How did Archimedes ACTUALLY calculate pi? Pi day 2021
TLDRIn this video, we explore the ancient Greek mathematician Archimedes' groundbreaking calculation of pi in 250 BCE. Utilizing the method of exhaustion, he approximated pi by inscribing and circumscribing polygons around a circle, iterating to refine the perimeters. His innovative approach, devoid of modern trigonometry, resulted in an impressively accurate estimate of pi, bounded between 3 and 10/71 and 3 and 1/7, showcasing the depth of mathematical understanding over 2300 years ago.
Takeaways
- π Humans have been calculating the perimeters, areas, and volumes of shapes for over 5000 years, with ancient civilizations like the Egyptians using geometry for practical purposes such as taxation and construction.
- π The ancient Greeks had a deep fascination with mathematics, particularly geometry, and their work laid the foundation for much of what we learn in high school geometry, as seen in Euclid's 'Elements'.
- β The circle has been a source of intrigue for mathematicians due to its prevalence in nature and its unique properties, such as pi, the ratio of a circle's circumference to its diameter.
- π€ Archimedes, a Greek mathematician, is renowned for his impressive approximation of pi in 250 BCE, which was the best approximation of the time.
- π Archimedes did not use trigonometry in the modern sense but employed a method known as the method of exhaustion, which has similarities to integral calculus.
- π The method of exhaustion involves circumscribing and inscribing polygons around a circle and calculating the ratio of their perimeters to the circle's diameter, with the polygons having more sides bringing the approximation closer to the actual circumference.
- πΊ Starting with a hexagon, Archimedes used the properties of equilateral triangles and the angle bisector theorem from Euclid's 'Elements' to iteratively refine the approximation of pi.
- π Archimedes' iterative process involved bisecting angles, applying the angle bisector theorem, and using the Pythagorean theorem to calculate the new ratios for each iteration.
- π After four iterations, Archimedes established an upper bound for pi as being less than three and one-seventh.
- π For the lower bound, Archimedes used a similar process with a 96-sided polygon, concluding that pi is greater than three and ten-seventy-firsts.
- π€© Archimedes' work on pi, which was conducted around 2300 years ago, is considered impressive even by today's standards, showcasing the ingenuity of ancient mathematicians.
Q & A
How long have humans been measuring shapes?
-Humans have been measuring, calculating, and recording the perimeters, areas, and volumes of shapes for over 5000 years.
Why were ancient Egyptians interested in geometry?
-Ancient Egyptians were interested in geometry because they needed to calculate the areas of land for tax purposes and to construct massive structures like the pyramids.
What is the significance of Euclid's 'Elements' in the history of geometry?
-Euclid's 'Elements', published a few centuries before the common era, is foundational in geometry and contains much of the material found in modern high school geometry curriculums.
Why is the circle a challenging shape for mathematicians?
-The circle is challenging because it is a naturally occurring shape that appears everywhere in nature, yet its properties, especially the ratio of its circumference to its diameter (pi), have been difficult to calculate precisely.
Who was Archimedes and what is his contribution to the calculation of pi?
-Archimedes was an ancient Greek mathematician who is known for his impressive approximation of pi using a method that involved circumscribing and inscribing polygons around a circle.
What method did Archimedes use to calculate an approximation of pi?
-Archimedes used the method of exhaustion, which involved inscribing and circumscribing polygons around a circle and calculating the perimeters to approximate the value of pi.
Why did Archimedes start with a hexagon for his calculation?
-Archimedes started with a hexagon because it can be made up of equilateral triangles, making the angles and calculations more manageable.
How did Archimedes find the ratio of the side of the hexagon to the radius of the circle?
-He used the fact that the angle at the center of the hexagon is one-third of a right angle and applied the angle bisector theorem from Euclid's 'Elements' along with Pythagoras's theorem to find the necessary ratios.
What was the significance of Archimedes' approximation of pi?
-Archimedes' approximation of pi was the best at the time, providing a bounded value of pi between 3 1/7 and 3 10/71, which was a significant achievement in the understanding of the circle's geometry.
