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Is the difference of two irrational numbers always irrational?
Yes, the difference of two irrational numbers is not always irrational. For example, the difference between the irrational number √2 and its negative (-√2) is a rational number (0). Therefore, the difference of two irrational numbers can be rational or irrational, depending on the specific numbers being subtracted. **
Is the difference between two irrational numbers always irrational?
No, the difference between two irrational numbers is not always irrational. For example, if you subtract the irrational number √2 from itself, you get 0, which is a rational number. So, it is possible for the difference between two irrational numbers to be rational. **
HARPERCOLLINS Predictably Irrational: The Hidden Forces that Shape Our Decisions by Dan Ariely
Why do smart people make irrational decisions every day? The answers will surprise you. Predictably Irrational is an intriguing, witty and utterly original look at why we all make illogical decisions. Why can a 50p aspirin do what a 5p aspirin can't? If an item is "free" it must be a bargain, right? Why is everything relative, even when it shouldn't be? How do our expectations influence our actual opinions and decisions? In this astounding book, behavioural economist Dan Ariely cuts to the heart of our strange behaviour, demonstrating how irrationality often supplants rational thought and that the reason for this is embedded in the very structure of our minds. Predicatably Irrational brilliantly blends everyday experiences with a series of illuminating and often surprising experiments, that will change your understanding of human behaviour. And, by recognising these patterns, Ariely shows that we can make better decisions in business, in matters of collective welfare, and in our everyday lives from drinking coffee to losing weight, buying a car to choosing a romantic partner.
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HARPERCOLLINS Dan Ariely 2 Books Collection Set By Dan Ariely (The Honest Truth About Dishonesty, Predictably Irrational)
The Honest Truth About Dishonesty Fascinating and provocative, Ariely's The Truth About Dishonesty is an insightful and brilliantly researched take on cheating, deception and willpower. Internationally bestselling author Ariely pulls no punches when it comes to home truths. Previous titles Predictably Irrational and The Upside Of Irrationality have becomes classics in their field, revealing unexpected and astonishing traits that run through modern humankind. Now acclaimed behavioural economist Dan Ariely delves deeper into the dark and murky recesses of contemporary psychology, daring to ask the big questions: What makes us cheat? How and why do we rationalise deception of ourselves and other people, and make ourselves 'wishfully blind' to the blindingly obvious? Predictably Irrational Why do smart people make irrational decisions every day? The answers will surprise you. Predictably Irrational is an intriguing, witty and utterly original look at why we all make illogical decisions.Why can a 50p aspirin do what a 5p aspirin can't? If an item is "free" it must be a bargain, right? Why is everything relative, even when it shouldn't be? How do our expectations influence our actual opinions and decisions? In this astounding book, behavioural economist Dan Ariely cuts to the heart of our strange behaviour, demonstrating how irrationality often supplants rational thought and that the reason for this is embedded in the very structure of our minds.
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HARPERCOLLINS The Last Kingdom: The Warrior Chronicles Tales Series 6 Books Collection Set by Bernard Cornwell Epic Historical Fiction Saxon TalesImmerse yourself in the brutal and heroic world of Uhtred of Bebbanburg with The Last Kingdom: The Warrior Chronicles / Saxon Tales Series 2 – 6 Books Collection Set by Bernard Cornwell. Set in 9th and 10th century England during the Viking invasions, this gripping series combines history, warfare, and unforgettable storytelling. Follow Uhtred — a Saxon raised by Vikings — as he battles for his homeland, torn between two worlds and two loyalties. With Cornwell’s masterful blend of historical accuracy and thrilling adventure, these novels are perfect for fans of epic sagas, medieval fiction, and the hit Netflix series The Last Kingdom. Whether you’re new to the series or expanding your collection, this 6-book set delivers powerful storytelling from one of Britain’s greatest historical novelists. Bernard Cornwell The Last Kingdom Series 2 (6 Books Collection Set Books 7-12) Warriors of the Storm The new novel in Bernard Cornwells number one bestselling series on the making of England and the fate of his great hero; Uhtred of Bebbanburg. BBC2s major TV show THE LAST KINGDOM is based on the first two books in the series. A fragile peace is about to be broken King Alfreds son Edward and formidable daughter; thelflaed; rule Wessex; Mercia and East Anglia. But all around the restless Northmen; eyeing the rich lands and wealthy churches; are mounting