You’ve probably heard the saying, “The pen is mightier than the sword.” This idea highlights how powerful words can be, sometimes even more so than physical force. Throughout history, authorities have banned books and texts that they found threatening. But did you know that numbers, too, have been considered dangerous enough to ban?
In ancient times, people counted using simple tally marks. As societies grew and agriculture developed, this method wasn’t enough. Numbers became more complex, and people began to explore their nature and uses. By around 600 B.C.E., in Ancient Greece, the study of numbers was well underway. Pythagoras, a famous mathematician, and his followers found fascinating patterns in numbers, shapes, music, and even the stars. They believed that mathematics held the secrets of the universe.
One of Pythagoras’s followers, Hippasus, discovered something unsettling: some numbers, like the diagonal of a square with sides of length one, couldn’t be expressed as whole numbers or fractions. These numbers are called irrational numbers. This discovery challenged the Pythagorean belief that the universe could be perfectly described with rational numbers. According to historians, Hippasus faced serious consequences for revealing this truth.
As mathematics evolved, it caught the attention of political and religious leaders. During the Middle Ages, Europe used Roman numerals, while other cultures had developed systems that included zero. When Arab travelers introduced this system to Italy, it was a game-changer for merchants and bankers. However, authorities were wary. Hindu-Arabic numerals were unfamiliar and could be easily altered, raising concerns about forgery. The concept of zero also introduced negative numbers and the idea of debt, which was suspicious at the time.
In the 13th century, Florence banned the use of Hindu-Arabic numerals for official records. Despite their usefulness, debates about zero and negative numbers continued for centuries. Negative numbers were considered absurd until the 19th century, and even prominent mathematicians avoided using zero, despite its potential to simplify complex equations.
Even today, some numbers are illegal for various reasons. They might be banned due to their symbolic significance, like dates of revolutions, or because they represent sensitive information. Almost any information—text, images, or even programs—can be encoded as numbers. This means that protected information, like copyrighted material or state secrets, can be represented as numbers, making their possession or distribution potentially illegal.
This issue became prominent in 2001 when a code to decrypt DVDs was shared as a large prime number. The idea of illegal numbers might seem strange, but like words, numbers can express powerful concepts and information. In our world, where calculations and algorithms play a huge role, the power of numbers continues to grow.
Investigate a historical instance where numbers were banned or considered controversial. Present your findings in a short report, focusing on the reasons behind the ban and its impact on society. Consider how the situation might be different today.
Participate in a class debate on whether it is ethical to ban numbers or information. Prepare arguments for both sides, considering the implications for freedom of speech and security. Reflect on how numbers can be both powerful and dangerous.
Work in groups to create a timeline that highlights key developments in the history of numbers, from ancient tally marks to modern-day encryption. Include significant events and figures, such as Pythagoras and the introduction of zero.
Conduct a hands-on activity to explore irrational numbers. Use geometric shapes to demonstrate why some numbers cannot be expressed as fractions. Discuss the historical context and the challenges faced by mathematicians like Hippasus.
Research how numbers are used in modern technology, such as encryption and data encoding. Present a case study on a specific technology, explaining how numbers are integral to its function and the potential ethical issues involved.
Here’s a sanitized version of the provided YouTube transcript:
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They say the pen is mightier than the sword, and authorities have often agreed. From outlawed religious texts and revolutionary manifestos to censored and burned books, we know the potential power of words to overturn the social order. Interestingly, some numbers have also been considered dangerous enough to ban.
Our ancestors initially counted objects using simple tally marks. However, as they developed agriculture and began living in larger groups, this method became insufficient. As numbers grew more complex, people began to not only use them but also contemplate their nature and functionality. By 600 B.C.E. in Ancient Greece, the study of numbers was well-established. The mathematician Pythagoras and his followers found numerical patterns in shapes, music, and the stars. For them, mathematics held the deepest secrets of the universe.
