and identifying the class with the highest frequency to determine the mode.
The document provides information about cumulative frequency curves and distributions:
1. A cumulative frequency curve is drawn from a cumulative frequency table by plotting the upper class boundary against the cumulative frequency.
2. The curve can be used to estimate measures of central tendency and spread, like the median and quartiles, by tracing the values on the y-axis that correspond to the fraction of total frequency, like the 25th percentile.
3. Examples show how to calculate values like the median and interquartile range from curves and tables, and how to determine probabilities from the curve values.
A bar graph uses bars of equal width separated by spaces to represent data. It shows the frequency distribution of scores from a student test. The document then provides solutions to 10 multiple choice questions about values that can be determined from the bar graph like the total number of students, mean score, median, range, and interquartile range.
This document provides information about measures of central tendency including the mode, mean, and median. The mode is the data value that occurs most frequently in a data set. The mean is the average of the values, found by summing all values and dividing by the total number of data points. The median is the middle value when data points are arranged in order. Examples are given of calculating the mode, mean, and median from data sets presented in tables.
The documents provide information about cumulative frequency curves and distributions:
1) Cumulative frequency curves are drawn from cumulative frequency tables and plot the upper class boundaries against the cumulative frequencies.
2) Examples of cumulative frequency tables are given showing the distribution of various data like exam marks and ages.
3) Measures of central tendency and spread like quartiles, median, and interquartile range can be estimated from a cumulative frequency curve by tracing specific fraction points on the y-axis.
4) Problems are worked out demonstrating how to calculate values from cumulative frequency tables, plot points on cumulative frequency curves, and estimate probabilities based on the curve distributions.
This document defines and explains various measures of spread for data sets, including range, interquartile range, mean deviation, variance, and standard deviation. It provides formulas to calculate each measure from discrete data sets and frequency tables. It also includes examples of problems calculating these measures from data sets and selecting the correct value of a measure.
This document defines and explains various measures of spread for data sets, including range, interquartile range, mean deviation, variance, and standard deviation. It provides formulas to calculate each measure from discrete data sets and frequency tables. Examples are given to demonstrate calculating measures such as standard deviation, variance, range, interquartile range, first and third quartiles, and median.
This document provides lesson notes on statistics and probability for students preparing for exams. It covers topics such as frequency distribution, measures of central tendency, measures of dispersion, and probability. For frequency distribution, it explains how to organize data into a table by arranging values in ascending order and counting frequencies. It also discusses various visual representations of data including pie charts, bar graphs, histograms, and frequency polygons. For measures of central tendency, it defines mean, median, and mode. For measures of dispersion, it covers range, interquartile range, variance, standard deviation, and mean deviation. The document also introduces experimental and theoretical probability, addition and multiplication rules for probabilities, and probabilities with and without replacement.
How to Make a Field Mandatory in Odoo 17Celine George
In Odoo, making a field required can be done through both Python code and XML views. When you set the required attribute to True in Python code, it makes the field required across all views where it's used. Conversely, when you set the required attribute in XML views, it makes the field required only in the context of that particular view.
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
How to Add Chatter in the odoo 17 ERP ModuleCeline George
In Odoo, the chatter is like a chat tool that helps you work together on records. You can leave notes and track things, making it easier to talk with your team and partners. Inside chatter, all communication history, activity, and changes will be displayed.
ISO/IEC 27001, ISO/IEC 42001, and GDPR: Best Practices for Implementation and...PECB
Denis is a dynamic and results-driven Chief Information Officer (CIO) with a distinguished career spanning information systems analysis and technical project management. With a proven track record of spearheading the design and delivery of cutting-edge Information Management solutions, he has consistently elevated business operations, streamlined reporting functions, and maximized process efficiency.
Certified as an ISO/IEC 27001: Information Security Management Systems (ISMS) Lead Implementer, Data Protection Officer, and Cyber Risks Analyst, Denis brings a heightened focus on data security, privacy, and cyber resilience to every endeavor.
His expertise extends across a diverse spectrum of reporting, database, and web development applications, underpinned by an exceptional grasp of data storage and virtualization technologies. His proficiency in application testing, database administration, and data cleansing ensures seamless execution of complex projects.
What sets Denis apart is his comprehensive understanding of Business and Systems Analysis technologies, honed through involvement in all phases of the Software Development Lifecycle (SDLC). From meticulous requirements gathering to precise analysis, innovative design, rigorous development, thorough testing, and successful implementation, he has consistently delivered exceptional results.
Throughout his career, he has taken on multifaceted roles, from leading technical project management teams to owning solutions that drive operational excellence. His conscientious and proactive approach is unwavering, whether he is working independently or collaboratively within a team. His ability to connect with colleagues on a personal level underscores his commitment to fostering a harmonious and productive workplace environment.
Date: May 29, 2024
Tags: Information Security, ISO/IEC 27001, ISO/IEC 42001, Artificial Intelligence, GDPR
-------------------------------------------------------------------------------
Find out more about ISO training and certification services
Training: ISO/IEC 27001 Information Security Management System - EN | PECB
ISO/IEC 42001 Artificial Intelligence Management System - EN | PECB
General Data Protection Regulation (GDPR) - Training Courses - EN | PECB
Webinars: https://pecb.com/webinars
Article: https://pecb.com/article
-------------------------------------------------------------------------------
For more information about PECB:
Website: https://pecb.com/
LinkedIn: https://www.linkedin.com/company/pecb/
Facebook: https://www.facebook.com/PECBInternational/
Slideshare: http://www.slideshare.net/PECBCERTIFICATION
This slide is special for master students (MIBS & MIFB) in UUM. Also useful for readers who are interested in the topic of contemporary Islamic banking.
Executive Directors Chat Leveraging AI for Diversity, Equity, and InclusionTechSoup
Let’s explore the intersection of technology and equity in the final session of our DEI series. Discover how AI tools, like ChatGPT, can be used to support and enhance your nonprofit's DEI initiatives. Participants will gain insights into practical AI applications and get tips for leveraging technology to advance their DEI goals.
LAND USE LAND COVER AND NDVI OF MIRZAPUR DISTRICT, UPRAHUL
This Dissertation explores the particular circumstances of Mirzapur, a region located in the
core of India. Mirzapur, with its varied terrains and abundant biodiversity, offers an optimal
environment for investigating the changes in vegetation cover dynamics. Our study utilizes
advanced technologies such as GIS (Geographic Information Systems) and Remote sensing to
analyze the transformations that have taken place over the course of a decade.
The complex relationship between human activities and the environment has been the focus
of extensive research and worry. As the global community grapples with swift urbanization,
population expansion, and economic progress, the effects on natural ecosystems are becoming
more evident. A crucial element of this impact is the alteration of vegetation cover, which plays a
significant role in maintaining the ecological equilibrium of our planet.Land serves as the foundation for all human activities and provides the necessary materials for
these activities. As the most crucial natural resource, its utilization by humans results in different
'Land uses,' which are determined by both human activities and the physical characteristics of the
land.
The utilization of land is impacted by human needs and environmental factors. In countries
like India, rapid population growth and the emphasis on extensive resource exploitation can lead
to significant land degradation, adversely affecting the region's land cover.
Therefore, human intervention has significantly influenced land use patterns over many
centuries, evolving its structure over time and space. In the present era, these changes have
accelerated due to factors such as agriculture and urbanization. Information regarding land use and
cover is essential for various planning and management tasks related to the Earth's surface,
providing crucial environmental data for scientific, resource management, policy purposes, and
diverse human activities.
Accurate understanding of land use and cover is imperative for the development planning
of any area. Consequently, a wide range of professionals, including earth system scientists, land
and water managers, and urban planners, are interested in obtaining data on land use and cover
changes, conversion trends, and other related patterns. The spatial dimensions of land use and
cover support policymakers and scientists in making well-informed decisions, as alterations in
these patterns indicate shifts in economic and social conditions. Monitoring such changes with the
help of Advanced technologies like Remote Sensing and Geographic Information Systems is
crucial for coordinated efforts across different administrative levels. Advanced technologies like
Remote Sensing and Geographic Information Systems
9
Changes in vegetation cover refer to variations in the distribution, composition, and overall
structure of plant communities across different temporal and spatial scales. These changes can
occur natural.
Exploiting Artificial Intelligence for Empowering Researchers and Faculty, In...Dr. Vinod Kumar Kanvaria
Exploiting Artificial Intelligence for Empowering Researchers and Faculty,
International FDP on Fundamentals of Research in Social Sciences
at Integral University, Lucknow, 06.06.2024
By Dr. Vinod Kumar Kanvaria
This presentation includes basic of PCOS their pathology and treatment and also Ayurveda correlation of PCOS and Ayurvedic line of treatment mentioned in classics.
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.
1. TABLE OF CONTENTS
List of Figures ..........................................................................................................................................2
Indices.....................................................................................................................................................3
First Law:.............................................................................................................................................3
Second Law: ........................................................................................................................................3
Third Law:............................................................................................................................................3
Fourth Law: .........................................................................................................................................4
Fifth Law:.............................................................................................................................................4
Fractional Indices:...............................................................................................................................4
Examples:............................................................................................................................................5
Logarithm................................................................................................................................................6
Relation between indices and logarithm............................................................................................7
Logarithm rules...................................................................................................................................7
Examples:............................................................................................................................................7
Multiplication using logarithm..........................................................................................................11
Division using logarithm....................................................................................................................11
Calculation of Power using Logarithm ..............................................................................................12
Calculation of Root using Logarithm.................................................................................................12
Anti-Logarithms.....................................................................................................................................13
Examples:..........................................................................................................................................15
Quiz ...................................................................................................................................................16
2. LIST OF FIGURES
Figure 1...................................................................................................................................................9
Figure 2.................................................................................................................................................10
Figure 3.................................................................................................................................................10
Figure 4.................................................................................................................................................13
Figure 5.................................................................................................................................................14
Figure 6.................................................................................................................................................14
3. INDICES
The plural of index is known as indices, where an index is used to compactly write a product
or a number.
In the expression , the index or exponent or power is “a” and the number ” is called the
base. Laws of indices are used to manipulate the expressions with indices. There are total
four laws of indices, each one of them is explained one by one as follow;
First Law:
When expressions with the same base are multiplied, the indices of all the expressions are to
be added.
In general,
Let’s a = 4, b = 2 and x = 3;
Then
Second Law:
When expressions with the same base are divided, the index of the expression in the
denominator is subtracted from the index of the expression in the numerator.
In general,
Let’s a = 4, b = 2 and x = 3;
Then
Third Law:
When the index of the expression is 0 then the whole expression become equal to 1.
In general,
80
= 1
4. Fourth Law:
When the index of an expression is equal to a negative number, then the same expression
with a positive index is written in a fraction form, with numerator equal to 1 and denominator
equal to that specified expression. It can be explained with the following example;
In general,
Let’s a = 4 and x = 3;
Fifth Law:
When the power of the base is raised to another power then indices are to be multiplied.
In general, (am
)n
= amn
.
Here the power or index “m” of the base “a” is raised to the power “n”, thus both the power
are multiplied with each other.
Let’s m = 2, n = 2 and a = 3;
Fractional Indices:
Fractional indices are represented by a base number with a fractional index or power or
exponent. In general, a fractional index is in the form of . The index laws are also
applicable to the fractional indices, for example squaring the expression means
multiplying the index by 2.
The fractional index is also known as positive square root and is represented by the symbol
√. For example the square root of 4 is equal to 2. It can further be explained as follow;
√
In order to simplify the expression,
5. Thus
√
Similarly √ , and so on.
Examples:
Simplify
As ;
Thus
Simplify
First of all we will simplify the expression in parenthesis;
=
=
=
Simplify
6. Simplify √
√
Simplify
LOGARITHM
In the simplest way, Logarithm definition provides the answer of the following question.
