Enhancing and Restoring Safety & Quality Cultures - Dave Litwiller - May 2024...
Business mathematics is a very powerful tools and analytic process that result in and optimal solution in spite of its limitation.
1.
2. Our Topics
Business mathematics is a very powerful tools and
analytical process that result in and optional
solution in spite of its limitation.
3. 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
6. 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).
7. 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}.
8. 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)!
9. 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.
12. 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.
13. 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.
14. 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
15. 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.
16. 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.
18. 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 .
19. 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}
20. 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.
21. Cont.
- "superset"
A B
A is a superset of B
Every element in B is also in A
Ax: x B x A
22. 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
23. 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 }
25. 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.
26. 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.
27. 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).
28. 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.
29. 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.
32. 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.
33. 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
34. 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
35. 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.
36. 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.
37. 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:
40. 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 .
41. 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
42. 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
43. 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
44. 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
45. 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
46. Operation On Matrix
Equality Matrices
Addition of matrices.
Subtraction of matrices.
Multiplication of matrices.
47. 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.
48. 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.
50. 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
51. 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.
52. 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%)
53. 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.
54. 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.
55. 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.
