Perfect Number

သခ်ၤာပညာရွင္ ေပါလ္ အဲဒစ္ရွ္ အပါအဝင္ သခ်ၤာသမား အမ်ားစု အတြက္ အလွတရား ဟူသည္ သာမာန္ သတ္မွတ္ထားေသာ အလွတရား ဟူသည့္ အဓိပၸာယ္ႏွင့္ မတူညီေပ။ ၎တို႔အတြက္ အလွတရား သည္ ကိန္းဂဏန္း ဆိုင္ရာ စဥ္းစားေတြးေခၚမႈ အပိုင္း၊ အသိဥာဏ္ ကြန္႔ျမဴးရမႈ ကိုသာ ဆိုလိုသည္။ တစ္ခ်ိန္က အဲဒစ္ရွ္ ေျပာၾကားခဲ့ဖူးသည့္ စကား တစ္ခြန္းရွိသည္။ “ေဘသိုဗင္ရဲ႕ ဆင္ဖိုနီ နံပါတ္ ၉ ကို လွတယ္လို႔ ေျပာ ၾကသလိုမ်ိဳး ပါပဲ။ သူ႔ေတးသြားက ဘယ္လိုလွတာလဲ လို႔ ေမးရင္ ျပန္ေျဖခ်င္မွ ေျဖတတ္မယ္။ အဲ့လိုမ်ိဳးပဲ က်ဳပ္တို႔ အတြက္ ကိန္းဂဏန္း ေတြဟာ လွတယ္။ ဒါကို မလွပါဘူးဗ်ာ လို႔ ေျပာမယ္ဆိုလို႔ ရွိရင္လည္း က်ေနာ္တုိ႔ ဘာမွ ျပန္ေျပာစရာလိုပါဘူး” တဲ့။

ဟိုးအရင္ တစ္ခ်ိန္။ ခရစ္မေပၚမီ ေျခာက္ရာစု ခန္႔က ျဖစ္မည္။ ဆမၼိဳျမိဳ႕က ပိုက္သာဂိုရ ဆိုသည့္ ပုဂၢိဳလ္မွာလည္း ဤကဲ့သို႔ ခံစားခ်က္ ရွိခဲ့ဖူးသည္။ ကိန္းဂဏန္း မ်ားသည္ တြက္ခ်က္ ၊ ေရတြက္ရာ တြင္သာ အသံုးျပဳသည္ မဟုတ္ဘဲ ျမင့္ျမတ္ ေသာ၊ ႏွစ္လိုဖြယ္ ေကာင္းေသာ ၊ ေကာင္းျမတ္ျခင္းကို ေဆာင္ၾကဥ္းတတ္သည့္ ( သို႔မဟုတ္ ) ဆိုးက်ိဳးကို ဖန္တီးတတ္သည့္ သေဘာ ရွိသည္ဟု ယူဆခဲ့သူ ျဖစ္သည္။ ထို႔ေၾကာင့္ပင္ ကိန္းဂဏန္းမ်ား၏ အတြင္းသေဘာ ထဲမွ ဘာသာေရး အဖြဲ႕အစည္း တစ္ရပ္ ကိုပင္ ဖြဲ႔စည္းခဲ့ေသးသည္။

ပိုက္သာဂိုရသည္ ကိန္းဂဏန္းမ်ား ႏွင့္ ပတ္သက္၍ ခြဲျခမ္းစိတ္ျဖာ ခဲ့သူ၊ အေစာပိုင္း သခ်ၤာပညာရွင္ဟုလည္း ေခၚဆိုႏိုင္ပါမည္။ တကယ္တမ္း တြင္ သခ်ၤာ၏ အေျခခံ အေပါင္း အႏုတ္ မ်ားအားျဖင့္ သာ ႏြယ္ယွက္၍ စဥ္းစား သြားၾကရေလရာ ယင္းတို႔မွ တစ္ဆင့္ ဂဏန္းစဥ္တို႔ ၏ ထူးျခား ခ်က္မ်ားကို ေတြ႔ရွိလာၾကရသည္။

Perfect Number ကို ျပည့္စံုကိန္းဟု ဘာသာျပန္ဆိုသည္ကို တစ္ခါက ၾကားဖူးသည္။ ျပီးျပည့္ဝေသာ ကိန္းတစ္ခုဟု ဆိုၾကသည္ ျဖစ္ေသာေၾကာင့္ သိဒၶိဝင္ ကိန္းဟု ဆိုရမည္လား မေျပာတတ္။ သိဒၶိဝင္ကိန္း ဟု ဆိုလ်ွင္ Prime Number ( သုဒၶကိန္း ) ႏွင့္ ေရာေထြး ႏိုင္ေသာေၾကာင့္ ျပည့္စံုကိန္း ဟုသာ ဆက္လက္ ေခၚဆိုပါမည္။

