Math Books

Amazon Guest Review: Stephen Hawking. Published in 1988, Stephen Hawking’s A Brief History of Time was a remarkable bestseller. It is a detailed exploration of physics that occupies the London Sunday Times Bestseller List for 237 consecutive weeks. A Briefer History in Time 1995, The Universe In a Nutshell and God Created The Integers: The Mathematics Breakthroughs That Changed History were some of the later successes. Stephen Hawking is the Lucasian Professor of Mathematical Sciences at the University of Cambridge. Leonard Mlodinow’s The Drunkards Walk provides a clear and easy-to-read guide on how randomness and mathematical laws affect our lives. He shows us how the signs of chance can be seen in the events around us. Understanding randomness has made profound changes in our perceptions of our world and the universe. This important branch of mathematics has been well explained by Leonard. Stephen Hawking –This text refers an alternate kindle_edition version. As a teenager, I can still remember looking through the Eyepiece of Randomness and seeing the yellow flames of the Sabbath candles dance randomly above the white paraffin tubes that powered them. Although I was too young for candlelight romance, I still found it magical because of the flickering images that the fire created. They changed and morphed, grew or waned without any apparent plan. I was certain that there had to be a rhyme and reason behind the flame. Scientists could easily predict and explain it with their mathematical equations. My father said to me, “Life isn’t like that.” Sometimes things don’t go as planned. He shared with me the story of when he stole bread from the bakery in Buchenwald, a Nazi concentration camp. The Gestapo was called in to gather all suspects and bring them together by the baker. “Who stole the bread?” The baker asked. He asked everyone to answer and told the guards that they would shoot each suspect one at a time until they either found someone guilty or all were dead. My father offered to help the others. He didn’t try to portray himself as heroic, but he told me that he did this because he knew he would be shot. My father was not killed by the baker, but he gave him a job as his assistant. My father called it “a chance event.” It had nothing to do you. But if it had happened differently, you wouldn’t have been born.” Then it occurred to me that Hitler is the reason I am alive. My father had lost his wife and two children to the Nazis, ending his previous life. My father would not have fled to New York without the war. He would have never met my mother, a refugee from Europe, and would not have given birth to me and my brothers. My father never spoke about the war. Although I didn’t know it at the time, years later, it became clear to me that he was sharing his experiences not because he wanted me learn more about life, but because he wanted me too. Although war is an extreme situation, the role of chance is not dependent on it. Our fate is determined by the outline of our lives. Life is difficult to predict and even harder to understand. As you might see Madonna and I in a Rorschach Blot, data from business, law, medicine or sports can also be read in many different ways. Interpreting the role of chance in an incident is different from interpreting a Rorschachblot. There are both right and wrong ways. When making decisions and making assessments in uncertain situations, we often use intuitive processes. These intuitive processes were a great evolutionary advantage, as we had to determine if a saber-toothed cat was smiling because it was happy and fat or because it was hungry and wanted us as its next meal. Modern life has changed the balance and these intuitive processes have their downsides. If we continue to use our old ways of thinking to manage today’s tigers, it is possible to make poor or incongruous decisions. This conclusion is not surprising to anyone who studies how the brain processes uncertainty. Many studies have shown a close link between brain parts that assess chance situations and those that deal with human emotion, which is often our primary source of irrationality. Functional magnetic resonance imaging shows, for instance, that parts of the brain responsible for motivational and emotional processes, such as reward and risk, are important in assessing reward and risk. Images also show that the amygdala is responsible for whi