One Pythagorean named Hippasus made a troubling discovery: some quantities, like the diagonal of a square with sides of length one, could not be expressed by any combination of whole numbers or fractions, regardless of how small. These numbers, known as irrational numbers, were seen as a threat to the Pythagorean belief in a perfect universe, which they thought could be described with rational numerical patterns. Historians note that Hippasus was exiled for revealing his findings, while legends suggest he faced dire consequences.
While irrational numbers troubled philosophers, later mathematical advancements attracted the attention of political and religious authorities as well. During the Middle Ages, while Europe was still using Roman numerals, other cultures had developed positional systems that included a symbol for zero. When Arab travelers introduced this system to the bustling maritime cities of Italy, its advantages for merchants and bankers became evident. However, authorities were cautious. Hindu-Arabic numerals were seen as easier to forge or alter, particularly since they were less familiar to customers than to merchants. The concept of zero also paved the way for negative numbers and the recording of debt, which was viewed with suspicion at the time.
In the 13th century, Florence prohibited the use of Hindu-Arabic numerals for record-keeping. Although these numerals soon proved too useful to ignore, debates over zero and negative numbers persisted for a long time. Negative numbers were dismissed as absurd well into the 19th century, and prominent mathematicians, like Gerolamo Cardano, avoided using zero, even though it would have simplified solving cubic and quartic equations.
Even today, certain numbers are illegal for various reasons. Some are banned due to their symbolic significance, such as dates of revolutions or associations with political figures or parties. Other numbers may be illegal because of the information they convey. Almost any information—text, images, video, or executable programs—can be represented as a string of numbers. This means that protected information, including copyrights, proprietary materials, or state secrets, can also be encoded as numbers, making the possession or publication of these numbers potentially a criminal offense.
This issue gained attention in 2001 when code that could decrypt DVDs was widely shared in the form of a large prime number. The notion of illegal numbers may seem absurd, but like words, written numbers serve as a means of expressing concepts and information. In a world where calculations and algorithms increasingly influence our lives, the mathematician’s pencil grows more powerful by the day.
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This version removes specific names and references that may not be suitable for all audiences while retaining the core ideas and themes of the original transcript.
Numbers – Symbols or words used to represent quantities and values in mathematics. – In mathematics, numbers are fundamental for performing calculations and solving equations.
Irrational – A type of number that cannot be expressed as a simple fraction, having a non-repeating, non-terminating decimal expansion. – The square root of 2 is an irrational number, as it cannot be precisely written as a fraction.
History – The study of past events, particularly in human affairs, which often influences the development of mathematical concepts. – The history of mathematics reveals how ancient civilizations contributed to modern mathematical theories.
Mathematics – The abstract science of number, quantity, and space, either as abstract concepts or as applied to other disciplines such as physics and engineering. – Mathematics is essential for understanding the principles of physics and engineering.
Pythagoras – An ancient Greek mathematician and philosopher known for the Pythagorean theorem, which relates the sides of a right triangle. – Pythagoras is credited with the discovery that the square of the hypotenuse is equal to the sum of the squares of the other two sides in a right triangle.
Zero – The integer denoting no quantity or null value, which plays a crucial role in mathematics as an additive identity. – The concept of zero was revolutionary in mathematics, allowing for the development of algebra and calculus.
Authority – The power or right to give orders, make decisions, and enforce obedience, often seen in historical contexts where mathematicians were recognized for their contributions. – In the history of mathematics, Euclid’s authority was established through his influential work, “The Elements.”
Ban – An official or legal prohibition, which in history, sometimes applied to certain mathematical ideas or texts. – During the Middle Ages, the Catholic Church imposed a ban on certain mathematical texts that contradicted religious teachings.
Negative – Less than zero; having a value that is less than zero, often used to describe numbers on the opposite side of zero on a number line. – In mathematics, negative numbers are used to represent values below zero, such as temperatures in winter.
Evolution – The gradual development of something, especially from a simple to a more complex form, often used to describe the historical progression of mathematical theories. – The evolution of calculus from its early beginnings to its modern form has greatly advanced scientific understanding.
