Rules for using of Log Table

Dear friends today I will write about log table and their different using rules. At my last post I had written about Algebraic Methods of Solving Simultaneous Equations, is totally explained the basic Simultaneous Equations according the Algebraic Methods. We use various symbols for business application like present value (PV), Future values (FV), Interest rate (i), Rate of depreciation (d), Number of Years (n), Scrap/salvage value (SV) and Annuity/Installment (A). So Let’s know the Rules of use of Log Table…

1. At least four decimal places are acceptable for logarithm values
2. Logarithm of negative number is not defined. So logarithm of negative number is always rejected.
3. The scientific calculator may be used in palace of logarithm tables for finding logarithm values.
4. In case of using scientific calculator in place of logarithm tables , negative numbers of logarithms must be covered into positive numbers with a bar donation.
5. Rules for converting negative of logarithms into positive numbers of logarithms are as follows:

• For example, let a scientific calculator gives the value of log 0.000786=-3.1046
• To convert this negative number into positive number we shall write one more value of integral part with a bar notation, and adding this number to the negative value for placing new functional results behind the bar value, i.e. 4.8954
• To find the value of antilogarithm of 4.8954 through the scientific calculator we must use the original negative number. Because simple scientific calculator does not receive bar value.

I think you understand the above article Rules for using of log table. It will be helpful for a newbie to learn more. I will write in next post about rules of Business Application.

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Algebraic Methods Of Solving Simultaneous Equations

As indicate earlier there can be both graphics as well as algebraic methods of solving simultaneous equations and therefore, employed quite often in linear and non-linear programming. The graphics method is also employed in case of inequalities. We illustrate below its use when the simultaneous equations consist of both linear and non-linear equations.


The point of intersection gives a common solution to the two equations, one of which is linear (y=3x+3) and other one cubic (y=x cube). The value of x=2.1 gives an approximate solution which is good enough for business decision making.

We now discus the algebraic method of solving such equation. The use of metrics for the same shall be dealt in relevant chapter. Various method are indicated depending on the combination of linear with linear or non linear equations.

There are three methods of Simultaneous Equation

(A) When Both Equations are Linear

• Method of substitution
• Method of Elimination
• Method of Cross Multiplication

(B) When one equation is linear and the other one is quadratic

(C) When both equations are quadratic


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Basic Concept Of Binary Composition (Groups, Rings and Fields) : Chapter 5

These are some special types of mathematical systems. The purpose of these is to help in performing certain mathematical operations on a set. In the first three chapters we studied the algebra of certain binary operations and in the fourth chapter we acquainted ourselves with the real number system. Now we take up some mathematical compositions with certain number systems and the binary operations defined on them. The two together form a mathematical system. Before coming to certain special algebraic structures like groups and fields we shall like to discuss binary compositions.

Structure

• Introduction
• Binary Composition
• Various types of Compositions
• Composition Table
• Group
• An Abelian Group
• Properties of a Group
• Modulo
• Rings
• Fields

Objectives: After studying this chapter you should be able to understand:

• Binary composition, various types of compositions, composition table
• Groups, Rings And Fields

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Tautologies and Fallacies With Example

Basic Concept Of Tautologies and Fallacies

Tautologies or theorems are like axioms which are true for all values. In a truth table of tautology there will be only T in the last column. For example p OR~p= p, p AND p=p, and ~ ~p=p are all tautologies. As against these the Fallacies are the contradictions which will never be truth and obviously there will be only F in the output column of a truth table. For example p AND ~p is a contradiction, how can a statement and its negative both be true. These truth tables will further reveal this fact.

Since tautology is always true its negation is a fallacy and is always false.

• Example 1: Verify the following statement by constructing truth tables:

 

Solution:
(i)

Since there is T for all values of p, q in column 5, it is a tautology.

(ii)

Since there is F all values of p, q in column 6, it is a Fallacy

• Example 2: Using truth tables, show that (p AND q) =p, and p = (p OR Q) are both tautologies, where p,q are any two statements.

 Solution : 

Since all the entries in column 4 and 6 are T, the given proposition is both tautologies.

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Negation Is The Consideration Of The Statement

Basic Concept Of Negation

To assert a statement is to say that it is true and to deny it is to say that, it is false. Negation is the consideration of the statement which may be either an assertion or denial. If p is a statement then ~ p (rear as not p) will be its Negation. But if ~ p is a statement then p will be its Negation which we can also express ~(~ p) p. The following Truth table shows how when p is true, ~ p is false and vice-versa is also there.

An electronic circuit showing this type of relationship is as follows:

The above circuits shows that the electronical impulse passes from S1 to S2 in case of p and it will pass to S2 (S2 bar) and not to S2 in case of ~ p.

Double Negation is positive which can be verified from the following Truth Table.

 

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