ေပါလ္အဲဒစ္ရွ္

ေက်ာင္းတက္ ခဲ့တာက ရူပေဗဒ ဆိုေပမယ့္ ဝါသနာ ပါတာက သူန႔ဲ ႏွီးႏြယ္ေနတဲ့ သခ်ာၤ သက္သက္ရယ္။

Paul Erdos

သခ်ာၤတစ္ခုတည္း ဆိုတဲ့အတိုင္း ေက်ာင္းတက္ေတာ့ အရူးအမူး အားသန္တာ အဲ့ဒီ သခ်ာၤသက္သက္။ အြန္လိုင္းမ်က္နွာစာေပၚ ေရာက္ေတာ့လည္း သခ်ၤာေတြသာ အဖတ္မ်ားတယ္။ သခ်ၤာအေၾကာင္း ေတာ္ေတာ္ မ်ားမ်ား ( အထူးသျဖင့္ သီဝရီပိုင္း၊ အီေကးရွင္း ပိုင္းေတြကို ) ဘေလာ့ေပၚ တင္ေရးမယ္ ဆိုျပန္ေတာ့ တင္လို႔ မရတဲ့ အခက္အခဲရွိပါတယ္။ သခ်ၤာေဖာင့္ အခက္အခဲေတြပါ ။ ကိုယ္လဲ ကြန္ပ်ဴတာ ဆိုတာ နကန္းတလံုး မသိလို႔ ျဖစ္ပါလိမ့္မယ္။ ပရိုဂရမ္းမင္း ဆို္လား ဂ်ာဗား ဆိုလား ကုတ္ဒင္ဆိုလား မသိလဲမသိ၊ ေရးလည္း မေရးတတ္ေတာ့ – မတင္တတ္တာနဲ႔ မတင္ျဖစ္ခဲ့တာ ၾကာျပီ။ စိတ္ကူးက ရွိေနဆဲပဲ။

အခုေနာက္ပိုင္းေတာ့ အလ်ဥ္းသင့္ ရင္ သင့္သလို သခ်ၤာ၊ ရူပေဗဒ ဆိုင္ရာေလး နည္းနည္းပါးပါး ( မေတာက္တေခါက္ ) ေရးၾကည့္ပါမယ္ ။ ပညာရပ္ဆိုင္ရာ ..အဲ..အဲ့လိုေျပာရင္ နဲနဲ ဂန္က်ယ္တယ္။ ဘာသာရပ္ဆိုင္ရာ Terms ေလးမ်ား ပါခဲ့ရင္ သည္းခံ ၾကေစခ်င္တယ္။ ေအာက္မွာ ေဖာ္ျပထားတဲ့ သခ်ၤာပညာရွင္ ေပါလ္အဲဒစ္ရွ္ အတၳဳပၸတၱိဟာ ဒီစိတ္ကူးအတြက္ စဦး မိတ္ဆက္သေဘာ သက္သက္ပါ။

…………………….
Paul Erdos တဲ့။ ေပါလ္အဲဒစ္ရွ္ ( Paul Air-dish ) လို႔ အသံထြက္တယ္။ ဂ်ဴးလူမ်ိဳး ၊ ဟန္ေဂရီယန္ ႏိုင္ငံသား ၊ သခ်ၤာပညာရွင္။ ရွိခဲ့သမ်ွ သခ်ၤာပညာရွင္ ေတာ္ေတာ္မ်ားမ်ား ထက္ သခ်ာၤပုစၦာအမ်ားစုကို ေတြးေခၚ တြက္ခ်က္ခဲ့သူ လို႔ အဆိုရွိတယ္။ အျခား သခ်ၤာပညာရွင္ေတြနဲ႔ မတူတာက သခ်ၤာ သီဝရီ အသစ္ေတြ ၾကံဆ ေဖာ္ထုတ္တာေတြ မရွိဘူး။ နည္းတယ္။ သခ်ၤာပုစၦာ ျပႆနာေတြ ေျဖရွင္းဖို႔နဲ႔ အဲ့ပုစၦာျပႆနာေတြ ေဖာ္ထုတ္ဖို႔ တင္ျပဖို႔ လုပ္ခဲ့တာ မ်ားပါတယ္။ အိမ္ေထာင္မရွိ၊ ကေလးမရွိ။ အျခားအသက္ ေမြးဝမ္းေက်ာင္း မရွိ။ အျခား ဘယ္ဟာမွ ဝါသနာ မပါတဲ့သူ။ ဘဝတစ္ေလ်ာက္လံုး သခ်ၤာနဲ႔သာ ေပ်ာ္ေမြ႕ခဲ့တဲ့သူ လို႔ ေျပာပါတယ္။