The emotional state of our emotions, particularly fear, is linked to ch. This is activated when we make uncertain decisions. Evolution factors, brain structure and personal experience, as well as emotion, are all involved in how people interpret situations that involve chance. The human response to uncertainty can be so complex that different brain structures may come up with different conclusions. They might even fight for the right to dominate. The “logical” left side of your brain will try to find patterns if your face grows to five times its usual size after eating shrimp three times out of four. Your brain’s “intuitive” right side will say “avoid shrimp.” Researchers found that this is what they did in less painful experiments. Probability guessing is the name of this game. Instead of playing with histamine and shrimp, the subjects are shown a series cards or lights. These can have either green or red colors. The colors are ordered so that they will appear with different probabilities, but not in a particular pattern. For example, red might appear twice as often as green in a sequence like red-red-green-red-green-red-red-green-green-red-red-red, and so on. After watching the sequence for a while, the subject must predict whether each member will be red or not. There are two main strategies to the game. The first is to guess the color you see more often. This is the preferred route by rats and other non-human animals. This strategy will guarantee you some success, but it also means that you can’t expect to do better. If green is present 75 percent of time, and you choose to guess green 75 percent of time, then you will be right 75 percent. Another strategy is to match your green and red guesses with the amount of green or red you have seen in the past. This strategy allows you to predict right every time if the reds and greens are in a pattern. If the colors are randomly appearing, the second strategy is better. The second strategy is only 6 times as likely to be correct in the case of green appearing randomly 75 percent of time. The pattern is something humans try to guess, and we often let ourselves be outperformed in this process by rats. There are some people who have a type of brain impairment called a split brain. This prevents the right and the left hemispheres from communicating with one another. This is an experiment on the right brain if the probability experiment is done on these patients so that they can only see the card or colored light with their left eyes and use only their left hand for their predictions. If the experiment involves only the right eye and righthand, it’s an experiment on their left brain. Researchers found that the right hemisphere chose the most frequent color, while the left hemisphere tried to predict the pattern. It is rare to make wise decisions and make good choices when faced with uncertainty. It can be learned, just like any other skill. In the pages to follow, I will discuss the role of chance within the world. The ideas that have been developed over time will help me understand this role and the factors that can lead us astray. Bertrand Russell, a British mathematician and philosopher, wrote that we all start with “naive reality,” which is the belief that things are as they appear. We believe that grass is green and stones are hard. Snow is cold. Physics tells us that the greenness and hardness and coldness in snow are not what we have experienced. We will see life through the eyes of randomness. Many of the events in our lives are not what they seem, but something quite different. A scientist named Daniel Kahneman was awarded the Nobel Prize in Economics in 2002 by the Nobel committee. Economists are responsible for many things, including explaining why teachers get so little and why football teams are so valuable. They also help to limit the size of hog farm farms. A hog excretes three-five times as much as a person, so a farm that has thousands of hogs often produces more waste than neighboring towns. Despite all the outstanding research by economists, Kahneman won the 2002 Nobel Prize because he is not an economist. Kahneman is a psychologist. He studied and clarified misperceptions about randomness for many decades with Amos Tversky.
Many of the common errors I will discuss in this book are fueled by them. Understanding the role of randomness is difficult because, although basic principles of randomness are derived from everyday logic many of the consequences that result from these principles are counterintuitive. Tversky and Kahneman’s studies were triggered by an unrelated event. In the middle of the 1960s, Kahneman was a junior psychology professor at Hebrew University and agreed to lecture to a group Israeli flight instructors about the conventional wisdom of behavior modification. Kahneman made the important point that rewarding positive behavior works and punishing errors does not. Kahneman was interrupted by one of his students, who offered an opinion that would guide his research for many decades. The flight instructor stated, “I have often praised people for their beautifully executed maneuvers, but the next time they do them worse,” “And I have screamed at people who do poorly executed maneuvers. Most of the time, they improve. You can’t tell me that punishment works and reward works. It is not my experience.” Other flight instructors also agreed. Kahneman believed that the experiences of flight instructors were true. Kahneman, on the other hand, believed animal experiments that showed reward works better than punishment. This paradox was what Kahneman pondered. Then it hit him: although the screaming preceded the improvement it didn’t cause it, contrary