raids. Uhtred of Bebbanburg; the kingdoms greatest warrior; controls northern Mercia from the strongly fortified city of Chester. But forces are rising up against him. Northmen allied to the Irish; led by the fierce warrior Ragnall Ivarson; are soon joined by the Northumbrians; and their strength could prove overwhelming. Despite the gathering threat; both Edward and thelflaed are reluctant to move out of the safety of their fortifications. But with Uhtred's own daughter married to Ivarson's brother; who can be trusted? In the struggle between family and loyalty; between personal ambition and political commitment; there will be no easy path. But a man with a warrior's courage may be able to find it. Such a man is Uhtred;and this may be his finest hour. The Pagan Lord The new novel in Bernard Cornwell's number one bestselling series The Warrior Chronicles; on the making of England and the fate of his great hero; Uhtred of Bebbanburg. Alfred the Great is dead and Edward his son reigns as king. Wessex survives but peace cannot hold. The Danes in the north; led by Viking Cnut Longsword; stand ready to invade and will never rest until the emerald crown is theirs. Uhtred; once Alfred's great warrior but now out of favour with the new king; must lead a band of outcasts north to recapture his old family home; that great Northumbrian fortress; Bebbanburg. Loyalties will be divided and men will fall; as every Saxon kingdom is drawn into the bloodiest battle yet with the Danes; a war which will decide the fate of every king; and the entire English nation. The Empty Throne The new novel in Bernard Cornwell's number one...24,99 £*Shipping: 2,99 £Secure redirect to the provider
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HARPERCOLLINS War Lord – Bernard Cornwell The Last Kingdom Series Final Book Historical Fiction & Viking AdventureExperience the epic conclusion to a legendary saga with War Lord by bestselling historical fiction author Bernard Cornwell. The final installment in The Last Kingdom series, War Lord follows Uhtred of Bebbanburg as he faces his greatest challenges yet in a land torn by war and ambition. With kingdoms at stake and enemies closing in, Uhtred must fight to secure his destiny in a dramatic and unforgettable conclusion. Packed with battle scenes, political intrigue, and richly detailed historical settings, this novel brings the Viking era to life with Cornwell’s signature storytelling style. Why Readers Love This Book: Final book in the bestselling Last Kingdom (Saxon Stories) series Epic historical fiction set in Anglo-Saxon England Action-packed battles and compelling characters Written by master storyteller Bernard Cornwell Perfect for fans of Viking history and adventure A powerful and satisfying finale, War Lord is a must-read for fans of historical fiction and the Last Kingdom series.6,99 £*Shipping: 2,99 £Secure redirect to the provider
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HARPERCOLLINS The History of Middle-earth (Boxed Set 1): The Silmarillion, Unfinished Tales, The Book of Lost Tales, Part One & Part Two: Book 1First in a series of hardback boxed sets celebrating the literary achievement of Christopher Tolkien, featuring double-sided dustjackets. Set 1 contains special editions of THE SILMARILLION and UNFINISHED TALES reproducing the first edition text, together with the two volumes of THE BOOK OF LOST TALES. The Silmarillion is an account of the Elder Days, of the First Age of Tolkien’s World. It is the ancient drama to which the characters in The Lord of the Rings look back, and in whose events some of them, such as Elrond and Galadriel, took part. The tales of The Silmarillion are set in an age when Morgoth, the first Dark Lord, dwelt in Middle-Earth, and the High Elves made war upon him for the recovery of the Silmarils, the jewels containing the pure light of Valinor. Unfinished Tales is a collection of narratives ranging in time from the Elder Days of Middle-earth and the rise of Númenor in the Second Age to the end of the War of the Ring, and provides those who have read The Hobbit and The Lord of the Rings with a whole collection of background and new stories from the twentieth century’s most acclaimed popular author. The Book of Lost Tales stands at the beginning of the entire conception of Middle-earth and Valinor for the Tales were the first form of the myths and legends that came to be called The Silmarillion. Embedded in English legend and English association, they are set in the narrative frame of a great westward voyage over the Ocean by a mariner named Eriol to the lonely Isle where the Elves dwelt; from them he learned their true history, the Lost Tales of Elfinesse. In the two volumes of the Tales are found the earliest accounts of Gods and Elves, Dwarves, Balrogs and Orcs; of the Silmarils and the Two Trees of Valinor; of the geography and cosmology of Tolkien’s invented world. Published together for the first time, these four books collect the beginning of Christopher Tolkien’s forty-year career devoted to presenting his father J.R.R. Tolkien’s writings on Middle-earth, a unique accomplishment that celebrates the greatest invented world in all of fantasy literature.67,89 £*Shipping: 0,00 £Secure redirect to the provider
