On Business Mathematics

1. On Business Mathematics Welcome To My Presentation

2. Special Thanks To Kazi Md. Nasir Uddin Assistant Professor Dept. Of AIS Faculty of Business studies Jagannath University.

3. Member

4. Presentation Topic “Business Mathematics is a very powerful tool and analytical process that results in and offers an optimal solution, in spite of its limitations”

5. Number system Number system is the mathematical notation for representing numbers of a given set by using digits or other symbols . These are the basic building blocks of mathematics.

6. Classification of number system: Real Numbers: It comprises a set of all rational and irrational number. The real numbers are “all the numbers” on the number line. Imaginary Numbers: Square root of negative numbers are called imaginary number, e.g. Complex numbers: |a+ib| is the complex number where a and b are real part & i is the imaginary part.

7. Classification of number system: Rational Numbers: Rational number are those numbers which can be expressed as a ratio between two integers. Irrational Numbers: Irrational numberis a number that cannot be expressed as a ratio. Integers: Integer are whole numbers positive, negative or Zero .e.g. -3, -2, -1, 0, +1, +2, +3. Whole Number: Whole Numbers are simply the numbers 0, 1, 2, 3, 4, 5 etc.

8. Theory of Sets Aset is a collection of objects. These objects are called elements or members of the set. The symbol for element is  .

9. Types of set Universal set: a universal set is a set which contains all objects, including itself. For identification it uses “U”. Subset: A set of which all the elements are contained in another set is called Subset.Example: The set {1,2,3} is a subset of the set {1,2,3,4,5}. Empty Set/ Null Set:The Null Set or Empty Set. This is a set with no elements, often symbolized by The Universal Set. It is represented by Or by { } (a set with no elements).

10. Types of set Equivalent Set: If the elements of one set can be put in to one to one correspondence with the elements of another set, the two sets are called equivalent. Example: A= {a,b,c,d} , B={1,2,3,4}. Equal Set: Two sets A & B, If every element of A is also an element of B, and every element of B also in an element A. Example: A= {3,5,5,9}, B= {9,5,3}. Singleton Set: Singleton Set is a set containing only one element. Example: A= {1}

11. Types of set Proper Subset: A proper subset of a set, is a subset is strictly contained in and so necessarily excludes at least one member of. If B is a proper subset of A, then all elements of B are in A but A contains at least one element that is not in B. Example: If A={1,3,5}, then B={1,5} is a proper subset of A. Multiplication of Two Sets: Where a set is multiple by another set, then it is called multiplication of two sets. Example: A= {1, 2}, B= {a, b} A*B= {1, 2}*{a, b} = { (1,a), (1,b), (2,a), (2,b) }

12. Venn Diagrams A Venn diagram is a pictorial representation. It was named after English logician John Venn to present pictorial representation. Example: For A = {1, 2}, B = {2, 3}, U = {1, 2, 3, 4}, find the following using a Venn diagram: A∪B is pronounced as: A union B the elements: {1,2}∪{2,3} Venn diagram:

13. Indices 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 they called the laws of Indices.

14. Laws of Indices: If a, b, then • x =

15. Types of Indices Positive Indices: If n is a positive integer, and “a” a real number. By an = a×a×……× to n factors where “n” is called the index or the exponent of base a. Zero and Unity Index: The general principle is that anything other than zero raised to the power zero is one, i.e.,

16. Types of Indices Fractional Index: In a positive fractional index the numerator represents the power and the denominator the root. For example:

17. Logarithm The logarithms of a number is a given base is the index or the power to which the base must be raised to produce the number, i.e. to make it equal to the given number. If ax = N then x is called the logarithm of N to the base a and is written as log aN Thus, ax = N => x = log aN Here, ax = N is exponential form and log aN = X is logarithm form.

18. Function of logarithm Natural logarithm: The logarithm of a number to the base ‘e’ is called natural logarithm or napierian logarithm. Example: Log e 3 . Common logarithm: The logarithm of a number to the base 10 is called common logarithm or briggslan logarithm. Example: log 10 3.

19. The laws of logarithm The three main laws are stated here: 1stLaw: log (m x n) = logm+ logn. 2ndLaw: log (m/n) = log m − log n. 3rdLaw: logm n = n log m

20. EQUATION Equation is a mathematical statement that the values of two mathematical expressions are equal. For example: the equation 3x+5=2x+7 is true only for x=2 and not for x=3.

21. Classification of Equation: Linear equation A linear equation is an equation for straight line. It made up of two expressions equal to each other. Example: y = 2x + 1 is a linear equation.

22. On-linear equation: Equation whose graph does not form a straight line (linear) is called a Nonlinear Equation. , the variables are either of degree greater than 1 or less than 1, but never 1. Example: 4x2 + 2y - 1 = 0 is the examples of nonlinear equations.

23. Quadratic equation A quadratic equation is one that can be written in the standard form of ax2 + bx + c=0.Where a, b and c are real number and a is not equal to zero.The highest power of quadratic equation is 2.

24. Cubic equation A cubic equation is an equation in which the highest power of the unknown is three. The general form of the cubic equation is ax3+bx2+cx+d=0.

25. Thank You