Few Random Questions On Basics

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1. What happens to the solar energy which earth receives in such a huge quantity?

We know conversation of energy theorem.  So, is the energy of the earth is constantly rising? A part of energy what we receive gets reflected back to universe, and then what happens to the other part of it?

Under the sun, we feel hot and that confirms that we are receiving the solar energy. In the same way, in night, we feel cold; reason is we are giving back the energy.

(If the energy is transferred from one body to another, than the body which is giving the energy will feel the coldness and others feel the hotness.)

2. Why multiplications of 2 negative numbers yield a positive number?

This is a very basic principle in mathematics which we all know and we are using it all the way from our childhood days.

  • How this came and why is it true?

Let’s prove it.

In fact negative numbers are the reflection of the positive numbers (Their magnitude) about the origin in the number line. And while multiplying 2 negative numbers, their magnitude is multiplied first. For 2 negative sign, two times reflection is taken resulting in the product to be a positive number.

3. What might’ve happened if the earth didn’t have gravity?

It’s impossible by the fact that anything which owes mass will have the property of gravity too.

4. Why concentration of air goes on decreasing as the altitude increases?

Atmosphere is present in the earth because of its gravitational power and as altitude increases, force decreases and hence the air concentration will also decrease.

5. What might’ve happened if the earth isn’t tilted to 23.5◦ to its axis of rotation?

Chinese Robo to moon…!

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Few photos for your thirst..!! 🙂

china moon rover robot

china moon rover robot

china moon rover robot

china moon rover robot

china moon rover robot

More info: IEEE Spectrum

Now even Robots can talk with each other…!!!

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If you’ve always feared that robots would one day start talking to each other and plotting the demise of the human race, you can start getting a bit scared right now. The folks over at the University of Queensland have come up with a way to get robots to communicate with each other. Instead of giving them a fixed language i.e. something we already know like English, researchers have given the robots the ability to come up with their own words to describe the situation it is in.  The robots generate “words” from images of what they see with their onboard cameras.

Called Lingodroids, these robots are designed to work together in a team to map out the area they are in. Whenever a robot encounters a new area, they store its location and give it a name using a word generated from its own language. The location and name is then shared with other Lingodroids in the area. The robots then add those discovered areas onto their own maps and are able to identify it by name. The robots can even travel to that particular location if given the instructions to from other robots.

The single task they have to complete is to build a map of the world they are capable of travelling around on their three wheels. In order to do that they have a camera, range finder and sonar, and audio capabilities for capturing (microphone) and sharing (speaker) information.

Whenever a Lingodroid encounters some new area they map it using SLAM (Simultaneous Localisation and Mapping) which involves making it memorable using a grid, landmark, and topological combination that remains unique.

Once SLAM has been used to log an area the robot decides on a word to represent it made up from a list of sylabbles it has in memory. That word is then passed along to the other robots. All the robots then use that word to represent a specific place with reinforcement happening through game playing where one robot says a known word and the others navigate to it. That word is linked to the unique SLAM area marker formed by the Lingodroid such as the examples given below:

Lingodroids

It’s very basic stuff in terms of communication, but it soon develops into a map with keywords the robots can say and the others can decide to travel to, or at least add to their own internal maps. It even extends to areas that cannot be accessed, and the robots can still give those areas a name.

The next stage is to increase the spatial intelligence of the robots by allowing them to learn and give directions to specific places. If that works we’ll have robots acting like a GPS, but in a language we don’t understand.

While right now, it’s only the ability to name locations, but we can imagine future where robots develop the language and come up with words for more functions or actions..!! Hope it will be soon :):)

for more: IEEE Spectrum

Advanced Google Search Tips…

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Customize Google

Google is indisputedbly the best search engine out there on the Internet, however there are a few ways by which you can customize Google and make it even better.