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Tales of Middle-Earth By J.R.R. Tolkien 5 Books Collection Set - Fiction - Paperback HarperCollins PublishersThis five-book collection is the perfect preparation for The Lord of the Rings: The Rings of Power. In 1917, long before he wrote The Hobbit, J.R.R. Tolkien laid the groundwork for his legendarium. These myths and legends were woven into The Lord of the Rings to evoke a deep history. His son Christopher Tolkien edited and published these tales of Middle-earth for its legions of fans and admirers. This collection starts with The Silmarillion (1977). The first section is a creation narrative, and the second describes the Valar and the main enemy, Melkor. The third and largest section, Quenta Silmarillion, tells of the First Age of Middle-Earth and the tragic saga of the Silmarils, three jewels containing the light of creation. The fourth section, Akallabêth, details the fall of the great island kingdom Numenor in the Second Age, and the fifth and final section places the events of The Lord of the Rings into this narrative. Unfinished Tales (1980) contains new text alongside alternate versions of tales in The Silmarillion, presented as Tolkien left them before his death. After completing the ambitious and comprehensive The History of Middle-earth in 12 volumes, Christopher Tolkien returned to the First Age to present three of its stories as standalone novels: The Children of Hurin, Beren and Luthien and The Fall of Gondolin. This set is perfect for the fantasy obsessive who wants to explore the narratives behind its greatest work. Grab this great value collection today, and learn of the dignity of elves, the frailty of man, and the rising of great evil. Titles in This Set: The Silmarillion Unfinished Tales The Children of Hurin Beren and Luthien The Fall of Gondolin Description: The Silmarillion The Silmarillion is an account of the Elder Days, of the First Age of Tolkien’s world. It is the ancient drama to which the characters in The Lord of the Rings look back, and in whose events some of them such as Elrond and Galadriel took part. The tales of The Silmarillion are set in an age when Morgoth, the first Dark Lord, dwelt in Middle-Earth, and the High Elves made war upon him for the recovery of the Silmarils, the jewels containing the pure light of Valinor. Unfinished Tales The book concentrates on the realm of Middle-earth and comprises such elements as Gandalf’s lively account of how it was that he came to send the Dwarves to the celebrated party at Bag-End, the emergence of the sea-god Ulmo before the eyes of Tuor on the coast of Beleriand, and an exact description of the military organization of the Riders of Rohan. Unfinished Tales also contains the only story about the long ages of Numenor before its downfall, and all that is known about such matters as the Five Wizards, the Palantiri and the legend of Amroth. The tales were collated and edited by JRR Tolkien’s son and literary heir, Christopher Tolkien, who provides a short commentary on each story, helping the reader to fill in the gaps and put each story into the context of the rest of his...32,99 £*Shipping: 2,99 £Secure redirect to the provider
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Why is the reciprocal of an irrational number also irrational?
The reciprocal of a number is the result of dividing 1 by that number. If the original number is irrational, then its reciprocal will also be irrational. This is because if the original number cannot be expressed as a simple fraction, then its reciprocal will also not be able to be expressed as a simple fraction. Therefore, the reciprocal of an irrational number will also be irrational. **
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Is the illusion irrational?
The illusion itself is not irrational, as it is simply a misperception or misinterpretation of sensory information. Illusions occur because our brains are constantly trying to make sense of the world around us, and sometimes they can be tricked by certain visual or auditory stimuli. However, it is important to recognize when we are experiencing an illusion and to use critical thinking to understand the true nature of the situation. In this way, we can overcome the illusion and make rational decisions based on accurate information. **
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Are emotions irrational?
Emotions are not inherently irrational, as they are a natural and important part of human experience. Emotions can be influenced by our thoughts, beliefs, and past experiences, but they also serve important functions in guiding our behavior and decision-making. While emotions can sometimes lead to irrational behavior, they can also provide valuable information about our needs and desires. It is important to acknowledge and understand our emotions in order to make rational decisions and effectively navigate our relationships and experiences. **
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What are irrational exponents?
