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Math Homework Help

Everything is Number

published on April 23rd, 2007 . by Vanaja

Pythagoras had the habit of thinking about the numbers always. He had a passion for numbers.
He mentioned each number has its own qualities like this:-

  • One: It represents reason. It is the source of all numbers
  • Two: Represents Man
  • Three: Represents Woman
  • Four: It represents justice because it is the product of two equal numbers
  • Five: Represents wedding. Because it is the sum of 2(Man) and 3(Woman)

Mathematician suggests extra dimensions are time-like

published on April 18th, 2007 . by Vanaja

In a recent study, mathematician George Sparling of the University of Pittsburgh examines a fundamental question pondered since the time of Pythagoras, and still vexing scientists today: what is the nature of space and time? After analyzing different perspectives, Sparling offers an alternative idea: space-time may have six dimensions, with the extra two being time-like.

Read More

Sourse: www.physorg.com

Area Problem

published on April 18th, 2007 . by Vanaja

John has a rectangular lawn of dimensions 50 meter by 40 meter.

It has two roads each 2 meter wide running in the middle of it one parallel to the length and one parallel to the breadth. He wants to tar the road.

Find area of the roads to be tarred.

Hint:

We have two roads to be tarred. One is horizontal and the other is vertical.

So, area to be tarred is the area of the two roads which are rectangular in shape.

Area of a rectangle is length x breadth.

Area of the horizontal road = 2 x 50=100 sq.m

Area of the vertical road = 2 x 40=80 sq.m

So,the total are of the two roads=100+80=180 sq.m

But, the roads are crossing at the middle. It is a square portion of sides 2m by 2m in dimensions and we can see that two times we added that area. So, we should subtract that area from the total area of the two rectangular roads.

Area of the square portion is 2 x 2=4 sq.m

Therefore, the area to be tarred = Total area of the two rectangles - The common area

Ans: 176 sq.m

Temparature Problem

published on April 17th, 2007 . by Vanaja

The average temperature of for Monday, Tuesday and Wednesday was 30ºC.
The average temperature for Tuesday, Wednesday and Thursday was 29º C. The Temperature on Thursday was 32º C.

What was the temperature on Monday?

Hint:

Total temperature on Mon,Tue and Wed = 30ºC*3 = 90ºC
Total temperature on Tue,Wed and Thu = 29ºC*3 = 87ºC

Total temperature on Tue and Wed = Total temperature on Tue, Wed and Thu -Temperature on Thu = 87ºC-32ºC= 55ºC

Now, Temperature on Monday can be calculated as follows

Temperature on Monday = Total temperature on Mon,Tue and Wed-Total temperature on Tue and Wed

Ans: 35ºC

Fundamental Theorem of Arithmetic

published on April 13th, 2007 . by Vanaja

Fundamental Theorem of Arithmetic :

Every composite number can be expressed ( factorised) as a product of primes, and this factorisation is unique, apart from the order in which the prime factors occur.

The Fundamental Theorem of Arithmetic says that every composite number can be factorised as a product of primes. Actually it says more. It says that given any composite number it can be factorised as a product of prime numbers in a‘unique’ way, except for the order in which the primes occur. That is, given any composite number there is one and only one way to write it as a product of primes,as long as we are not particular about the order in which the primes occur.

So, for example, we regard 2 × 3 × 5 × 7 as the same as 3 × 5 × 7 × 2, or any other possible order in which these primes are written. This fact is also stated in the following form:

The prime factorisation of a natural number is unique, except for the order of its factors.

In general, given a composite number x, we factorise it as x = p1p2 … pn, where p1, p2,…, pn are primes and written in ascending order

If we combine the same primes, we will get powers of primes.

Once we have decided that the order will be ascending, then the way the number is factorised, is unique.

The Fundamental Theorem of Arithmetic has many applications, both within mathematics and in other fields.

The Gong

published on April 11th, 2007 . by Vanaja

If a clock takes 6 seconds to strike 6. How long the same clock take to strike 12.


Can you find the answer?

Euclid’s Division Lemma

published on April 10th, 2007 . by Vanaja

Euclid’s Division Lemma
Given positive integers a and b, there exist unique integers q and r satisfying a = bq + r, 0?r

This result was perhaps known for a long time, but was first recorded in Book VII of Euclid’s Elements. Euclid’s division algorithm is based on this lemma.
Euclid’s division algorithm is a technique to compute the Highest Common Factor (HCF) of two given positive integers. Recall that the HCF of two positive integers a and b is the largest positive integer d that divides both a and b.
Let us see how the algorithm works, through an example first. Suppose we need to find the HCF of the integers 455 and 42.

We start with the larger integer, that is, 455.
Then we use Euclid’s lemma to get 455 = 42 × 10 + 35.
Now consider the divisor 42 and the remainder 35, and apply the division lemma to get 42 = 35 × 1 + 7.
Now consider the divisor 35 and the remainder 7, and apply the division lemma to get 35 = 7 × 5 + 0.

Notice that the remainder has become zero, and we cannot proceed any further. We claim that the HCF of 455 and 42 is the divisor at this stage, i.e., 7. You can easily verify this by listing all the factors of 455 and 42.Why does this method work? It works because of the following result. So, let us state Euclid’s division algorithm clearly.To obtain the HCF of two positive integers, say c and d, with c > d, follow the steps below:

Step 1 : Apply Euclid’s division lemma, to c and d. So, we find whole numbers, q and r such that c = dq + r, 0 ?rStep 2 : If r = 0, d is the HCF of c and d. If r ? 0, apply the division lemma to d and r.
Step 3 : Continue the process till the remainder is zero. The divisor at this stage will be the required HCF.

This algorithm works because HCF (c, d) = HCF (d, r) where the symbol HCF (c, d) denotes the HCF of c and d, etc.Euclid’s division algorithm is not only useful for calculating the HCF of very large numbers, but also because it is one of the earliest examples of an algorithm that a computer had been programmed to carry out.

Remarks :
1. Euclid’s division lemma and algorithm are so closely interlinked that people often call former as the division algorithm also.
2. Although Euclid’s Division Algorithm is stated for only positive integers, it can be extended for all integers except zero, i.e., b ? 0.

Algorithm & Lemma

published on April 3rd, 2007 . by Vanaja

An algorithm is a series of well defined steps
which gives a procedure for solving a type of
problem?

The word algorithm comes from the name
of the 9th century Persian mathematician
al-Khwarizmi. In fact, even the word ‘algebra’
is derived from a book, he wrote, called Hisab
al-jabr w’al-muqabala.

A lemma is a proven statement used for
proving another statement.

Theoretical cloaking device is created

published on April 3rd, 2007 . by Vanaja

U.S. scientists, taking a tip from Star Trek, have used nanotechnology to create a theoretical optical “cloaking” device that can make objects invisible.
The Purdue University engineers, following mathematical guidelines devised by British physicists, created the theoretical device that can render objects invisible by guiding light around anything placed inside the “cloak.”
The design uses an array of tiny needles radiating outward from a central spoke. The device would bend light around the object being cloaked. Background objects would be visible, but not the object surrounded by the cylindrical array of nano-needles, said Vladimir Shalaev, a professor of electrical and computer engineering.
The design does, however, have a major limitation: It works only for any single wavelength, and not for the entire frequency range of the visible spectrum, Shalaev said.
“But this is a first design step toward creating an optical cloaking device that might work for all wavelengths of visible light,” he said.
The research is detailed in a paper appearing this month in the journal Nature Photonics.

Copyright 2007 by United Press International. All Rights Reserved.

Ref: science daily