In ancient Greece, there was a philosopher named Hippasus who made a groundbreaking discovery in mathematics. Unlike the heroes in Greek myths who often faced the wrath of the gods for their actions, Hippasus’s “crime” was uncovering something about numbers that changed how people understood the world. He discovered irrational numbers, which was a big deal back then.
Hippasus was part of a group called the Pythagorean mathematicians. These people believed that numbers were the foundation of everything in the universe. They thought that everything, from the stars to music, could be explained using numbers and their ratios. For example, they could express the number 5 as 5/1 or 0.5 as 1/2. Even numbers with long decimal expansions could be written as fractions, like 34/45. These are what we call rational numbers.
Hippasus’s discovery started with a simple shape: a square with each side measuring one unit. According to the Pythagorean Theorem, the diagonal of this square would be the square root of 2. However, Hippasus found that he couldn’t express this diagonal as a ratio of two whole numbers, which was a problem for the Pythagorean belief system.
Instead of giving up, Hippasus decided to prove that it was impossible to express the square root of 2 as a fraction. He assumed that it could be written as a fraction p/q, where p and q are whole numbers with no common factors. By working through the math, he showed that both p and q would have to be even, meaning they would have a common factor, which contradicted his initial assumption. This type of proof is called a “proof by contradiction.”
Even though we can’t write irrational numbers like the square root of 2 as fractions, we can still find them on the number line. For example, if you draw a right triangle with two sides each measuring one unit, the hypotenuse will be the square root of 2. You can use this method to find other irrational numbers, like the square root of 3, by extending the line further.
Another famous irrational number is pi, which represents the ratio of a circle’s circumference to its diameter. While we often use approximations like 22/7 or 355/113, these fractions will never exactly equal pi.
Hippasus’s discovery of irrational numbers changed mathematics forever. Even though we don’t know exactly what happened to him, his work paved the way for future mathematicians to explore new ideas. So, don’t be afraid to dive into the world of numbers and discover the impossible!
Draw a number line on a large sheet of paper. Mark the integers and rational numbers you know. Then, using a ruler and compass, find and mark the position of the square root of 2. This activity will help you visualize where irrational numbers fit on the number line.
Try to prove that the square root of 3 is irrational using a proof by contradiction. Assume it can be expressed as a fraction p/q and follow the steps similar to Hippasus’s proof for the square root of 2. This will enhance your understanding of mathematical proofs.
Use a string to measure the circumference and diameter of various circular objects. Calculate the ratio of circumference to diameter and compare it to the value of pi. Discuss why these ratios are approximations and not exact values.
Research and present another irrational number, such as the golden ratio. Explain its significance and where it appears in nature or art. This will broaden your understanding of the role of irrational numbers beyond mathematics.
Create a storyboard or comic strip that illustrates the story of Hippasus and his discovery of irrational numbers. Use this creative exercise to explore the historical and mathematical impact of his work.
Sure! Here’s a sanitized version of the transcript:
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Like many heroes of Greek myths, the philosopher Hippasus was rumored to have faced severe consequences from the gods. But what was his transgression? Did he harm others or disrupt a sacred ritual? No, Hippasus’s offense was a mathematical proof: the discovery of irrational numbers. Hippasus was part of a group known as the Pythagorean mathematicians, who held a deep reverence for numbers. Their belief that “All is number” suggested that numbers were fundamental to the Universe, and they thought that everything, from cosmology and metaphysics to music and morals, followed eternal rules describable as ratios of numbers. Thus, any number could be expressed as such a ratio—5 as 5/1, 0.5 as 1/2, and so on. Even an infinitely extending decimal could be expressed exactly as 34/45. All of these are what we now call rational numbers. However, Hippasus discovered one number that contradicted this harmonious rule, one that was not supposed to exist.
The issue began with a simple shape: a square with each side measuring one unit. According to the Pythagorean Theorem, the diagonal length would be the square root of two, but Hippasus could not express this as a ratio of two integers. Instead of giving up, he decided to prove it couldn’t be done. Hippasus began by assuming that the Pythagorean worldview was correct, that the square root of 2 could be expressed as a ratio of two integers. He labeled these hypothetical integers p and q. Assuming the ratio was in its simplest form, p and q could not have any common factors. To prove that the square root of 2 was not rational, Hippasus just had to show that p/q cannot exist.
He multiplied both sides of the equation by q and squared both sides, leading to a new equation. Multiplying any number by 2 results in an even number, so p² had to be even. This could not be true if p was odd, as an odd number times itself is always odd, so p must be even as well. Thus, p could be expressed as 2a, where a is an integer. Substituting this into the equation and simplifying gave q² = 2a². Once again, two times any number produces an even number, so q² must also be even, meaning q must be even as well, making both p and q even. However, if that were true, then they would have a common factor of two, which contradicted the initial assumption. This is how Hippasus concluded that no such ratio exists. This method is known as a proof by contradiction.
Interestingly, even though we can’t express irrational numbers as ratios of integers, it is possible to precisely plot some of them on the number line. Take the square root of 2. All we need to do is form a right triangle with two sides each measuring one unit. The hypotenuse has a length of the square root of 2, which can be extended along the line. We can then form another right triangle with a base of that length and a height of one unit, and its hypotenuse would equal the square root of 3, which can also be extended along the line. The key here is that decimals and ratios are merely ways to express numbers. The square root of 2 simply represents the hypotenuse of a right triangle with sides of length one. Similarly, the famous irrational number pi is always equal to exactly what it represents: the ratio of a circle’s circumference to its diameter. Approximations like 22/7 or 355/113 will never precisely equal pi.
While we may never know what truly happened to Hippasus, we do know that his discovery revolutionized mathematics. So, regardless of the myths, don’t hesitate to explore the impossible.
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This version maintains the essence of the original content while ensuring clarity and appropriateness.
Irrational – A number that cannot be expressed as a simple fraction, meaning its decimal form is non-repeating and non-terminating. – The number √2 is irrational because it cannot be written as a fraction.
Numbers – Mathematical objects used to count, measure, and label. – In algebra, we often solve equations to find unknown numbers.
Square – The result of multiplying a number by itself. – The square of 5 is 25, because 5 × 5 = 25.
Root – A value that, when multiplied by itself a certain number of times, gives the original number. – The cube root of 27 is 3, since 3 × 3 × 3 = 27.
Fraction – A way of expressing a number that is not whole, using two integers, one above the other, separated by a line. – The fraction 3/4 represents three parts out of four equal parts of a whole.
Proof – A logical argument that demonstrates the truth of a mathematical statement. – We used a proof to show that the sum of the angles in a triangle is always 180 degrees.
Contradiction – A situation in which a mathematical statement is shown to be false by assuming the opposite and arriving at an illogical conclusion. – By assuming the opposite, we reached a contradiction, proving that the original statement must be true.
Triangle – A polygon with three edges and three vertices. – In geometry class, we learned how to calculate the area of a triangle using its base and height.
Pi – The ratio of the circumference of a circle to its diameter, approximately equal to 3.14159. – We used the value of pi to calculate the circumference of a circle in our math problem.
Mathematics – The abstract science of number, quantity, and space, either as abstract concepts or as applied to other disciplines. – Mathematics helps us understand patterns and solve problems in everyday life.