Irrational exponents are exponents that are not rational numbers, meaning they cannot be expressed as a fraction of two integers. For example, the square root of 2 is an irrational number, so raising a number to the power of the square root of 2 would result in an irrational exponent. When dealing with irrational exponents, we often use approximations or special techniques to evaluate expressions, as they cannot be represented as simple fractions or whole numbers. **
HARPERCOLLINS Wilbur Smith 6 Book Collection Set Golden Lion, Predator, Desert God, War Cry, The Tiger’s Prey & Pharaoh Historical Adventure Fiction
Experience sweeping historical adventures, dangerous journeys and action-packed storytelling with this Wilbur Smith 6 Books Collection Set from bestselling novelist Wilbur Smith. This exciting collection includes: Golden Lion Predator Desert God War Cry The Tiger’s Prey Pharaoh From ancient Egypt and battles for power to seafaring adventures and conflicts across Africa, these novels showcase the epic storytelling, dramatic settings and larger-than-life characters that have made Wilbur Smith a favourite among generations of adventure-fiction readers. Filled with warfare, survival, revenge, political intrigue, exploration and family conflict, this six-book collection takes readers across different historical periods and locations. Each novel delivers its own gripping story while showcasing Smith's signature blend of historical detail and fast-moving adventure. Perfect for existing Wilbur Smith fans or readers discovering his novels for the first time, this collection makes an excellent addition to any historical fiction, action-adventure or thriller library. Key Features Includes 6 Wilbur Smith novels Features Golden Lion, Predator, Desert God, War Cry, The Tiger’s Prey and Pharaoh Epic historical fiction and action-adventure storytelling Stories spanning Ancient Egypt, Africa, war and seafaring adventure Packed with action, conflict, intrigue and suspense Ideal for fans of historical adventure novels Excellent collection for binge reading or gifting
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The Paulo Coelho Classics 10 Books Collection Box Set - Fiction - Paperback HarperCollins Publishers
Titles In this set 1. The Alchemist 2. The Valkyries 3. The Pilgrimage 4. The Fifth Mountain 5. Veronika Decides to Die 6. Manual of the Warrior of Light 7. Eleven Minutes 8. The Zahir 9. The Witch of Portobello 10. Brida Description The Alchemist Santiago is an Andalusian shepherd boy who dreams of travelling the world in search of a worldly treasure as fabulous as any ever found. From his home in Spain he journeys to the markets of Tangiers, and from there into the Egyptian desert, where a fateful encounter with the alchemist awaits him. The Valkyries This is a modern-day adventure story featuring Paulo's supernatural encounter with angels who appear as warrior women and travel through the Mojave desert on their motorbikes. Haunted by a devastating curse, Paulo is instructed by his mysterious spiritual master to embark upon a journey to find and speak to his guardian angel in an attempt to confront and overcome his dark past. The Valkyries is a compelling account of this forty day quest into the searing heat of the Mojave Desert, where Paulo and his wife, Chris, encounter the Valkyries warrior women who travel the desert on motorcycles, spreading the word of angels. This exotic spiritual odyssey is a rare combination of truth, myth, imagination and inspiration. Ultimately it is a story about being able to forgive our past and believe in our future. The Pilgrimage An extraordinary mixture of adventure story and guide to self-knowledge, this book recounts the spectacular trials of Paulo and his mentor, Petrus, as they journey across Spain in search of a miraculous sword. A compelling tale that delivers a powerful brew of magic and insight, The Pilgrimage recounts the amazing trials of Paulo Coelho and his mysterious mentor, Petrus, as they journey across Spain in search of a miraculous sword -- on a legendary road travelled by pilgrims of San Tiago since the Middle Ages. Paulo's visionary blend of mysticism, magical realism and folklore makes this an adventure story with a difference. The Fifth Mountain The Fifth Mountain, is set in the 9th century BC. Elijah is a young man struggling to maintain his sanity amidst a chaotic world of tyranny and war. Forced to flee his home, then choose between his newfound love and security and his overwhelming sense of duty, this is a moving and inspiring story about how we can transcend even the most terrible ordeals by keeping faith and love alive. Veronika Decides to Die The new novel from internationally acclaimed author Paulo Coelho -- a dramatic story of love, life and death that shows us all why every second of our existence is a choice we all make between living and dying. Veronika has everything she could wish for. She is young and pretty, has plenty of boyfriends, a steady job, a loving family. Yet she is not happy; something is lacking in her life, and one morning she decides to die. She takes an overdose of sleeping pills, only to wake up some time later in the local hospital. There she is told that...
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Are religious beliefs irrational?
Religious beliefs are not necessarily irrational. While they may not be based on empirical evidence or scientific proof, they are often deeply rooted in personal experiences, cultural traditions, and moral values. Many people find meaning, purpose, and comfort in their religious beliefs, and these beliefs can provide a framework for understanding the world and guiding ethical behavior. Ultimately, whether religious beliefs are considered rational or not may depend on one's perspective and the criteria used to evaluate rationality. **
How can one recognize irrational numbers and are all negative numbers irrational?