How did Archimedes ensure the accuracy of his pi approximation?
-Archimedes ensured accuracy by iteratively bisecting angles and using the angle bisector theorem and Pythagoras's theorem to refine the ratios of the sides of the polygons he was inscribing and circumscribing around the circle.
What is the modern-day significance of Archimedes' method of exhaustion?
-The method of exhaustion used by Archimedes has a conceptual similarity to integral calculus, showing a deep understanding of mathematical principles that would not be formalized for many centuries.
Outlines
π The Historical Quest for Pi
This paragraph delves into the historical significance of geometry and the measurement of shapes, highlighting the ancient Egyptians' and Greeks' contributions. It introduces the challenge of calculating pi, the ratio of a circle's circumference to its diameter, and sets the stage for Archimedes' method of exhaustion to approximate pi's value in 250 BCE. The paragraph emphasizes the absence of trigonometry in the ancient Greek sense and the ingenious approach Archimedes took to tackle the problem.
π Archimedes' Method of Exhaustion
The second paragraph explains Archimedes' method of exhaustion in detail. It describes how he used a hexagon as a starting point and iteratively bisected angles to inscribe polygons with increasing numbers of sides around a circle. By applying the angle bisector theorem and Pythagoras' theorem, Archimedes was able to calculate the perimeters of these polygons and use them to establish an upper bound for pi, concluding that pi is less than three and one-seventh.
π Archimedes' Lower Bound for Pi
The final paragraph continues the exploration of Archimedes' work on pi by focusing on finding its lower bound. It outlines the process of inscribing polygons within a circle and using similar triangles to establish the relationship between the sides. Through a series of iterations, Archimedes determined that pi is greater than three and ten-sevenths, thus providing a range for the value of pi. The paragraph concludes by reflecting on the impressive nature of Archimedes' mathematical achievements over 2300 years ago and invites viewers to engage with the content.
Mindmap
Keywords
π‘Measurement
π‘Geometry
π‘Ancient Egyptians
π‘Ancient Greeks
π‘Euclid's Elements
π‘Circle
π‘Pi (Ο)
π‘Archimedes
π‘Method of Exhaustion
π‘Integral Calculus
π‘Polygon
Highlights
Humans have been measuring shapes for over 5000 years, essential for ancient civilizations like the Egyptians for calculating taxes based on land area.
The ancient Greeks had a profound understanding of geometry, evident in Euclid's 'Elements', a foundational text for high school geometry.
The circle has been a challenging shape for mathematicians due to its ubiquity in nature and its mathematical properties.
Archimedes is credited with calculating an impressive approximation of pi, the ratio of a circle's circumference to its diameter, in 250 BCE.
Archimedes' method did not rely on trigonometry as understood today but on the method of exhaustion, an early form of integral calculus.
The method of exhaustion involves inscribing and circumscribing polygons around a circle to approximate its circumference.
Archimedes started with a hexagon, using its properties of equilateral triangles for easier calculation.
He used the angle bisector theorem from Euclid's 'Elements' to find new ratios for iterative calculations.
Pythagoras's theorem was essential in calculating the lengths needed for the iterative process of approximating pi.
Archimedes iteratively bisected angles and used geometric ratios to progressively refine his approximation of pi.
After four iterations, Archimedes determined that the perimeter of a 96-sided polygon was a close approximation to the circle's circumference.
He established an upper bound for pi, concluding it to be less than three and one-seventh.
For the lower bound, Archimedes used a similar approach with a triangle inscribed in a semicircle.
By bisecting angles and using similar triangles, Archimedes found a ratio that led to an approximation greater than three and ten seventy-ones.
Archimedes' final approximation of pi was between three and a fraction more than ten seventy-ones and less than three and ten seventieths.
The mathematical techniques used by Archimedes over 2300 years ago are considered impressive even by today's standards.
The video provides a detailed explanation of Archimedes' method, showcasing the historical significance and mathematical brilliance of his work.
Transcripts
5.0 / 5 (0 votes)
Thanks for rating: