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# Business mathematics is a very powerful tools and analytic process that result in and optimal solution in spite of its limitation.

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### Business mathematics is a very powerful tools and analytic process that result in and optimal solution in spite of its limitation.

1. 1. Our Topics Business mathematics is a very powerful tools and analytical process that result in and optional solution in spite of its limitation.
2. 2. Our Group Members Name of the Students ID Md. Asaduzzaman M140201769 Md. Rubel Hossain M140201745 Md. Monjur Kader M140201722 Md. Kamruzzaman M140201776 Md. Al-Amin Mullah M140201762 Nur-Nabi M140201768 Mamunur Rashid M140201776
3. 3. Md. Asaduzzaman ID: M140201769
4. 4. Permutations & Combinations
5. 5. Permutations  A permutation is an arrangement of all or part of a set of objects, with regard to the order of the arrangement. Arrangement Variation Order Permutation – the arrangement is important Permutation is an ordered arrangement of items that occurs when:  No item is used more than once.  The order of arrangement makes a difference.  Ex: There are six permutations of the set {1,2,3}, namely (1,2,3), (1,3,2), (2,1,3), (2,3,1), (3,1,2), and (3,2,1).
6. 6. Combinations  A combination is a way of selecting se veral things out of a larger group, where (unlike permutations) order does not matter.  Arrangement is not important  The items are selected from the same group.  No item is used more than once.  The order of items makes no difference. Example: You are making a sandwich. How many different combinations of 2 ingredients can you make with cheese, mayo and ham? Answer: {cheese, mayo}, {cheese, ham} or {mayo, ham}.
7. 7. Combinations Choices Grouping selection  Remarks:  Permutation problems involve situations in which order matters.  Combination problems involve situations in which the order of items makes no difference.  The number of possible combinations if r items are taken from n items is: N C r = n! / r!*(n-r)!
8. 8. Difference between Permutations and Combinations Permutations  Arranging people, digits, numbers, alphabets, letters, and colors.  Keywords: Arrangements, arrange,…  Order is important Combinations  Selection of menu, food, clothes, subjects, teams.  Keywords: Select, choice  Order is not important.
9. 9. Md. Rubel Hossain ID: M140201745
10. 10. Number Systems
11. 11. Number Systems Number theory is one of the oldest branches of pure mathematics and focusses on the study of natural numbers. Arithmetic is taught in schools where children begin with learning numbers and number operations. The first set of numbers encountered by children is the set of counting numbers or natural numbers. In mathematics, a number system is a set of numbers. As mentioned earlier, children begin by studying the natural numbers: 1,2,3, ... with the four basic operations of addition, subtraction, multiplication and division. Later, whole numbers 0,1,2, .... are introduced, followed by integers including the negative numbers.
12. 12. Number Systems  The Natural Numbers The natural (or counting) numbers are 1, 2, 3, 4, 5, etc. There are infinitely many natural numbers. The set of natural numbers, {1, 2, 3, 4, 5, ...}, is sometimes written N for short.  The whole numbers are the natural numbers together with 0.  The Integers  The integers are the set of real numbers consisting of the natural numbers, their additive inverses and zero.  {..., -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5...}  The set of integers is sometimes written J or Z for short.
13. 13. Types of Number Systems  Real Number  A real number refers to any number that you would expect to find on the number line. It is a number whose name will be the "address" of a point on the number line. Its absolute value will name the distance of that point from 0.  Properties of Real Numbers  All of the numbers that you use in everyday life are real numbers.  Each real number corresponds to exactly one point on the number line, and every point on the number line represents one real number.  Real numbers can be classified  Rational Numbers  Irrational Numbers
14. 14. Properties of Real Numbers  The Rational Numbers  The rational numbers are those numbers which can be expressed as a ratio between two integers. For example, the fractions 1/3 and –1111/8 are both rational numbers. All the integers are included in the rational numbers, since any integer z can be written as the ratio z/1.  The Irrational Numbers  An irrational number is a number that cannot be written as a ratio (or fraction). In decimal form, it never ends or repeats. The ancient Greeks discovered that not all numbers are rational; there are equations that cannot be solved using ratios of integers.
15. 15. Number Systems  Imaginary Number  Square roots of negative numbers are called imaginary numbers because, the square of any number is positive only.  Complex Number  The complex numbers are the set {a + bi | a and b are real numbers}, where i is the imaginary unit, –1. (click here for more on imaginary numbers and operations with complex numbers.  The complex numbers include the set of real numbers. The real numbers, in the complex system, are written in the form a + 0i = a. a real number.
16. 16. Md. Monjur Kader ID:M140201722
17. 17. Set Theory Sets are used to define the concepts of relations and functions. The study of geometry, sequences, probability, etc. requires the knowledge of sets.The theory of sets was developed by German mathematician Georg Cantor (1845-1918). He first encountered sets while working on “problems on trigonometric series”.  Definition  A set is any collection of objects specified in such a way that we can determine whether a given object is or is not in the collection.  In other words A set is a collection of objects. These objects are called elements or members of the set. The symbol for element is  .
18. 18. There are three methods used to indicate a set  Description : Description means just that, words describing what is included in a set.  For example, Set M is the set of months that start with the letter J.  Roster Form : Roster form lists all of the elements in the set within braces {element 1, element 2, …}.  For example, Set M = { January, June, July}  Set-Builder Notation: Set-builder notation is frequently used in algebra.  For example, M = { x x  is a month of the year and x starts with the letter J}
19. 19. Properties of Set  Sub-sets  A is a subset of B if every element of A is also contained in B. This is written A  B.  For example, the set of integers { …-3, -2, -1, 0, 1, 2, 3, …} is a subset of the set of real numbers.  Formal Definition:  A  B means “if x  A, then x  B.”  Empty set  Set with no elements  { } or Ø.
20. 20. Cont.  - "superset"  A B  A is a superset of B  Every element in B is also in A  Ax: x B x A
21. 21. Cont.  - "proper subset"  A B - A is a proper subset of B (A  B)  Every element in A is also in B and  A  B  ( x: x A x B) A  B  - "proper superset"  A B - A is a proper superset of B (A  B)  Every element in B is also in A and  A  B  (Ax: x B x A) A  B  Example: N Z Q R
22. 22. Set Operators  Union of two sets A and B is the set of all elements in either set A or B.  Written A  B.  A  B = {x | x  A or x  B}  Intersection of two sets A and B is the set of all elements in both sets A or B.  Written A  B.  A  B = {x | x  A and x  B}  Difference of two sets A and B is the set of all elements in set A which are not in set B.  Written A - B.  A - B = {x | x  A and x  B}  also called relative complement  Complement of a set is the set of all elements not in the set.  Written Ac  Need a universe of elements to draw from.  Set U is usually called the universal set.  Ac = {x | x  U - A }
23. 23. Md. Kamruzzaman ID: M140201776
24. 24. Linear Programming: Model Formulation and Graphical Solution Linear programming is a widely used mathematical modeling technique to determine the optimum allocation of scarce resources among competing demands. Resources typically include raw materials, manpower, machinery, time, money and space.
25. 25. Characteristics of Linear Programming Problems  A decision amongst alternative courses of action is required.  The decision is represented in the model by decision variables.  The problem encompasses a goal, expressed as an objective function, that the decision maker wants to achieve.  Restrictions (represented by constraints) exist that limit the extent of achievement of the objective.  The objective and constraints must be definable by linear mathematical functional relationships.
26. 26. Assumptions of Linear Programming Model  Proportionality - The rate of change (slope) of the objective function and constraint equations is constant.  Additively - Terms in the objective function and constraint equations must be additive.  Divisibility -Decision variables can take on any fractional value and are therefore continuous as opposed to integer in nature.  Certainty - Values of all the model parameters are assumed to be known with certainty (non-probabilistic).
27. 27. Advantages of Linear Programming Model  It helps decision - makers to use their productive resource effectively.  The decision-making approach of the user becomes more objective and less subjective.  In a production process, bottle necks may occur. For example, in a factory some machines may be in great demand while others may lie idle for some time. A significant advantage of linear programming is highlighting of such bottle necks.