One can recognize irrational numbers by their inability to be expressed as a simple fraction or as a repeating or terminating decimal. They are non-repeating and non-terminating. Not all negative numbers are irrational. For example, -1 can be expressed as a simple fraction (-1/1) and is therefore rational. Irrational numbers can be positive or negative, such as the square root of 2 or pi. **
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HARPERCOLLINS Predictably Irrational: The Hidden Forces that Shape Our Decisions by Dan ArielyWhy do smart people make irrational decisions every day? The answers will surprise you. Predictably Irrational is an intriguing, witty and utterly original look at why we all make illogical decisions. Why can a 50p aspirin do what a 5p aspirin can't? If an item is "free" it must be a bargain, right? Why is everything relative, even when it shouldn't be? How do our expectations influence our actual opinions and decisions? In this astounding book, behavioural economist Dan Ariely cuts to the heart of our strange behaviour, demonstrating how irrationality often supplants rational thought and that the reason for this is embedded in the very structure of our minds. Predicatably Irrational brilliantly blends everyday experiences with a series of illuminating and often surprising experiments, that will change your understanding of human behaviour. And, by recognising these patterns, Ariely shows that we can make better decisions in business, in matters of collective welfare, and in our everyday lives from drinking coffee to losing weight, buying a car to choosing a romantic partner.1,99 £*Shipping: 1,99 £Secure redirect to the provider
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HARPERCOLLINS Dan Ariely 2 Books Collection Set By Dan Ariely (The Honest Truth About Dishonesty, Predictably Irrational)The Honest Truth About Dishonesty Fascinating and provocative, Ariely's The Truth About Dishonesty is an insightful and brilliantly researched take on cheating, deception and willpower. Internationally bestselling author Ariely pulls no punches when it comes to home truths. Previous titles Predictably Irrational and The Upside Of Irrationality have becomes classics in their field, revealing unexpected and astonishing traits that run through modern humankind. Now acclaimed behavioural economist Dan Ariely delves deeper into the dark and murky recesses of contemporary psychology, daring to ask the big questions: What makes us cheat? How and why do we rationalise deception of ourselves and other people, and make ourselves 'wishfully blind' to the blindingly obvious? Predictably Irrational Why do smart people make irrational decisions every day? The answers will surprise you. Predictably Irrational is an intriguing, witty and utterly original look at why we all make illogical decisions.Why can a 50p aspirin do what a 5p aspirin can't? If an item is "free" it must be a bargain, right? Why is everything relative, even when it shouldn't be? How do our expectations influence our actual opinions and decisions? In this astounding book, behavioural economist Dan Ariely cuts to the heart of our strange behaviour, demonstrating how irrationality often supplants rational thought and that the reason for this is embedded in the very structure of our minds.8,39 £*Shipping: 2,99 £Secure redirect to the provider
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HARPERCOLLINS The Last Kingdom: The Warrior Chronicles Tales Series 6 Books Collection Set by Bernard Cornwell Epic Historical Fiction Saxon TalesImmerse yourself in the brutal and heroic world of Uhtred of Bebbanburg with The Last Kingdom: The Warrior Chronicles / Saxon Tales Series 2 – 6 Books Collection Set by Bernard Cornwell. Set in 9th and 10th century England during the Viking invasions, this gripping series combines history, warfare, and unforgettable storytelling. Follow Uhtred — a Saxon raised by Vikings — as he battles for his homeland, torn between two worlds and two loyalties. With Cornwell’s masterful blend of historical accuracy and thrilling adventure, these novels are perfect for fans of epic sagas, medieval fiction, and the hit Netflix series The Last Kingdom. Whether you’re new to the series or expanding your collection, this 6-book set delivers powerful storytelling from one of Britain’s greatest historical novelists. Bernard Cornwell The Last Kingdom Series 2 (6 Books Collection Set Books 7-12) Warriors of the Storm The new novel in Bernard Cornwells number one bestselling series on the making of England and the fate of his great hero; Uhtred of Bebbanburg. BBC2s major TV show THE LAST KINGDOM is based on the first two books in the series. A fragile peace is about to be broken King Alfreds son Edward and formidable daughter; thelflaed; rule Wessex; Mercia and East Anglia. But all around the restless Northmen; eyeing the rich lands and wealthy churches; are mounting raids. Uhtred of Bebbanburg; the kingdoms greatest warrior; controls northern Mercia from the strongly fortified city of Chester. But forces are rising up against him. Northmen allied to the Irish; led by the fierce warrior Ragnall Ivarson; are soon joined by the Northumbrians; and their strength could prove overwhelming. Despite the gathering threat; both Edward and thelflaed are reluctant to move out of the safety of their fortifications. But with Uhtred's own daughter married to Ivarson's brother; who can be trusted? In the struggle between family and loyalty; between personal ambition and political commitment; there will be no easy path. But a man with a warrior's courage may be able to