Wednesday, February 22, 2012

What is Maths?

Readers,

You should spend some time to watch this video, cheer! =)


History of Complex Numbers a.k.a istory of Imaginary Numbers(i)

            
           i is as amazing number.  It is the only imaginary number.  However, when you square it, it becomes real.  Of course, it wasn’t instantly created. It took several centuries to convince certain mathematicians to accept this new number. Eventually, though, a section of numbers called “imaginary” was created (which also includes complex numbers, which are numbers that have both a real and imaginary part), and people now used in everyday math.

            was created due to the fact that people simply needed it.  At first, solving problems such as “-39” and “x2+1=0” were thought to be impossible.  However, mathematicians soon came up with the idea that such a number to solve these equations could be created.  Today, the number is√-1, more commonly known as i.  It’s a good thing that scientists, mathematicians who didn’t want a new numbers created, and other non-believers finally allowed (and complex numbers) in the number system.  Today, is very useful to the world. Engineers use it to study stresses on beams and to study resonance.  Complex numbers help us study the flow of fluid around objects, such as water around a pipe.  They are used in electric circuits, and help in transmitting radio waves.  So, if it weren’t fori, we might not be able to talk on cell phones, or listen to the radio!  Imaginary numbers also help in studying infinite series.  Lastly, every polynomial equation has a solution if complex numbers are used. Clearly, it is good that was created.

            The very first mention of people trying to use imaginary numbers dates all the way back to the 1stcentury.  In 50 A.D., Heron of Alexandria studied the volume of an impossible section of a pyramid. What made it impossible was when he had to take√81-114.  However, he deemed this impossible, and soon gave up.  For a very long time, no one tried to manipulate imaginary numbers.  Although, it wasn’t for a lack of trying.  Once negative numbers were “invented”, mathematicians tried to find a number that, when squared, could equal a negative one.  Not finding an answer, they gave up.  In the 1500’s, some speculation about square roots of negative numbers was brought back.  Formulas for solving 3rd and 4th degree polynomial equations were discovered, and people realized that some work with square roots of negative numbers would occasionally be required.  Naturally, they didn’t want to work with that, so they usually didn’t.  Finally, in 1545, the first major work with imaginary numbers occurred.

            In 1545, Girolamo Cardano wrote a book titled Ars Magna.  He solved the equation
x(10-x)=40, finding the answer to be 5 plus or minus√-15.  Although he found that this was the answer, he greatly disliked imaginary numbers.  He said that work with them would be, “as subtle as it would be useless”, and referred to working with them as “mental torture.”  For a while, most people agreed with him.  Later, in 1637, Rene Descartes came up with the standard form for complex numbers, which is a+bi.  However, he didn’t like complex numbers either.  He assumed that if they were involved, you couldn’t solve the problem.  Lastly, he came up with the term “imaginary”, although he meant it to be negative.  Issac Newton agreed with Descartes, and Albert Girad even went as far as to call these, “solutions impossible”.  Although these people didn’t enjoy the thought of imaginary numbers, they couldn’t stop other mathematicians from believing that might exist.

            Rafael Bombelli was a firm believer in complex numbers.  He helped introduce them, but since he didn’t really know what to do with them, he mostly wasn’t believed.  He did understand that itimes should equal -1, and that –times should equal one.  Most people did not believe this fact either.  Lastly, he did have what people called a “wild idea”- the idea that you could use imaginary numbers to get the real answers.  Today, this is known as conjugation.  Although Bombelli himself did not have much of an impact at the time, he helped lead the way for imaginary numbers.

Over decades, many people believed that complex numbers existed, and set out to make them understood and accepted.  One of the ways they wanted to make them accepted was to be able to plot them of a graph.  In this case, the X-axis is would be real numbers, and the Y-axis would be imaginary numbers.  If the number were purely imaginary (like 2i), it would just be on the Y-axis.  If the number was purely real, it would just be on the X-axis.  The first person who considered this kind of graph was John Wallis.  In 1685, he said that a complex number was just a point on a plane, but he was ignored.  More than a century later, Caspar Wessel published a paper showing how to represent complex numbers in a plane, but was also ignored. In 1777, Euler made the symbol stand for -1, which made it a little easier to understand.  In 1804, Abbe Buee thought about John Wallis’s idea about graphing imaginary numbers, and agreed with him.  In 1806, Jean Robert Argand wrote how to plot them in a plane, and today the plane is called the Argand diagram.  In 1831, Carl Friedrich Gauss made Argand’s idea popular, and introduced it to many people.  In addition, Gauss took Descartes’ a+binotation, and called this a complex number.  It took all these people working together to get the world, for the most part, to accept complex numbers.

Mathematicians kept working to make sure that imaginary and complex numbers were understood.  In 1833, William Rowan Hamilton expressed complex numbers as pairs of real numbers (such as 4+3i being expresses as (4,3)), making them less confusing and even more believable.  After this, many people, such as Karl Weierstrass, Hermann Schwarz, Richard Dedekind, Otto Holder, Henri Poincare, Eduard Study, and Sir Frank Macfarlane Burnet all studied the general theory of complex numbers.  Augustin Louis Cauchy and Niels Henrik Able made a general theory about complex numbers accepted.  August Mobius made many notes about how to apply complex numbers in geometry.  All of these mathematicians helped the world better understand complex numbers, and how they are useful.

Clearly, complex numbers are amazing. They have many uses, more than we realize.  They have a fascinating history, full of some mathematicians not believing in them and others desperately trying to prove their existence.  i is also fascinating, being the only imaginary number.  Many mathematicians brought together as much proof as they could that imaginary numbers should exist, and we have them to thank today that we can use iwhenever we please, without being questioned about it.

A very good writing on the history of complex number!! Well done to the person who gathered all the pieces and come out with this writing, brilliant! 

Wednesday, February 08, 2012

亂寫

事實是殘酷的,這是人類改變不了的事~ 我不後悔認識你,因為你我看清很多東西~ 我明白了許多,甚至可以接受及體諒一些事情~ 原來是那麼的苦,那麼的無奈,那麼的難~ 那麼的。。。

心中的仇與恨,放下了嗎?不完全,但已經放下很多~ 原來,自己之前的堅持不完全是對的~ 對於錯真的那麼重要嗎?現在的我會說,平安開心較為重要~ 以前的事情,不再重要了~已經過去了,開心或不開心的已經成了我人生的一部分~ 

人之所以會煩,是因為他在意那人事物,不然的話會有那麼煩嗎?我不知道你是抱著什麼心態來打訊息的,但老實說我不喜歡這種感覺~ 所以我選擇面對你,也趁此機會說個明白,好讓大家有個明白,不必在猜疑~ 這樣我想是最好的了~

學業上的事,就很多人喜歡過問~ 幾時畢業啊?總平均拿幾分啊?要繼續念下去哦?我不愛回答這些人,因為沒有這個必要~ 我的事情,我只需向自己交代就夠了~ 就算真的要交代,也只有我父母有這個權利~ 其他的人,管你們屁事哦!?管好你們自個兒的事先吧~哼!!