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Math Homework Help

Mobius Strip

published on August 20th, 2006 . by Vanaja

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.

Paradox

published on August 19th, 2006 . by Vanaja

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

published on August 14th, 2006 . by Vanaja

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.