find it. Such a man is Uhtred;and this may be his finest hour. The Pagan Lord The new novel in Bernard Cornwell's number one bestselling series The Warrior Chronicles; on the making of England and the fate of his great hero; Uhtred of Bebbanburg. Alfred the Great is dead and Edward his son reigns as king. Wessex survives but peace cannot hold. The Danes in the north; led by Viking Cnut Longsword; stand ready to invade and will never rest until the emerald crown is theirs. Uhtred; once Alfred's great warrior but now out of favour with the new king; must lead a band of outcasts north to recapture his old family home; that great Northumbrian fortress; Bebbanburg. Loyalties will be divided and men will fall; as every Saxon kingdom is drawn into the bloodiest battle yet with the Danes; a war which will decide the fate of every king; and the entire English nation. The Empty Throne The new novel in Bernard Cornwell's number one...24,99 £*Shipping: 2,99 £Secure redirect to the provider
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HARPERCOLLINS War Lord – Bernard Cornwell The Last Kingdom Series Final Book Historical Fiction & Viking AdventureExperience the epic conclusion to a legendary saga with War Lord by bestselling historical fiction author Bernard Cornwell. The final installment in The Last Kingdom series, War Lord follows Uhtred of Bebbanburg as he faces his greatest challenges yet in a land torn by war and ambition. With kingdoms at stake and enemies closing in, Uhtred must fight to secure his destiny in a dramatic and unforgettable conclusion. Packed with battle scenes, political intrigue, and richly detailed historical settings, this novel brings the Viking era to life with Cornwell’s signature storytelling style. Why Readers Love This Book: Final book in the bestselling Last Kingdom (Saxon Stories) series Epic historical fiction set in Anglo-Saxon England Action-packed battles and compelling characters Written by master storyteller Bernard Cornwell Perfect for fans of Viking history and adventure A powerful and satisfying finale, War Lord is a must-read for fans of historical fiction and the Last Kingdom series.6,99 £*Shipping: 2,99 £Secure redirect to the provider
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Is the difference of two irrational numbers always irrational?
Yes, the difference of two irrational numbers is not always irrational. For example, the difference between the irrational number √2 and its negative (-√2) is a rational number (0). Therefore, the difference of two irrational numbers can be rational or irrational, depending on the specific numbers being subtracted. **
-
Is the difference between two irrational numbers always irrational?
No, the difference between two irrational numbers is not always irrational. For example, if you subtract the irrational number √2 from itself, you get 0, which is a rational number. So, it is possible for the difference between two irrational numbers to be rational. **
-
Why is the reciprocal of an irrational number also irrational?
The reciprocal of a number is the result of dividing 1 by that number. If the original number is irrational, then its reciprocal will also be irrational. This is because if the original number cannot be expressed as a simple fraction, then its reciprocal will also not be able to be expressed as a simple fraction. Therefore, the reciprocal of an irrational number will also be irrational. **
-
Is the illusion irrational?
The illusion itself is not irrational, as it is simply a misperception or misinterpretation of sensory information. Illusions occur because our brains are constantly trying to make sense of the world around us, and sometimes they can be tricked by certain visual or auditory stimuli. However, it is important to recognize when we are experiencing an illusion and to use critical thinking to understand the true nature of the situation. In this way, we can overcome the illusion and make rational decisions based on accurate information. **
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HARPERCOLLINS The History of Middle-earth (Boxed Set 1): The Silmarillion, Unfinished Tales, The Book of Lost Tales, Part One & Part Two: Book 1First in a series of hardback boxed sets celebrating the literary achievement of Christopher Tolkien, featuring double-sided dustjackets. Set 1 contains special editions of THE SILMARILLION and UNFINISHED TALES reproducing the first edition text, together with the two volumes of THE BOOK OF LOST TALES. The Silmarillion is an account of the Elder Days, of the First Age of Tolkien’s World. It is the ancient drama to which the characters in The Lord of the Rings look back, and in whose events some of them, such as Elrond and Galadriel, took part. The tales of The Silmarillion are set in an age when Morgoth, the first Dark Lord, dwelt in Middle-Earth, and the High Elves made war upon him for the recovery of the Silmarils, the jewels containing the pure light of Valinor. Unfinished Tales is a collection of narratives ranging in time from the Elder Days of Middle-earth and the rise of Númenor in the Second Age to the end of the War of the Ring, and provides those who have read The Hobbit and The Lord of the Rings with a whole collection of background and new stories from the twentieth century’s most acclaimed popular author. The Book of Lost Tales stands at the beginning of the entire conception of Middle-earth and Valinor for the Tales were the first form of the myths and legends that came to be called The Silmarillion. Embedded in English legend and English association, they are set in the narrative frame of a great westward voyage over the Ocean by a mariner named Eriol to the lonely Isle where the Elves dwelt; from them he learned their true history, the Lost Tales of Elfinesse. In the two volumes of the Tales are found the earliest accounts of Gods and Elves, Dwarves, Balrogs and Orcs; of the Silmarils and the Two Trees of Valinor; of the geography and cosmology of Tolkien’s invented world. Published together for the first time, these four books collect the beginning of Christopher Tolkien’s forty-year career devoted to presenting his father J.R.R. Tolkien’s writings on Middle-earth, a unique accomplishment that celebrates the greatest invented world in all of fantasy literature.67,89 £*Shipping: 0,00 £Secure redirect to the provider