“How many of one number do we multiply to get another number?”
For example: how many 4s do we multiply to get 64?
Answer: 4x4x4 = 64, so we needed to multiply 3 of the 4s to get 64.
Logarithm represents the above answer in the following format: log4 (64) = 3
The number we are multiplying is called the base. Thus we can rephrase our answer as, the
logarithm of 64 with base 4 is equal to 3 or the log base 4 of 64 is 3.
All Logarithms with the base 10 are known as common logarithms. There is no need to
mention the base.
For example: log 100 = 2
7. In other words, it can also be written as log10 100 = 2
Relation between indices and logarithm
As square root of a number is the inverse process to squaring a number; in the same way,
taking a logarithm is the inverse process to taking a power.
Considering previous example; log4(64) = 3 in exponential form can be written as 43
= 64.
4x4x4 = 43
= 64, the index of base 4 is 3, and it is equal to 64, while the same index 3 is the
answer of log base 4 of 64.
Logarithm rules
Total three rules of logarithm will be discussed here.
1. log (a*b) = log a + log b, where the base will remains the same on both sides of equation.
2. log (a/b) = log a - log b, where the base will remains the same on both sides of equation.
3. Log (ab
) = b*log a, where the base will remains the same on both sides of equation.
Examples:
Find the value of x;
log2 32 = x
In exponential form;
32 = 25
Thus
Log2 32 = 5, where x = 5
log2 x = 7
In exponential form;
27
= 128
Thus
log2 128 = 7, where x = 128
8. logx 25 = 2
In exponential form;
52
= 25
Thus
Log5 25= 2, where x = 5
Log x = 3
As the base is not mentioned, so it is understood to be a common logarithm, hence it can be
rewritten as;
Log10 x = 3
It can be transformed as 103
= x
In exponential form;
103
= 100
Thus the value of x=100
If we want to calculate the value of log105, then it can be written in a form of 10x
= 5, where x
lies between 0 and 1. In order to calculate such problems, we either need a calculator or a
logarithmic table.
Log table is based on common logarithm. Before discussing the log table, lets first discus the
properties of logarithm, then we will move on to the log tables, because some of the
properties are required for solving questions based on log table.
Let's say we want to find the log base 10 of 15 on a common log table. 15 lies between 10
(101
) and 100 (102
), so its logarithm will lie between 1 and 2, or be 1.something. 150 lies
between 100 (102
) and 1000 (103
), so its logarithm will lie between 2 and 3, or be
2.something. The .something is called the mantissa. What comes before the decimal point (1
in the first example, 2 in the second) is known as characteristic. In figure 1, the leftmost
column will show the first two or, for some large log tables, three digits of the number whose
logarithm you're looking up.
9. Figure 1
If you're looking up the log of 15.27 in a normal log table, go to the row marked 15 in figure
1. If you're looking up the log of 2.57, go to the row marked 25. Sometimes the numbers in
this row will have a decimal point, so you'll look up 2.5 rather than 25. You can ignore this
decimal point, as it won't affect your answer. Also ignore any decimal points in the number
whose logarithm you're looking up, as the mantissa for the log of 1.527 is no different from
that of the log of 152.7.
On the appropriate row, slide your finger over to the appropriate column. This column will be
the one marked with the next digit of the number whose logarithm you're looking up. For
example, in figure 2, if you want to find the log of 15.27, your finger will be on the row
marked 15. Slide your finger along that row to the right to find column 2. You will be
pointing at the number 1818. Write this down.
10. Figure 2
In figure 3, the log table has a mean difference table; hence slide your finger over to the
column in that table marked with the next digit of the number you're looking up. For 15.27,
this number is 7. Your finger is currently on row 15 and column 2. Slide it over to row 15 and
mean differences column 7. You will be pointing at the number 20. Write this down.
Figure 3
11. Add the numbers found in the two preceding steps together. For 15.27, you will get 1838.
This is the mantissa of the logarithm of 15.27.