1. OptimizeGoogle:

Using OptimizeGoogle, you can get rid of text ads from Google search results, add links from about 10 other search engines, add position counter, product results and more. You can even filter your search results to see dead websites (using WayBack Machine) and remove click tracking so that you can search anonymously. Here is a list of some of the other useful features of this plugin.

1. Use Google suggest (get word suggestion while typing)

2. Add more security by using https wherever necessary

3. Filter spammy websites from search results page

4. Option to remove SideWiki

5. Add links to bookmark your favorite result

6. Add links to other news and product search sites

To install the plugin, just visit the OptimizeGoogle page from your Firefox browser and click on the Download button. Once installed, you can enable or disable the customization options from the Tools -> OptimizeGoogle Options.

After you customize Google to improve the search results, how about adding a feature that provides a way to preview the website in the search results itself? Here is a miracle Firefox extension to this job for us.

2.  SearchPreview:

SearchPreview (formerly GooglePreview) will insert thumbnail view of the webpage into the Google search results page itself so that you can take the guess work out of clicking a link. Just install the plugin, reload Firefox and you will have the SearchPreview at work. You can install this plugin from the following link.

SearchPreview Download

OptimizeGoogle and SearchPreview have made our search results smarter and faster. Now how about safer? Well you have another Firefox plugin to make your search results safer as well. Here we go.

3. McAfee SiteAdvisor:

This is a free browser plugin that gives safety advice about websites on the search results page before you actually click on the links. After you install the SiteAdvisor plugin, you will see a small rating icon next to each search result which will alert you about suspecious/risky websites and help you find safer alternatives. These ratings are derived based on various tests conducted by McAfee.

Based on the quality of links, SiteAdvisor may display Green, Yellow, Red or Grey icon next to the search results. Green means that the link is completely safe, Yellow means that there is a minor risk, Red means a mojor risk and Grey means that the site is not yet rated. These results will guide you to Web safety.

SiteAdvisor works on both Internet Explorer and Firefox which you can download from the following link:

McAfee SiteAdvisor Download

Custom Google Search Results

SEARCH Techniques:

1.Google for getting Resources:

Using Google it is possible to gain access to an email repository containing CV of hundreds of people which were created when applying for their jobs. The documents containing their Address, Phone, DOB, Education, Work experience etc. can be found just in seconds.

intitle:”curriculum vitae” “phone * * *” “address *” “e-mail”

You can gain access to a list of .xls (excel documents) which contain contact details including email addresses of large group of people. To do so type the following search query and hit enter.

filetype:xls inurl:”email.xls”

Also it’s possible to gain access to documents potentially containing information on bank accounts, financial summaries and credit card numbers using the following search query

intitle:index.of finances.xls

2. Using Google to access Free Stuffs

 Ever wondered how to get free music or ebooks. Well here is a way to do that. To download free music just enter the following query on google search box and hit enter.
“?intitle:index.of?mp3 akon“

Now you’ll gain access to the whole index of akon album where in you can download the songs of your choice. Instead of eminem you can subtitute the name of your favorite album. To search for the ebooks all you have to do is replace “akon” with your favorite book name. Also replace “mp3″ with “pdf” or “zip” or “rar”.

Google Search:-[ intitle: ]

The “intitle:” syntax helps Google restrict the search results to pages containing that word in the title. For example, “intitle: login password” (without quotes) will return links to those pages that has the word “login” in their title, and the word “password” anywhere in the page.

Similarly, if one has to query for more than one word in the page title then in that case “allintitle:” can be used instead of “intitle” to get the list of pages containing all those words in its title. For example using “intitle: login intitle: password” is same as querying “allintitle: login password”.

Google Search:-[ inurl: ]

The “inurl:” syntax restricts the search results to those URLs containing the search keyword. For example: “inurl: passwd” (without quotes) will return only links to those pages that have “passwd” in the URL.

Similarly, if one has to query for more than one word in an URL then in that case “allinurl:” can be used instead of “inurl” to get the list of URLs containing all those search keywords in it. For example: “allinurl: etc/passwd“ will look for the URLs containing “etc” and “passwd”. The slash (“/”) between the words will be ignored by Google.