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Tales of Middle-Earth By J.R.R. Tolkien 5 Books Collection Set - Fiction - Paperback HarperCollins PublishersThis five-book collection is the perfect preparation for The Lord of the Rings: The Rings of Power. In 1917, long before he wrote The Hobbit, J.R.R. Tolkien laid the groundwork for his legendarium. These myths and legends were woven into The Lord of the Rings to evoke a deep history. His son Christopher Tolkien edited and published these tales of Middle-earth for its legions of fans and admirers. This collection starts with The Silmarillion (1977). The first section is a creation narrative, and the second describes the Valar and the main enemy, Melkor. The third and largest section, Quenta Silmarillion, tells of the First Age of Middle-Earth and the tragic saga of the Silmarils, three jewels containing the light of creation. The fourth section, Akallabêth, details the fall of the great island kingdom Numenor in the Second Age, and the fifth and final section places the events of The Lord of the Rings into this narrative. Unfinished Tales (1980) contains new text alongside alternate versions of tales in The Silmarillion, presented as Tolkien left them before his death. After completing the ambitious and comprehensive The History of Middle-earth in 12 volumes, Christopher Tolkien returned to the First Age to present three of its stories as standalone novels: The Children of Hurin, Beren and Luthien and The Fall of Gondolin. This set is perfect for the fantasy obsessive who wants to explore the narratives behind its greatest work. Grab this great value collection today, and learn of the dignity of elves, the frailty of man, and the rising of great evil. Titles in This Set: The Silmarillion Unfinished Tales The Children of Hurin Beren and Luthien The Fall of Gondolin Description: The Silmarillion The Silmarillion is an account of the Elder Days, of the First Age of Tolkien’s world. It is the ancient drama to which the characters in The Lord of the Rings look back, and in whose events some of them such as Elrond and Galadriel took part. The tales of The Silmarillion are set in an age when Morgoth, the first Dark Lord, dwelt in Middle-Earth, and the High Elves made war upon him for the recovery of the Silmarils, the jewels containing the pure light of Valinor. Unfinished Tales The book concentrates on the realm of Middle-earth and comprises such elements as Gandalf’s lively account of how it was that he came to send the Dwarves to the celebrated party at Bag-End, the emergence of the sea-god Ulmo before the eyes of Tuor on the coast of Beleriand, and an exact description of the military organization of the Riders of Rohan. Unfinished Tales also contains the only story about the long ages of Numenor before its downfall, and all that is known about such matters as the Five Wizards, the Palantiri and the legend of Amroth. The tales were collated and edited by JRR Tolkien’s son and literary heir, Christopher Tolkien, who provides a short commentary on each story, helping the reader to fill in the gaps and put each story into the context of the rest of his...32,99 £*Shipping: 2,99 £Secure redirect to the provider
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HARPERCOLLINS Wilbur Smith 6 Book Collection Set Golden Lion, Predator, Desert God, War Cry, The Tiger’s Prey & Pharaoh Historical Adventure FictionExperience sweeping historical adventures, dangerous journeys and action-packed storytelling with this Wilbur Smith 6 Books Collection Set from bestselling novelist Wilbur Smith. This exciting collection includes: Golden Lion Predator Desert God War Cry The Tiger’s Prey Pharaoh From ancient Egypt and battles for power to seafaring adventures and conflicts across Africa, these novels showcase the epic storytelling, dramatic settings and larger-than-life characters that have made Wilbur Smith a favourite among generations of adventure-fiction readers. Filled with warfare, survival, revenge, political intrigue, exploration and family conflict, this six-book collection takes readers across different historical periods and locations. Each novel delivers its own gripping story while showcasing Smith's signature blend of historical detail and fast-moving adventure. Perfect for existing Wilbur Smith fans or readers discovering his novels for the first time, this collection makes an excellent addition to any historical fiction, action-adventure or thriller library. Key Features Includes 6 Wilbur Smith novels Features Golden Lion, Predator, Desert God, War Cry, The Tiger’s Prey and Pharaoh Epic historical fiction and action-adventure storytelling Stories spanning Ancient Egypt, Africa, war and seafaring adventure Packed with action, conflict, intrigue and suspense Ideal for fans of historical adventure novels Excellent collection for binge reading or gifting14,99 £*Shipping: 2,99 £Secure redirect to the provider