1818 + 20 = 1838
Since 15 is between 10 and 100 (101
and 102
), the log of 15 must be between 1 and 2, so
1.something, hence the characteristic is 1. Combine the characteristic with the mantissa to get
your final answer. Finally the log of 15.27 is 1.1838.
Multiplication using logarithm
The sum of the logarithms of two different numbers is the logarithm of the product of those
numbers. In general,
For example, if you want to multiply 15.27 and 48.54, you would find the log of 15.27 to be
1.1838 and the log of 48.54 to be 1.6861.
In this example, add 1.1838 and 1.6861 to get 2.8699. This number is the logarithm of your
answer.
You can do this by finding the number in the body of the table closest to the mantissa of this
number (8699). The more efficient and reliable method, however, is to find the answer in the
table of anti-logarithms, as described in the method above. For this example, you will get
741.1.
Division using logarithm
The difference of the logarithms of two different numbers is the logarithm of the division of
those numbers. In general,
For example, if you want to divide 15.27 over 48.54, you would find the log of 15.27 to be
1.1838 and the log of 48.54 to be 1.6861.
In this example, subtract 1.1838 from 1.6861 to get 0.5023. This number is the logarithm of
your answer.
The more efficient and reliable method, however, is to find the answer in the table of anti-
logarithms, as described in the method above. For this example, you will get 3.179.
12. Calculation of Power using Logarithm
In general, log (a) = n or 10n
= a; In order to calculate the answer of 21.532
, first we take the
log of the number without its exponent or power using log table;
log ( 21.53) = 1.333
Now in exponential form, the expression becomes;
101.333
= 21.53
Squaring both sides of the equation;
(101.333
)2
= 21.532
Using the indices property; (ax
)y
= ax*y
Thus
102.666
= 21.532
Now the anti-log of 2.666 will give us the final answer of 21.532
, which is 463.44.
Calculation of Root using Logarithm
The same method of calculating power using logarithm is applicable to root calculation. A
simple root (√) is equal to power 1/2, while ( is equal to power 1/3 and so on.
Let’s have an example of using logarithm to calculate the value of 850.
First of all convert 850 into exponential form as;
850 = 8501/4
Now calculate the logarithm of 850 using log table;
log (850) = 2.9294
In exponential form, log (850) = 2.9294 can be written as 102.9294
= 850
Thus
850 = 8501/4
= (102.9294
)1/4
=102.9294
*1/4
= 100.73235
Finally the anti-log of 0.73235 is 5.3994, which is the final answer of 850.
13. ANTI-LOGARITHMS
Use Anti-log table, when you have the log of a number but not the number itself. In the
formula 10n
= x, n is the common log, or base-ten log, of x. If you have x, find n using the log
table. If you have n, find x using the anti-log table.
Characteristic is the number before the decimal point. If you're looking up the anti-log of
2.8699, the characteristic is 2. Mentally remove it from the number you're looking up, but
make sure to write it down so you don't forget it.
Find the row that matches the first part of the mantissa. In 2.8699, the mantissa is .8699. Most
anti-log tables, like most log tables, have two digits in the leftmost column as shown in figure
4, so run your finger down that column until you find .86.
Figure 4
Slide your finger over to the column marked with the next digit of the mantissa. For 2.8699,
slide your finger along the row marked .86 to find the intersection with column 9. This is
shown as 7396 in figure 5. Write this down.
14. Figure 5
The anti-log table shown in figure 6 has a table of mean differences; thus slide your finger
over to the column in that table marked with the next digit of the mantissa. Make sure to keep
your finger in the same row.
Figure 6
15. In this case, you will slide your finger over to the last column in the table, column 9. The
intersection of row .86 and mean differences column 9 is 15. Write that down.
Add the two numbers from the two previous steps. In our example, these are 7396 and 15.
Add them together to get 7411.
Use the characteristic to place the decimal point. Our characteristic was 2. This means that
the answer is between 102
and 103
, or between 100 and 1000. In order for the number 7411 to
fall between 100 and 1000, the decimal point must go after three digits, so that the number is
about 700 rather than 70, which is too small, or 7000, which is too big. So the final answer is
741.1.