World’s First Hack free software!!!

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World’s ‘first hack-free’ software developed. Scientists have developed what they claim is the world’s first hack-free software which can protect systems from failure or malicious attacks.The ‘seL4’ micro kernel has been developed by a team led by Australia’s ICT Research Center of Excellence’s spin out company — Open Kernel Labs(OK Labs). It is a small operating system kernel which regulates access to a computer’s hardware.

Its unique feature is that it has been mathematically proven to operate correctly, enabling it to separate trusted from untrusted software, protecting critical services from a failure or a malicious attack, say the scientists.

In future applications, seL4 could ensure that trusted financial transaction software from secure sources like banks or stock exchanges can operate securely on a customer’s mobile phone alongside “untrusted” software, such as games downloaded from the Internet, according to its developers.

It could also provide a secure and reliable environment for mission-critical defense data, operating on the same platform as everyday applications like email. Or, it could protect the life-supporting functions of an implanted medical device, such as a pacemaker, from hacking, they say.

“Our seL4 micro kernel is the only operating system kernel in existence whose source code has been mathematically proven to implement its specification correctly. Under the assumptions of the proof, the seL4 kernel for ARM11 will always do precisely what its specification says it will do,” lead scientist Gerwin Klein said.

Golden Rectangle and Golden Ratio

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What is the Golden Ratio

Uppercase and lowercase Greek letter phi, the ...

Golden rectangle

In a golden rectangle, it can be divided into a square and a smaller rectangle. The ratio of the width of the small rectangle to the width of the square is the same as that of the width of the square to the entire rectangle. In fact, when you divide a golden rectangle into a square and smaller rectangle, the smaller rectangle is a golden rectangle as well.

Golden ration in Nature

nautilus shell showing the chambe...

The most common example of the golden ratio is the nautilus shell (see image). As it spirals in on itself, the spirals get smaller and smaller in the same proportion to each other as they do to the whole. You can also see the ratio in things like sunflower petals, and the curvature of fern fronds (see image).

Let’s look at the golden ratio, a little bit. You can verify these observations algebraically.

  1. What is one over the golden ratio? 1 / 1.61803398875 . . .=0.61803398875 . . . Does that answer look familiar? It is one less than the golden ratio.
  2. What is the golden ratio squared? (1.61803398875 . . .)2=2.61803398875 . . . Does that answer look familiar? It is one more than the golden ratio.
  3. I was trying to investigate chaos, and I drew the graph of y=x2-1. This parabola intersects the graph of the line y=x in two points. What are the coordinates of those two points? The points are (-0.61803398875 . . .,-0.61803398875 . . .) and (1.61803398875 . . .,1.61803398875 . . .). All four of those numbers should look familiar by now. Of course, the x and y coordinates are identical, for both points, as the points are on the line y=x.

Here is an infinite series which evaluates to the golden ratio:

ø=1+1/1·1-1/1·2+1/2·3-1/3·5+1/5·8-1/8·13+…

Notice the alternating signs. Each denominator is the product of two consecutive Fibonacci numbers. Where did I get this series? I deduced it from the fact that the ratio of consecutive Fibonacci numbers approaches the golden ratio, as the numbers get larger. The sequence of ratios of consecutive Fibonacci numbers is 1/1, 2/1, 3/2, 5/3, 8/5, 13/8,… We get the second element of this sequence with this series: 1+1/1·1. The third element is 1+1/1·1-1/1·2. The fourth is 1+1/1·1-1/1·2+1/2·3, and so on. This series was known long before I discovered it.

The golden ratio can be represented as the simplest continued fraction (as shown on the left). This fraction is 1+1/a where a is 1+1/a. You probably evaluate such a continued fraction from the right side, and work toward the left, except that it has infinitely many terms. We can get a sequence from this by starting at the left, and evaluating only part of the continued fraction: 1, 2/1, 3/2, 5/3, 8/5, 13/8… These are the ratios of consecutive Fibonacci numbers.