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The Paulo Coelho Classics 10 Books Collection Box Set - Fiction - Paperback HarperCollins PublishersTitles In this set 1. The Alchemist 2. The Valkyries 3. The Pilgrimage 4. The Fifth Mountain 5. Veronika Decides to Die 6. Manual of the Warrior of Light 7. Eleven Minutes 8. The Zahir 9. The Witch of Portobello 10. Brida Description The Alchemist Santiago is an Andalusian shepherd boy who dreams of travelling the world in search of a worldly treasure as fabulous as any ever found. From his home in Spain he journeys to the markets of Tangiers, and from there into the Egyptian desert, where a fateful encounter with the alchemist awaits him. The Valkyries This is a modern-day adventure story featuring Paulo's supernatural encounter with angels who appear as warrior women and travel through the Mojave desert on their motorbikes. Haunted by a devastating curse, Paulo is instructed by his mysterious spiritual master to embark upon a journey to find and speak to his guardian angel in an attempt to confront and overcome his dark past. The Valkyries is a compelling account of this forty day quest into the searing heat of the Mojave Desert, where Paulo and his wife, Chris, encounter the Valkyries warrior women who travel the desert on motorcycles, spreading the word of angels. This exotic spiritual odyssey is a rare combination of truth, myth, imagination and inspiration. Ultimately it is a story about being able to forgive our past and believe in our future. The Pilgrimage An extraordinary mixture of adventure story and guide to self-knowledge, this book recounts the spectacular trials of Paulo and his mentor, Petrus, as they journey across Spain in search of a miraculous sword. A compelling tale that delivers a powerful brew of magic and insight, The Pilgrimage recounts the amazing trials of Paulo Coelho and his mysterious mentor, Petrus, as they journey across Spain in search of a miraculous sword -- on a legendary road travelled by pilgrims of San Tiago since the Middle Ages. Paulo's visionary blend of mysticism, magical realism and folklore makes this an adventure story with a difference. The Fifth Mountain The Fifth Mountain, is set in the 9th century BC. Elijah is a young man struggling to maintain his sanity amidst a chaotic world of tyranny and war. Forced to flee his home, then choose between his newfound love and security and his overwhelming sense of duty, this is a moving and inspiring story about how we can transcend even the most terrible ordeals by keeping faith and love alive. Veronika Decides to Die The new novel from internationally acclaimed author Paulo Coelho -- a dramatic story of love, life and death that shows us all why every second of our existence is a choice we all make between living and dying. Veronika has everything she could wish for. She is young and pretty, has plenty of boyfriends, a steady job, a loving family. Yet she is not happy; something is lacking in her life, and one morning she decides to die. She takes an overdose of sleeping pills, only to wake up some time later in the local hospital. There she is told that...25,95 £*Shipping: 2,99 £Secure redirect to the provider
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Are emotions irrational?
Emotions are not inherently irrational, as they are a natural and important part of human experience. Emotions can be influenced by our thoughts, beliefs, and past experiences, but they also serve important functions in guiding our behavior and decision-making. While emotions can sometimes lead to irrational behavior, they can also provide valuable information about our needs and desires. It is important to acknowledge and understand our emotions in order to make rational decisions and effectively navigate our relationships and experiences. **
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What are irrational exponents?
Irrational exponents are exponents that are not rational numbers, meaning they cannot be expressed as a fraction of two integers. For example, the square root of 2 is an irrational number, so raising a number to the power of the square root of 2 would result in an irrational exponent. When dealing with irrational exponents, we often use approximations or special techniques to evaluate expressions, as they cannot be represented as simple fractions or whole numbers. **
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Are religious beliefs irrational?
Religious beliefs are not necessarily irrational. While they may not be based on empirical evidence or scientific proof, they are often deeply rooted in personal experiences, cultural traditions, and moral values. Many people find meaning, purpose, and comfort in their religious beliefs, and these beliefs can provide a framework for understanding the world and guiding ethical behavior. Ultimately, whether religious beliefs are considered rational or not may depend on one's perspective and the criteria used to evaluate rationality. **
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How can one recognize irrational numbers and are all negative numbers irrational?
One can recognize irrational numbers by their inability to be expressed as a simple fraction or as a repeating or terminating decimal. They are non-repeating and non-terminating. Not all negative numbers are irrational. For example, -1 can be expressed as a simple fraction (-1/1) and is therefore rational. Irrational numbers can be positive or negative, such as the square root of 2 or pi. **
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