Math Homework Help

It is very important for a math student to learn the basic concepts in mathematics. It is a common fact that most children find math is very hard and in particular Geometry. The main reason behind this is that they don’t have the basics in maths.
and in Geometry the concepts are more abstract. If the students get some good basic help in math, I am sure most of them do better in maths. So let us learn some basics of Geometry today.
There are three basic concepts of geometry. These are “point“, “line” and “plane“. I am not attempting to define them as it is not possible to define them precisely. We can however, have a good idea of these three by considering examples. A fine dot made by a sharp pencil on a sheet of paper, resembles a geometrical point very closely. The sharper the pencil, the closer is the dot to the concept of a point.

The surface of a sheet of paper or the surface of a smooth table are examples of plane. But these surfaces limited in extent. The geometrical plane extends endlessly in all directions.
A straight line, drawn on a sheet of paper with a sharp pencil, is a close example of a geometrical straight line. A geometrical line is a set of points and extends endlessly in both the directions. To emphasize this we use two arrowheads.
It is impossible to find exact example for point,line and plane in nature. The geometrical point, line and plane are ideal concepts. but for practical purpose it is enough to deal with close examples.
So, a plane is a set of points, line is a subset of plane. Moreover all the other figures in geometry are sets of points. But they are not just set of points. They are special set of points possessing some properties.

Notation
We use capital letters such as A, B, C, P, Q, R, X, Y, Z etc. to denote points.
We use small letters (lower case) such as l,m,n,p,q,r etc. to denote lines.

Yesterday we have learned what is a function.
Today let’s discuss about range and domain of a function.

Answers

Try to do more problems from your text.

Today we can discuss a topic from functions.

Definitions:

Function
Let A anb B be two non empty sets. A function “f” from a set A to a set B is a rule so that to each element x in A there corresponds exactly one element y in B, under f ,then we say that f is a functin from A to B and write
f:A -> B

y is called the image of x under f and is denoted by f(x). x and y are respectively called the independent variable and the dependent variable. We also say that y is a function of x and write
y=f(x)

Examples:

1. In the family of circles, the area A of the circle is a function of radius r of the circle.
   Here radius r is the independent variable and area A is the dependent variable
2. The speed of a chemical reaction increases 2 times with the addition of every 5 milligrams of a catalyst. Here the amount of catalyst is the independent variable and speed of the chemical reaction is the dependent variable.

A sphere has two surfaces. A sheet of paper has two surfaces. The first one sided surface was discovered by A.F Mobius and bears his name Mobius strip.

A model can be obtained by taking a paper strip ,giving it a half twist(180 degree), and then attaching the two ends. If you start drawing a line down its center along a line parallel to its edge of the strip, you’ll end up right back where you started. You will never cross an edge. To compare this you can make a cylinder by attaching the ends of the paper without any twist and draw a line like you did before. See the difference.In Euclidean space there are two types of mobius strips depending upon the direction(clockwise or anticlockwise) of the twist.

This illustration shows interlocked gears along the length of a mobius strip.

The mobius strip has several properties. Try cutting the strip along the line you drew. Instead of getting two separate strips you get one strip with two half twists. Then draw a center line around the resulting band and cut along it. You will get two strips wound around each other.

In my last post I mentioned about paradox. Let’s see what a paradox is. A paradox is a situation in which something seems both true and false. in other words a paradox is a sentence or a sentiment that is seemingly contradict or opposed to common sense and is yet perhaps true in fact.

Examples:

1. When we increase our knowledge, we know how little we know and there are lots of things we really don’t know about. So when we really know a lot we say “I know that I know nothing”
2. 
   A Mobius strip (wait for my next post to know what a Mobius strip is) is a topological paradox.Actually a mobius strip is a surface with only one side and one boundary; but it seems that it has two surfaces and two boundaries.

Paradoxes are as old as humankind. The ancient Greeks studied them intensely and resulted in the discovery of irrational numbers.

Mathematical logic is branch of mathematics. The idea of logic was first given by an English man George Boole in 1854 in his book ‘An investigation of the Laws of Thoughts. Logic is the study of reasoning.

In practical life, we express our ideas by means of sentences. A sentence may be either true of false or neither.In Mathematics we are concerned with only those sentences which are either true or false. These sentences are called logical statements.
A sentence which is both true and false simultaneously is not a sentence, rather, it is a paradox.(I will give details about paradox in my next post)

Example:

1. The angles of a rectangle are right angle-Since this sentence is true it is a statement.
2. 3+6=10-Since this sentence is false, it is a statement.
3. x is a prime number-this is true for x=2,3,5 etc but false for x=1,4,6 etc.So this sentence is not a logical statement.

Locus: When a point moves subject to certain specified conditions, the path traced out by it is called locus.

Equation of a locus: Equation to a locus is the algebraic relation that exists between x and y coordinates of a general point on the locus.

Example:
Find the equation of the locus of points such that the sum of its distances from (0,3) and (0,-3) is 8.

Solution:

(To solve this problem you must know Distance Formula)

Let P(x,y) be any point of the locus and A and B be the points (0,3) and
(0,-3) respectively.