to what appears. How could that be? Regression toward the mean is a phenomenon that explains this phenomenon. This means that in any sequence of random events, an extraordinary event is more likely to be followed by an ordinary one. This is how it works: All the student pilots had some personal ability to fly fighter aircrafts. It took a lot of practice and many factors to raise their skill level. The result was not immediately noticeable. It was mostly luck that a pilot would perform exceptionally well or poorly. If a pilot makes a landing that is exceptional, it’s likely that he will perform better than his normal performance the next day. Even if his instructor had given him praise, it would seem that the praise was not of any benefit. If a pilot makes a terrible landing, such as running the plane off the runway and into the vats of corn chowder at the base cafeteria, then the chances are that he will perform better the next day. It would seem that his critics were helpful if his instructor was a fan of calling out “you clumsy monkey” when a student did poorly. This would lead to an obvious pattern: student performs well, praise doesn’t do any good; student performs poorly; instructor compares student with a lower primate at high volume and student improves. These experiences led Kahneman’s instructors to conclude that screaming could be a powerful learning tool. It made no difference. Kahneman was prompted to think about this error of intuition. Kahneman wondered if such misperceptions are universal. Are we like the flight instructors who believe harsh criticism can improve our children’s behavior and our employees’ performance? Are there other mistakes we make when confronted with uncertainty? Kahneman understood that humans use certain strategies to reduce the complexity and difficulty of judgments. He also believed that intuition about probabilities is an important part of that process. Do you feel sick after eating the delicious-looking street vendor’s seviche tostada? Although you don’t remember all the similar food stands you’ve visited, you can count how many times you have drunk Pepto-Bismol the next night and make a numerical estimation. Your intuition does the rest. Research in the 1950s and 1960s showed that people’s intuitions about randomness were not working in these situations. Kahneman wondered how widespread this misinterpretation of uncertainty was. What are the implications of this misunderstanding for human decision-making? Amos Tversky was a junior professor who Kahneman invited to give a lecture at one his seminars. Kahneman later shared his ideas with Tversky at lunch. Tversky and Kahneman discovered that even in highly-skilled subjects, people’s intuition and beliefs often failed them when dealing with random processes. Let’s say that four publishers rejected your novel about war, love, and global warming. Your gut instinct and the bad feeling in your stomach are mig
It is possible to conclude that all the publishing experts have rejected your manuscript. Is your intuition right? Is your novel unsellable? It is not a matter of whether a few coins come up heads. We know this from personal experience. It could be that publishing success can be so unpredictable that many publishers might miss the point and send letters saying thanks but not thanks, even if the novel is on the bestseller list. Publishers rejected a 1950s book. They responded with comments like “very dull,” “a dreary report of typical family bickering and petty annoyances, and adolescent emotion,” and “even though the work came to light five years ago when the topic [World War II] was appropriate, I don’t think there would have been an opportunity for it.” The Diary of a Young Girl, Anne Frank’s book, has been a bestseller, selling 30 million copies. It is now one of the most popular books in history. Sylvia Plath was also rejected because she “never had enough genuine talent to warrant our attention,” George Orwell for Animal Farm because it is “impossible to sell animal stories in America,” and Isaac Bashevis Singer because it’s “Poland and the rich Jews” again. Tony Hillerman, his agent, dumped him before he made it big. He advised him to “get rid all that Indian stuff.” This text refers to an alternative kindle_edition edition.
Leonard Mlodinow’s The Drunkard’s Walk – How Randomness Rules Your Lives is an entertaining and eye-opening guide to understanding the random world. Everything we do is surrounded by uncertainty and randomness. Why is it that we are so poor at understanding them? These tools can also be used to understand the random paths of molecules. They can be applied to randomness that affects many aspects of our daily lives. The Drunkard’s Walk exposes the psychological illusions that keep us from understanding everything, stock-picking to wine tasting – you must read it or risk being another victim of chance. Stephen Hawking, author of A Brief History of Time, describes this book as “a wonderfully readable guide to the mathematical laws of randomness that affect our lives.”