The golden ratio can be represented as a repeated square root, as in the diagram at the right. Evaluate this from right to left, too.

Magic of special Numbers.

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153

We begin this page of special numbers with one of my all-time favorites: 153. The reason I like it so much, I guess, is that it was the first one I learned about many years ago from a number category called narcissistic. Its unique quality, and what qualifies it as narcissistic, can be stated in these words: it is equal to the sum of the cubes of its digits. Observe:

153 = 13 + 53 + 33

= 1 + 125 + 27

= 153

Now, how’d you like that? Clever, huh! Well, there are only three other numbers that share this same quality. They are also three-place numbers. Can you find them?

Now, here are two more reasons why 153 is special.

If you add up all the whole numbers from 1 to 17, you get 153. This can be expressed mathematically in this way:

153 = 1 + 2 + 3 + 4 + … + 16 + 17

This is the same as saying that 153 is the 17th triangular number.

The other reason that 153 is interesting uses the concept of factorial. Recall that n! means “n(n-1)(n-2)…2×1.” So look at this expression:

1! + 2! + 3! + 4! + 5!

What do you think it equals? Right! 153.

extra curious fact about 153 can be shown…

sqrt(153 – SoD(153)) = 15 – 3.

where SoD(n) means the “Sum of the Digits of n”.

1634

Continuing with the theme of 153, with but one slight and rather important change, I will show why 1634 is rather special too. The change this time is that we will use the fourth powers of the digits. This gives this equation:

1634 = 14 + 64 + 34 + 44

= 1 + 1296 + 81 + 256

= 1634

Will numbers ever cease to amaze me? I hope not. There are two more cases like this one. Again we challenge you to find them.

Mathematicians call such numbers as those we’ve just discussed above as PDI’s (Perfect Digital Invariants). A PDI is a number that can be expressed as some combination of its digits and various operations, powers, or whatnot. The two numbers discussed so far used the cubes and 4th powers of the digits. Numbers that use higher powers also exist; some examples include:

54,748 and fifth powers,

548,834 and sixth powers,

1,741,725 and seventh powers.

Do you care to verify these cases, or do you trust me by now? 😉

1729

This special number, which involves cubes in a different way than described above, has a special story to go along with it. Some time ago in the early part of this century, a famous young mathematician from India was visiting the noted English mathematician Hardy in London. Ramanujan, the Indian, had become sick and Hardy had gone to see him. Hardy later wrote of the incident, saying: “I had ridden in a taxi-cab No. 1729, and remarked to my guest that the number seemed to me a rather dull one, and that I hoped it was not an unfavorable omen. ‘No’, he remarked, ‘it is a very interesting number; it is the smallest number expressable as the sum of two cubes in two different ways.'”

Here is what Ramanujan meant:

1729 = 13 + 123 = 93 + 103

= 1 + 1728 = 729 + 1000

By way of asking an easier question so that you may better appreciate this concept:

What is the smallest number that is expressable as

the sum of two distinct squares in two different ways?

Three final notes about 1729 that I think are equally interesting:

1st: If you factor it into primes, you get

1729 = 7 × 13 × 19

and those primes form an arithmetical sequence with a common difference of 6, making them “special” primes .

2nd: Then by combining 7 and 13 into one factor and rearranging the order, we have

1729 = 19 × 91

That’s sort of a palindromic arrangement of the digits, right?

3rd: If we note that the sum of the digits of 1729 is 19, then this is sufficient to declare that “1729 is a Niven number!”

Wow! 1729 is some fantastic number, for sure.

635,318,657

I can hear you saying on this one: “Now how can that monster of a number be special?” It’s really quite simple if I tell you to use Ramanujan’s idea just explained above. Believe it or not, but it is the smallest number that is express-able as the sum of two 4th powers in two different ways.

