Products related to Irrationality:
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Irrationality and Transcendence in Number Theory
Irrationality and Transcendence in Number Theory tells the story of irrational numbers from their discovery in the days of Pythagoras to the ideas behind the work of Baker and Mahler on transcendence in the 20th century.It focuses on themes of irrationality, algebraic and transcendental numbers, continued fractions, approximation of real numbers by rationals, and relations between automata and transcendence.This book serves as a guide and introduction to number theory for advanced undergraduates and early postgraduates.Readers are led through the developments in number theory from ancient to modern times.The book includes a wide range of exercises, from routine problems to surprising and thought-provoking extension material. FeaturesUses techniques from widely diverse areas of mathematics, including number theory, calculus, set theory, complex analysis, linear algebra, and the theory of computationSuitable as a primary textbook for advanced undergraduate courses in number theory, or as supplementary reading for interested postgraduatesEach chapter concludes with an appendix setting out the basic facts needed from each topic, so that the book is accessible to readers without any specific specialist background
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The Age of Magical Overthinking : Notes on Modern Irrationality
In the modern information age, our brain’s coping mechanisms have been overloaded, and our irrationality turned up to eleven.Amanda Montell blends cultural criticism and personal narrative to explore our modern cognitive biases and the power, disadvantages and highlights of magical overthinking. "Magical thinking" can be broadly defined as the belief that one's internal thoughts can affect unrelated events in the external world.Whether that's "manifesting" their way out of poverty, staving off cancer with positive vibes, or transforming an unhealthy relationship to a glorious one through loyalty alone. In a series of razor sharp and introspective chapters, Montell delves into cognitive biases that run rampant in our brains, from how the "halo effect" cultivates worship (and hatred) of larger-than-life celebrities, to how the "sunk cost fallacy" can keep us in detrimental relationships long after they no longer serve us. Told with her signature brilliance and wit, Montell's prevailing message is one of hope and forgiveness for our anxiety riddled human self.If you have all but lost faith in our ability to reason through the noise of information overload, this book aims to make sense of the senseless.To crack open a window in our minds and let the fresh air in.To help quiet the cacophony for a while, and maybe, even hear a melody in it.
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Reasoning in Psychopathology : Rationality and Irrationality in Mental Disorders
Reasoning in Psychopathology adopts a pragmatic conception of reasoning, demonstrating how people with mental disorders develop characteristic strategies of reasoning depending on the particular disorder they have and the emotions they experience. The book argues that these strategies are perfectly rational, as the individuals are using reasoning as a tool at the service of their goals.Through the analysis of the typical reasoning styles of very different psychopathologies, from anxiety disorders to obsessive-compulsive disorder, from schizophrenia to depression and paranoia, the book argues that mental disorders can affect common sense, or social cognition, while rationality is usually preserved.Supported by recent research, the authors claim that people with mental disorders follow the same rules as healthy people, and that in some cases, when the specific topic of their disorder is at stake, they can be even more logical than healthy people. It is a must-read for all researchers and students of rationality from cognitive psychology, psychiatry, and philosophy backgrounds.
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The Upside of Irrationality : The Unexpected Benefits of Defying Logic at Work and at Home
Behavioral economist and New York Times bestselling author of Predictably Irrational Dan Ariely returns to offer a much-needed take on the irrational decisions that influence our dating lives, our workplace experiences, and our general behaviour, up close and personal. In The Upside of Irrationality, behavioral economist Dan Ariely will explore the many ways in which our behaviour often leads us astray in terms of our romantic relationships, our experiences in the workplace, and our temptations to cheat.Blending everyday experience with groundbreaking research, Ariely explains how expectations, emotions, social norms and other invisible, seemingly illogical forces skew our reasoning abilities. Among the topics Dan explores are:• What we think will make us happy and what really makes us happy;• How we learn to love the ones we are with;• Why online dating doesn’t work, and how we can improve on it;• Why learning more about people make us like them less;• Why large bonuses can make CEOs less productive;• How to really motivate people at work;• Why bad directions can help us;• How we fall in love with our ideas;• How we are motivated by revenge; and• What motivates us to cheat. Drawing on the same experimental methods that made Predictably Irrational such a hit, Dan will emphasize the important role that irrationality plays in our day-to-day decisionmaking—not just in our financial marketplace, but in the most hidden aspects of our lives.
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What is the meaning of mathematics and irrationality?
Mathematics is the study of numbers, quantities, shapes, and patterns, and it provides a framework for understanding the world around us. Irrationality, on the other hand, refers to numbers that cannot be expressed as a simple fraction or ratio of two integers. These numbers, such as the square root of 2 or pi, have decimal representations that go on forever without repeating. Understanding irrational numbers is important in mathematics as they challenge our traditional notions of rationality and provide a deeper insight into the complexity and beauty of numbers.
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Doesn't the irrationality of people often upset you?
As an AI, I don't have emotions, so the irrationality of people doesn't upset me. However, I understand that dealing with irrational behavior can be frustrating for many people. It's important to remember that everyone has their own perspectives and reasons for their actions, and practicing empathy and understanding can help navigate through these situations. It's also important to focus on what can be controlled and to find constructive ways to address and manage irrational behavior.
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What is the significance of mathematics and irrationality?
Mathematics is a fundamental tool for understanding the world around us, providing a precise language to describe and analyze complex phenomena. Irrational numbers, such as the square root of 2 or pi, are essential in mathematics as they cannot be expressed as simple fractions and have infinite non-repeating decimal expansions. The existence of irrational numbers challenges our intuition and highlights the infinite and mysterious nature of mathematics, pushing the boundaries of our understanding and paving the way for new discoveries and insights. Embracing irrationality in mathematics allows for a deeper exploration of the complexities and beauty of the mathematical universe.
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What is a proof of the irrationality of...?
The proof of the irrationality of the square root of 2 is a classic mathematical result that demonstrates that the square root of 2 cannot be expressed as a fraction of two integers. This proof involves assuming the opposite (that the square root of 2 can be expressed as a fraction) and then using a contradiction to show that this assumption leads to a logical inconsistency. This proof is a fundamental result in number theory and has important implications in mathematics and beyond.
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How can one prove the irrationality of a number?
One way to prove the irrationality of a number is by contradiction. Assume that the number is rational, and then show that this assumption leads to a contradiction. This can be done by expressing the number as a fraction and then showing that the numerator and denominator have a common factor, which contradicts the assumption that the number is in its simplest form. Another method is to use the properties of algebraic numbers and show that the number cannot be expressed as the root of a polynomial with integer coefficients. Both of these methods can be used to prove the irrationality of a number.
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How do you prove the irrationality of √2 + ∛3?
To prove the irrationality of √2 + ∛3, we can use the method of contradiction. Assume that √2 + ∛3 is rational, meaning it can be expressed as a fraction a/b where a and b are integers with no common factors. Then we can manipulate the equation to show that both √2 and ∛3 are also rational, which is a contradiction. This contradiction arises because we know that √2 and ∛3 are irrational numbers. Therefore, our initial assumption that √2 + ∛3 is rational must be false, and thus √2 + ∛3 is irrational.
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What is the irrationality of a logarithm of a number?
The irrationality of a logarithm of a number refers to the property that the value of the logarithm cannot be expressed as a simple fraction or ratio of two integers. In other words, the result of taking the logarithm of a number is not a rational number. This is because logarithms involve the use of exponents and can produce non-terminating, non-repeating decimal values, making them irrational. For example, the logarithm of 2 to the base 10 is an irrational number, approximately equal to 0.30103.
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What is the assumption of coprimality in the proof of the irrationality of square root 2?
The assumption of coprimality in the proof of the irrationality of square root 2 is that the square root of 2 can be expressed as a fraction in its simplest form, meaning that the numerator and denominator have no common factors other than 1. This assumption is used to derive a contradiction by squaring both sides of the equation and showing that the resulting expression leads to a contradiction, thus proving that the original assumption of expressing the square root of 2 as a fraction is false. This contradiction then leads to the conclusion that the square root of 2 is irrational.
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