Thus the anti-log of 2.8699 is 741.1, i.e.
Anti-log (2.8699) = 741.1
Or
102.8699
= 741.1
Examples:
Calculate the anti-log of 2.80.
The characteristic here is 2, while the mantissa is 80. The .80 comes across 6310 under column 0. Use
the characteristic to place the decimal point. Our characteristic was 2. This means that the
answer is between 102
and 103
, or between 100 and 1000. In order for the number 6310 to fall
between 100 and 1000, the decimal point must go after three digits, thus the final answer is
631.0.
Calculate the anti-log of 1.943.
The characteristic here is 1, while the mantissa is 943. The .94 comes across 8770 under column 3.
Use the characteristic to place the decimal point. Our characteristic was 1. This means that
the answer is between 101
and 102
, or between 10 and 100. In order for the number 8770 to
fall between 10 and 100, the decimal point must go after two digits, thus the final answer is
87.70.
16. Calculate the anti-log of 0.9244.
The characteristic here is 0, while the mantissa is 9244. The .92 comes across 8395 under column 4.
For an anti-log table with another table of mean differences, the value that comes across the mantissa
fourth digit 4 in the same row of .92 is 8. Now addition of 8395 and 8 gives 8403. Use the
characteristic to place the decimal point. Our characteristic was 0. This means that the answer
is between 100
and 101
, or between 1 and 10. In order for the number 8403 to fall between 1
and 10, the decimal point must go after one digit, thus the final answer is 8.403.
Calculate the anti-log of 1.8375.
The characteristic here is 1, while the mantissa is 8375. The .83 comes across 6871 under column 7.
For an anti-log table with another table of mean differences, the value that comes across the mantissa
fourth digit 5 in the same row of .83 is 8. Now addition of 6871 and 8 gives 6879. Use the
characteristic to place the decimal point. Our characteristic was 1. This means that the answer
is between 101
and 102
, or between 10 and 100. In order for the number 6879 to fall between
10 and 100, the decimal point must go after two digits, thus the final answer is 68.79.
Calculate the anti-log of 3.8888.
The characteristic here is 3, while the mantissa is 8888. The .88 comes across 7727 under column 8.
For an anti-log table with another table of mean differences, the value that comes across the mantissa
fourth digit 8 in the same row of .88 is 14. Now addition of 7727 and 14 gives 7741. Use the
characteristic to place the decimal point. Our characteristic was 3. This means that the answer
is between 103
and 104
, or between 1000 and 10000. In order for the number 7741 to fall
between 1000 and 10000, the decimal point must go after four digits, thus the final answer is
7741.
Quiz
1. find the value of x;
√
(a) 8 (b) 4 (c) 2 (d) 16 (e) 32
2. √ √
(a) 14 (b) 12 (c) 10 (d) 16 (e) 20
17. 3. The simplified form of
(a) (b) (c) (d) (e)
4. Which of the following represents
(a) (b) √ (c) √ (d) √ (e) √
5. Which of the following is the correct value of x in ?
(a) 50 (b) 100 (c) 10 (d) 1000 (e) 500
6. Using the log table, what is the value of 5.30?
(a) 0.7250 (b) 0.7242 (c) 0.7240 (d) 0.7252 (e) 0.7245
7. Solve for correct value of x.
(a) 15/2 (b) 15 (c) 2 (d) 2/15 (e) 3/15
8. Using anti-log table, which of the following is the correct value of 1.59333?
(a) 39 (b) 39.2 (c) 3.9 (d) 3.92 (e) 3.0
9. Using anti-log table, which of the following is the correct value of 0.21?
(a) 1.62 (b) 1.72 (c) 1.82 (d) 1.92 (e) 2.02
10. Using anti-log table, which of the following is the correct value of 0.1234?
(a) 1.3268 (b) 1.3286 (c) 1.32 (d) 1.86 (e) 1.8632