အဲဒစ္ရွ္ကို ၁၉၁၃ ၊ မတ္လ ၂၆ ရက္ေန႔က ဟန္ေဂရီႏိုင္ငံ၊ ေဘာ္ဒါပက္ ျမိဳ႕ မွာ ေမြးတယ္။ သူ႔မိဘ ႏွစ္ပါးစလံုးက အထက္တန္းေက်ာင္း သခ်ၤာဘာသာရပ္ စာျပ ဆရာဆရာမေတြ ျဖစ္ျပီး သူ႔သားျဖစ္တဲ့ အဲဒစ္ရွ္ကို ငယ္စဥ္ကတည္းက ေကာင္းေကာင္း လမ္းေၾကာင္းခဲ့ တယ္။ အဲဒစ္ရွ္ကလည္း ငယ္စဥ္ကတည္းက အစြမ္းအစ ရွိတဲ့သူ လို႔ ေျပာပါတယ္။ ေလးႏွစ္သား အရြယ္က သူ႔အေမကို ေျပာဖူးတယ္။ တကယ္လို႔ ေဒၚလာတစ္ရာထဲက ေဒၚလာ ၂၅၀ ႏႈတ္ရင္ သုညေအာက္မွာ ေဒၚလာ ၁၅၀ က်န္လိမ့္မယ္ တဲ့။ သခ်ၤာဆိုင္ရာ အႏႈတ္ကိန္း အေၾကာင္းကို ေလးႏွစ္သား ကတည္းက ေျပာႏိုင္ခဲ့တယ္ ။ အဲ့လို ျဖစ္ရပ္မ်ိဳး ဆိုတာ ျမန္မာျပည္မွာေတာ့ မလြယ္လွတဲ့ ဒ႑ာရီ ပံုျပင္လိုပါပဲ။ ( သခ်ၤာနဲ႔ ဆိုင္တာ ေျပာတာပါ။ အျခားဟာေတြေတာ့ ရွိတယ္ ၾကားတယ္။ ဥပမာ

The Concept “Infinity”

Does one concentrate purely on the mathematical aspects of the topic or does one consider the philosophical and even religious aspects?
The most famous among those is the problem of Infinity.The concept ” Infinity ” cause the great difficulties for many thinkers over centuries. If u was a professional thinker,u might thought that at least . Because it is correspond not only to the reason of human thought ,but also even religious aspects.

Different Way of Using the Mind

Mathematics has something to teach us, all of us, whether or not we like mathematics or use it very much. This lesson has to do with thinking, the way we use our minds to draw conclusions about the world around us. When most people think about mathematics they think about the logic of mathematics. They think that mathematics is characterized by a certain mode of using the mind, a mode I shall henceforth refer to as “algorithmic.” By this I mean a step-by-step, rule-based procedure for going from old truths to new ones through a process of logical reasoning. But is this really the only way that we think in mathematics? Is this the way that new mathematical truths are brought into being? Most people are not aware that there are, in fact, other ways of using the mind that are at play in mathematics.

The Parallels between Modern Physics and Eastern Mysticism

I am so interested in not only Physics ,but also Eastern Philosophy,especially so-called Mysticism. The following paragraphs are copied from The Tao of Physics;it explained the current Modern Physics in the view of Eastern Religions;Taoism,Zen Buddhism and India Mysticism. I wanna translate it to Burma Language, but don’t have enough time to do .Hence,Sorry for now on, merely describing one of its chapters for my friends with so keen mindedness in these subjects just like me.

Thank you,
Zayya

Knowing and Seeing;The Parallels between Modern Physics and Eastern Mysticism

From the unreal lead me to the real!
From darkness lead me to light!
From death lead me to immortality!
Brihad-aranyaka Upanishad

Before studying the parallels between modern physics and Eastern mysticism, we have to deal with the question of how we can make any comparison at all between an exact science, expressed in the highly sophisticated language of modern mathematics, and spiritual disciplines which are mainly based on meditation and insist on the fact that their insights cannot be communicated verbally. What we want to compare are the statements made by scientists and Eastern mystics about their knowledge of the world.

To establish the proper framework for this comparison, we must firstly ask ourselves what kind of ‘knowledge’ we are talking about; does the Buddhist monk from Angkor Wat or Kyoto mean the same thing by ‘knowledge’ as the physicist from Oxford or Berkeley? Secondly, what kind of statements are we going to compare? What are we going to select from the experimental data, equations and theories on the one side, and from the religious scriptures, ancient myths, or philosophical treatises on the other?