28. 28. Limitations of Linear Programming Model  Factors such as uncertainty, and time are not taken into consideration.  Parameters in the model are assumed to be constant but in real life situations they are not constants.  Linear programming deals with only single objective , whereas in real life situations may have multiple and conflicting objectives.  In solving a LPP there is no guarantee that we get an integer value. In some cases of no of men/machine a non-integer value is meaningless.
29. 29. Nur Nabi ID: M140201768
30. 30. Logarithm and their Properties
31. 31. Logarithm  Discovered by the Scottish Laird, John Napier of Merchiston He was a mathematician, astronomer, astrologer and physicist He believed in black magic and used to travel with a spider in a box and his familiar spirit was a black rooster He introduced logarithms as a way to simplify calculations.  Definition  If ax = N then x is called the logarithm of N to the base a and is written as log a N thus,  XX = N X = log a N  (The logarithm of ‘N’ to the base ‘a’ is X)  Here; ax = N is exponential form and log a N = X is logarithm form.
32. 32. Function of logarithm Natural logarithm: When base ‘e’ then the logarithm function is called natural logarithm function.  Example: Log e 3 Common logarithm:  When base is 10, and then it is called common logarithm.  Example: log 10 3
33. 33. Applications 1. Logarithmic scale  Scientific quantities are often expressed as logarithms of other quantities, using a logarithmic scale. For example, the decibel is a unit of measurement associated with logarithmic-scale quantities.  It is based on the common logarithm of ratios—10 times the common logarithm of a power ratio or 20 times the common logarithm of a voltage ratio.  It is used to quantify the loss of voltage levels in transmitting electrical signals, to describe power levels of sounds in acoustics, and the absorbance of light in the fields of spectrometry and optics.  The signal-to-noise ratio describing the amount of unwanted noise in relation to a (meaningful) signal is also measured in decibels
34. 34. Cont. 2. Fractals  Logarithms occur in definitions of the dimension of fractals are geometric objects that are self-similar: small parts reproduce, at least roughly, the entire global structure.  The Sierpinski triangle (pictured) can be covered by three copies of itself, each having sides half the original length. This makes the Hausdorff dimension of this structure log(3)/log(2) ≈ 1.58.  Another logarithm-based notion of dimension is obtained by counting the number of boxes needed to cover the fractal in question.
35. 35. Indices  Introduction  Indices are a useful way of more simply expressing large numbers. They also present us with many useful properties for manipulating them using what are called the Law of Indices.  What are Indices?  The expression 25 is defined as follows:  We call "2" the base and "5" the index.
36. 36. The Law of Indices  Rule 1:  Any number, except 0, whose index is 0 is always equal to 1, regardless of the value of the base.  Rule 2:  Rule 3:  Rule 4:
37. 37. Md. Al-Amin Mullah ID:M140201762
38. 38. Matrix and Application
39. 39. Introduction: The knowledge of matrices is necessary in various branches of mathematics. Matrices are one of the most powerful tools in mathematics. This mathematics tool simplifies our work to a great extent when compared with other straight forward method. The evolution of concept of matrices is the result of an attempt to obtain compact and simple method of salving systems of linear equations. Definition:  A matrices is an ordered rectangular array of numbers or functions. The numbers or functions are called the elements or the entries of the matrix.  A matrix is a rectangular array (arrangement) of numbers real or imaginary or functions kept inside braces () or [ ]subject to certain rules of operations .
40. 40. Types of Matrix  Rectangular Matrix  Square Matrix  Row Matrix  Column Matrix  Diagonal Matrix  Scalar Matrix  Unit Matrix or Identity Matrix  Zero Matrix or Null Matrix
41. 41. Types of Matrix  Rectangular Matrix A Matrix in which number of rows is not equal to number of columns it is called as rectangular matrix. A = 3 2 1 4 3 2 1 2 3  Square Matrix: A matrix with equal number of rows and columns (i.e. m = n) is called as square matrix. A = 1 2 1 2 1 2 1 2 1
42. 42. Cont.  Row Matrix: A matrix with a single row and any number of columns is called a row matrix. Example: A = 1 2 3 4 5  Column Matrix: A matrix with a single columns and any number of rows is called a column matrix. Example: 1 A= 2 3
43. 43. Cont.  Diagonal Matrix: A diagonal matrix is a square matrix in which all the elements except those on the leading are zero. Example: 2 0 0 A = 0 5 0 0 0 7  Scalar Matrix: A diagonal matrix in which all the diagonal elements are equal is called the scalar matrix. Example: 3 0 0 A = 0 3 0 0 0 3
44. 44. Cont.  Unit matrix or Identity matrix: When the diagonal elements are one and no diagonal elements are zero then the matrix is called as unit matrix or identity matrix. A unit matrix is always as square matrix. Example: 1 0 0 A = 0 1 0 0 0 1  Zero matrix or Null matrix: A matrix in which every element is zero is called a zero matrix or null matrix. Example: 0 0 0 0 0 0 0 0 0
45. 45. Operation On Matrix  Equality Matrices  Addition of matrices.  Subtraction of matrices.  Multiplication of matrices.