गणित के खेल –
कुछ मान्यताओं के अनुसार गणित एक ऐसा विषय माना जाता है जिसमें मनोरंजन की सम्भावना नहीं खोजी जा सकती। वास्तव में यह बहुत से लोगों के लिए एक मुश्किल विषय है जो उनके मस्तिष्क को बोझिल बना देता है। इसमें किसी प्रकार के खेल या मनोरंजन की कल्पना करना सरल नहीं लगता बल्कि असम्भव प्रतीत होता है। लेकिन इसका दूसरा पक्ष यह है कि दरअसल ऊपर से मुश्किल दिखाई देने वाला विषय भी रोचकता के साथ समझा जा सकता है और उसमें भी मनोरंजन खोजा जा सकता है। ‘गणित के खेल’ नामक यह पुस्तक उनके लिए उपयोगी है जो तथाकथित प्रचलित मान्यताओं के कारण अब तक ऐसा मानते आ रहे हैं। यह किताब गणित को सीखने की दिशा, नया फ्रेम वर्क प्रस्तुत करती है जिस पर भरोसा किया जा सकता है। इसे समझने के लिए पाठकों के भीतर तार्किक गुण विकसित करना भी इस पुस्तक का मूलभूत उद्देश्य है। इसे लोकप्रिय विज्ञान और गणित के रूसी विद्वान या.ई.पेरेलमान ने तैयार किया है पुस्तक को पठनीय के साथ-साथ दर्शनीय भी बनाने के लिए बीच-बीच में कार्टून शैली के चित्र भी दिये गये हैं। इसमें गणित की नाना पहेलियाँ संकलित हैं जिनमें से कई पहेलियों को छोटी कहानियों की तरह पढ़ा जा सकता है।

ज्यामिति –
गणित सबके लिए- योजना के अन्तर्गत चार पुस्तकें तैयार की गयी हैं। अंकगणित, बीजगणित ज्यामिति व त्रिकोणमिति।
ये पुस्तकें दो उद्देश्यों को ध्यान में रखकर तैयार की गयी हैं। पहला स्पर्शज्या क्या है, प्रतिशत कैसे निकालते हैं। वर्ग समीकरण का मूल ज्ञात करने के लिए कौन-से सूत्र हैं आदि सूचनाएँ इसमें जल्द से जल्द मिल सकती हैं। सभी के साथ उदाहरण भी दिये गये हैं। साथ ही किन-किन परिस्थितियों में कौन-सा नियम लागू होगा यह बताया गया है। दूसरा उद्देश्य यह भी रहा है कि पुस्तक सरल रूप में पाठकों को उपलब्ध हो जाये। गणित को एक दुसाध्य विषय माना जाता है। परन्तु यदि इसे पढ़ने की प्रक्रिया क्या हो, साथ ही पुस्तक की शैली क्या हो, यदि लेखक और पाठक ध्यान में रखें तो विषय बेहद सरल है, सुगम्य है।
स्कूली पाठ्य पुस्तक में, विशेषकर यदि वह उच्च कक्षाओं के लिए लिखी गयी है, मुख्य भूमिका विवेचना की होती है, तथ्यपरक सामग्री तर्क के बोझ से दबी रहती है। कम से कम विद्यार्थियों को ऐसा ही लगता है, प्रस्तुत पुस्तक में तथ्यपरक सामग्री की भूमिका मुख्य है। इसका मतलब यह नहीं है कि इसमें विवेचना का तर्क है ही नहीं। कहीं-कहीं सूत्र की स्थापना का तार्किक आधार भी दर्शाया गया है, पर यह सिर्फ़ विशेष परिस्थितियों में।
पुस्तक के हर भाग में ऐतिहासिक सर्वेक्षण भी दिये गये हैं इन्हें ध्यानपूर्वक पढ़ लेना अत्यन्त लाभदायक है। ये पुस्तक के आवश्यक अंग हैं, इन्हें आत्मसात का ही पुस्तकें आगे पढ़ी जायें तो बेहतर होगा।