This chapter is intended to clarify these two points: the nature of the knowledge involved and the language in which this knowledge is expressed. Throughout history, it has been recognized that the human mind is capable of two kinds of knowledge, or two modes of consciousness, which have often been termed the rational and the intuitive, and have traditionally been associated with science and religion, respectively. In the West, the intuitive, religious type of knowledge is often devalued in favour of rational, scientific knowledge, whereas the traditional Eastern attitude is in general just the opposite. The following statements about knowledge by two great minds of the West and the East typify the two positions. Socrates in Greece made the famous statement ‘I know that I know nothing’, and Lao Tzu in China said, ‘Not knowing that one knows is best.’ In the East, the values attributed to the two kinds of knowledge are often already apparent from the names given to them. The Upanishads, for example, speak about a higher and a lower knowledge and associate the lower knowledge with various sciences, the higher with religious awareness. Buddhists talk about ‘relative’ and ‘absolute’ knowledge, or about conditional truth’ and ‘transcendental truth’. Chinese philosophy, on the other hand, has always emphasized the complementary nature of the intuitive and the rational and has represented them by the archetypal pair yin and yang which form the basis of Chinese thought.

Some Mathematics Notes and PDF files

Dear Lovers of Maths,My Friends,

As u know, I love Mathematics. Whatever u think what is life,only the life being with Maths is so peaceful for me. For just now, then, I would like to introduce you to a blog written about Mathematics. The blogger of it ,Daniel Murfet was a postdoc at the Hausdorff center of Mathematics.

He also love his subject,too,I think. You can see about him here. And let’s go to his job,I mean ; to his Blog. Thank you all !  😉

With love,
Zayya

Memoir to Ramanujan

Srinivasa Ramanujan

အခု ဒီဇင္ဘာ ၂၂ – Srinivasa Ramanujan, သခ်ာၤပညာဆိုင္ရာ ကမၻာ့ ဥာဏ္ၾကီးရွင္တစ္ပါးရဲ႕ ့ Birthday ျဖစ္ပါတယ္။ ဆရာကြယ္လြန္သြားခဲ႔တာ ႏွစ္ကိုးဆယ္ ရွိပါျပီ။ ဒါေပမယ့္ သူ ခ်ထား ေပးခဲ႔တဲ႔ သခ်ာၤသီဝရီေတြနဲ႔ စဥ္းစားစရာ ပုစၦာေတြက ကေန႔ထက္တိုင္ ဆက္လက္ အေျဖရွာေနရဆဲ ၊ ေလ့လာ သံုးသပ္ ေနရဆဲပါ။ ဆရာ ေဖာ္ထုတ္သြားခဲ႔တဲ႔ Number Theory ( အထူးသျဖင့္ – Ramanujan theta function ၊ Landau–Ramanujan constant ) ကို က်ေနာ္ တို႔ သခ်ၤာသမားမ်ား ေလ့လာေနရဆဲ ၊ ၾကိဳးစားေနရဆဲ ျဖစ္ပါတယ္။ ဒီညမွာ ဆရာ့အတြက္ အမွတ္တရ အျဖစ္ ဒီစာကို ေရးပါတယ္။

A.M 4:30 / 23.12.2010

………………………………..

“The mathematician’s patterns, like the painter’s or the poet’s, must be beautiful. The ideas, like the colours or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in this world for ugly mathematics.”
သခ်ာၤ၏ စံပံုစံတို႔ မည္သည္ ပန္းခ်ီဆရာ၏ စုတ္ခ်က္ကဲ႔သို႔ လည္းေကာင္း၊ ကဗ်ာဆရာ၏ စကားလံုးမ်ား ကဲ႔သို႔လည္းေကာင္း လွပစြာ သီဖြဲ႕ထားၾကသည္ခ်ည္း ျဖစ္သည္။ အေရာင္မ်ား ၊ စကားလံုးမ်ားကဲ႔သို႔ပင္ ေတြးဆခ်က္ စိတ္ကူးမ်ားသည္

My Master, the one I want to be.

“He loves only numbers”

My Master

As soon as I heard this in Dagon University, I wanna know who he was , so keenly. At that time, – to the present – my favorite hobby is Think-Hard. So , I studied  Logic, Physics, Philosophy, Psychology, especially Mathematics.I tried to study more than usual as in this country, Myanmar , i.e., within my ability to grasp the core of them according to my life-style. I took no place the social affairs ,other business and so- called ethical behaviour in my consciousness , but ……….

When I started mathematics – course , I felt I liked this subject more than others , because the pure thought can display and end in that field itself, that I wanted to be. And so forth, I saw my way that flows on my life.

In the pile of Mathematics Books, I met many great mathematician , professor  , reader and scholar of Mathematics. Among them, I admire the one,only one. He  was  Paul Erdos  , the lover of number only.

The following paragraphs  might be the last part of his life.

Dr. Paul Erdos, a legendary mathematician who was so devoted to his subject that he lived as a mathematical pilgrim with no home and no job, died Friday (20-Sep-1996 )  in Warsaw, Poland. He was 83.