46. 46. Cont. Equality Matrices : Two Matrices are said to be equal if they have the same order & all the corresponding elements are equal. Addition of matrices: The sum of two matrices of the same order is the matrix whose elements are the sum of the corresponding elements of the given matrices.
47. 47. Cont…. Subtraction of matrices: Subtraction of the matrices is also done in the same manner of addition of matrices. When the matrix B is to be subtracted from matrix A, the elements in matrix B are subtracted from corresponding elements in matrix A. Multiplication of matrices: A matrix may be multiplied by any one number or any other matrix. Multiplication of a matrix by any one number is called a scalar multiplication .One matrix may also be multiplied by other matrix.
48. 48. Mamunur Rashid ID: M140201778
49. 49. Mathematics of Finance  Annuity  A regular periodic payment made by an insurance company to a policyholder for a specified period of time.  Factors affecting Annuities  Investment amount  Gender, age, health  Choice of benefit options
50. 50. Types of annuities The two basic annuities are  1. Immediate Annuity  2. Deferred Annuity  Immediate annuity  Immediate annuity returns payment immediately after an initial investment is made. The income starts within one year after the initial investment is made.The following factors are to be noted while choosing immediate annuity.  Deferred annuity  Deferred annuity accumulates money until the investment period and the accumulated sum is withdrawn during the retirement period.Deferred annuity is suitable to those who need a steady income after their retirement period.This annuity earning is tax deferred which means that the tax need not to be paid on all gains during the investment period.
51. 51. Annuity  Need of Annuities  1. The payment of tax is deferred  2. Annuity provides large amount which is more helpful for retiring persons  3. The annuity income and payments are guaranteed  Disadvantages of Annuities  1. When you are starting annuity for the first type, you need to provide commission to insurance brokers(from 10%) 2. Surrender charges need to be paid when you are withdrawing annuity(from 7%)
52. 52. Sinking Fund A fund into which a company sets aside money over time, in order to retire its preferred stock, bonds or debentures. In the case of bonds, incremental payments into the sinking fund can soften the financial impact at maturity. Investors prefer bonds and debentures backed by sinking funds because there is less risk of a default.  Use sinking fund in a sentence  “Many investors prefer investing in bonds that are backed by sinking funds because it reduces the organization's credit risk.”  Discount Rate:  1.Banking: Rate at which a bill of exchange or an accounts receivable is paid (discounted) before its maturity date.
53. 53. Mathematics of Finance  Present Value:  Present value describes how much a future sum of money is worth today.  The formula for present value is:  PV = CF/(1+r)n  Future value (FV)  Refers to a method of calculating how much the present value (PV) of an asset or cash will be worth at a specific time in the future.  Simple interest  A quick method of calculating the interest charge on a loan. Simple interest is determined by multiplying the interest rate by the principal by the number of periods. 
54. 54. Mathematics of Finance  Compound Interest: Interest which is calculated not only on the initial principal but also the accumulated interest of prior periods. Compound interest differs from simple interest in that simple interest is calculated solely as a percentage of the principal sum. The equation for compound interest is: P = C(1+ r/n)nt  True discount  If interest is deducted at the time a loan is obtained it is called true discount if the amount received plus the interest equals the amount to be paid at the maturity of the obligation.  Banker's discount  The difference between the amount shown on a bill of exchange, etc. that a customer sends to a bank for payment, and the amount that the customer receives, after the bank has taken its payment.
55. 55. Any Question?