If I present this in symbols of algebra, it would look like this:

635,318,657 = A4 + B4 = C4 + D4

where A, B, C, and D are distinct whole numbers.

Of course, you’d like to know what those letters stand for, wouldn’t you? To give them out to you directly is not the way of “Trotter Math”; but I will give you some clues, okay?

Clue #1: One pair of numbers are consecutive, in the 130-140 range.

Clue #2: One number of the second pair is 59.

Now you can find the numbers.

An historical fact about this number is that it was discovered by the brilliant Swiss mathematician Leonhard Euler in the 18th century.

145

This is a very special number, even though at first glance there seems to be nothing unique about it. In fact, it’s another narcissistic number but in a way different than what was discussed earlier. To understand why, we need to know the meaning of a special mathematical symbol: “!” That’s it; the ordinary exclamation point.

But in math, it is called the factorial symbol. It is used in this way:

5! = 5 × 4 × 3 × 2 × 1 = 120.

We read that as “5-factorial”. So it means nothing more nor less than the product of the given whole number with all the smaller ones down to 1.

So what does this have to do with 145, you ask? Well, just observe this neat little statement:

145 = 1! + 4! + 5!

= 1 + 24 + 120

= 145

See? We are back to 145 again. Now that’s special, wouldn’t you agree?

24

In addition to being two dozen and 4! (4-factorial), the number 24 has another interesting characteristic. If you select any prime number, greater than 3, square it, then diminish that by 1, then 24 is always a divisor (factor) of the result.

Here is an example:

1. Choose the prime 17.

2. Its square is 289.

3. Subtracting 1 gives 288.

4. Then 288 divided by 24 gives exactly 12.

See? It works. I kid you not. Now you might wish to test a few more primes of your own choosing, just so you believe it more firmly. (Then later you can amaze your friends with your new found knowledge.)

A good activity here for students of Algebra is to PROVE that it always works. Just using many examples is not considered as proof in mathematics. (However, if many examples do work out, it is, I suppose, an indication that something “might” be true.)

142857

Probably you recognize this number; it’s the 6-digit period of the rational number 1/7.

1

— = 0.142857142857142857…

7

It is a favorite of math buffs because it has some unusual characteristics when multiplied by other numbers. But one that seems to be overlooked in most books is the fact that the missing digits are 0, 3, 6, and 9, which are the multiples of 3. (This is an observation that becomes more important when one investigates the periods of other fractions.)

But back to our multiplying idea… Note these products.

142857 × 2 = 285714

142857 × 3 = 428571

142857 × 4 = 571428

142857 × 5 = 714285

142857 × 6 = 857142

When you multiply by 7, you get a surprise. Do it!

Here is another trick to show how special the number is. Separate it into two halves, and then add.

142857

142 + 857 = 999

Next split it into thirds before adding.

142857

14 + 28 + 57 = 99

Finally, let’s square the number before splitting and adding.

1428572 = 20,408,122,449

What did you find, my friend?

92

The year 1992 has been gone for some time now, but the short form of the year ’92 is a number that produces a lot of 9’s (remember what happened to the previous entry just above?). (I found this idea in a book titled “Every Number Is Special” by Boyd Henry.) It is done in this unusual manner:

Multiply 92 by 8, then that product by 8, then that product by 8, etc. List the products one under the other, shifting the digits two places to the right as shown below. Continue indefinitely. Add. The sum converges to a string of 9s.

92

736

5888

47104

376832

3014656

24117248

9999999…

Not to let the full year number be left out, WTM takes pleasure in pointing out that 1992 is sorta nice by itself. Look:

1992 = 8 × 3 × 83

WTM now hopes that you will take a new interest in the world of numbers and try to uncover their inner personalities. They’re a little like we people, each unique in some way, yet at the same time, sharing common properties with others.

Thanks to http://www.trottermath.net

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