इन पुस्तकों को पढ़ते समय कई पाठक उदाहरणों पर विशेष ध्यान देंगे तो निश्चय ही गणित जैसे विषय को रुचिकर बल्कि पहेलियों जैसा सरस पायेंगे। गणित को समझने के लिए पुस्तक में गणित प्रतीक उनकी प्रणाली, लिपि आदि भी दिये गये हैं।
गणित के पठन-पाठन के लिए भारतीय टिप्पणियों के साथ-साथ यूरोपीय प्राचीन और नवीन परम्पराओं का वर्णन भी कोष्ठकों में किया गया है। पाठक उनको ध्यान देकर पढ़ें। इस पुस्तक के अनेको संस्मरण प्रकाशित हुए हैं यह इस पुस्तक की सफलता का प्रतीक है।

त्रिकोणमिति –
गणित सबके लिए- योजना के अन्तर्गत चार पुस्तकें तैयार की गयी हैं। अंकगणित, बीजगणित ज्यामिति व त्रिकोणमिति।
ये पुस्तकें दो उद्देश्यों को ध्यान में रखकर तैयार की गयी हैं। पहला स्पर्शज्या क्या है, प्रतिशत कैसे निकालते हैं। वर्ग समीकरण का मूल ज्ञात करने के लिए कौन-से सूत्र हैं आदि सूचनाएँ इसमें जल्द से जल्द मिल सकती हैं। सभी के साथ उदाहरण भी दिये गये हैं। साथ ही किन-किन परिस्थितियों में कौन-सा नियम लागू होगा यह बताया गया है। दूसरा उद्देश्य यह भी रहा है कि पुस्तक सरल रूप में पाठकों को उपलब्ध हो जाये। गणित को एक दुसाध्य विषय माना जाता है। परन्तु यदि इसे पढ़ने की प्रक्रिया क्या हो, साथ ही पुस्तक की शैली क्या हो, यदि लेखक और पाठक ध्यान में रखें तो विषय बेहद सरल है, सुगम्य है।
स्कूली पाठ्य पुस्तक में, विशेषकर यदि वह उच्च कक्षाओं के लिए लिखी गयी है, मुख्य भूमिका विवेचना की होती है, तथ्यपरक सामग्री तर्क के बोझ से दबी रहती है। कम से कम विद्यार्थियों को ऐसा ही लगता है, प्रस्तुत पुस्तक में तथ्यपरक सामग्री की भूमिका मुख्य है। इसका मतलब यह नहीं है कि इसमें विवेचना का तर्क है ही नहीं। कहीं-कहीं सूत्र की स्थापना का तार्किक आधार भी दर्शाया गया है, पर यह सिर्फ़ विशेष परिस्थितियों में।
पुस्तक के हर भाग में ऐतिहासिक सर्वेक्षण भी दिये गये हैं इन्हें ध्यानपूर्वक पढ़ लेना अत्यन्त लाभदायक है। ये पुस्तक के आवश्यक अंग हैं, इन्हें आत्मसात का ही पुस्तकें आगे पढ़ी जायें तो बेहतर होगा।
इन पुस्तकों को पढ़ते समय कई पाठक उदाहरणों पर विशेष ध्यान देंगे तो निश्चय ही गणित जैसे विषय को रुचिकर बल्कि पहेलियों जैसा सरस पायेंगे। गणित को समझने के लिए पुस्तक में गणित प्रतीक उनकी प्रणाली, लिपि आदि भी दिये गये हैं।
गणित के पठन-पाठन के लिए भारतीय टिप्पणियों के साथ-साथ यूरोपीय प्राचीन और नवीन परम्पराओं का वर्णन भी कोष्ठकों में किया गया है। पाठक उनको ध्यान देकर पढ़ें। इस पुस्तक के अनेको संस्मरण प्रकाशित हुए हैं यह इस पुस्तक की सफलता का प्रतीक है।