Eratosthenes of Cyrene: Genius Scholar and Mathematician
Eratosthenes of Cyrene, a prominent Greek scholar, excelled in various fields such as mathematics, astronomy, and history. He is best known for his Sieve method to isolate prime numbers and accurately determining the Earth's circumference. His contributions to prime number theory and the Fundamental Theorem of Arithmetic are significant aspects of his legacy. Learn about his life, achievements, and impact in the ancient world.
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Eratosthenes of Cyrene (275-194 B.C) Eratosthenes was a prominent Greek scholar who spent his early life in Athens. He was a friend and contemporary of Archimedes and excelled in many areas, notably mathematics, astronomy, geography, history, poetry and athletics. He was a universal genius who was known to his friends as Beta, because he was regarded as the second best in almost all the fields he studied. He eventually went to Alexandria (Egypt) where he became the 3rd librarian at the great university as well as private tutor to the son of Ptolemy III. It was Eratosthenes who suggested a calendar (later adopted by the Romans) of 365 days with an additional day every 4th year. During old age he went blind and ended his life by drinking poison. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 He is best remembered today for two notable achievements: The use of his Sieve to isolate prime numbers The use of his Sieve to isolate prime numbers His ingenious method for determining the distance around His ingenious method for determining the distance around the Earth the Earth with a high degree of accuracy. with a high degree of accuracy.
1 2 3 4 5 6 7 8 9 10 Eratosthenes Sieve 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 1 1st st Cross out 1 Cross out 1 2 2nd nd Circle 2 then cross out all the multiples of 2 Circle 2 then cross out all the multiples of 2 3 3rd rd Circle 3 then cross out all the remaining Circle 3 then cross out all the remaining multiples of 3 multiples of 3 4 4th th Repeat this process, starting with circling the Repeat this process, starting with circling the next available number until all numbers are next available number until all numbers are either crossed out or circled either crossed out or circled
Prime 1 2 3 4 5 6 7 8 9 10 The circled values are Prime Numbers. Numbers have exactly two factors. 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100
The Fundamental Theorem of Arithmetic The Basic Idea The Basic Idea is that any integer above 1 is either a Prime Number, or can be made by multiplying Prime Numbers together, also known as Prime Number Decomposition 2 is prime 3 is prime 4 = 2 2 5 is prime 6 = 2 3 7 is prime Make the numbers 8 20 with your prime factor tiles
8 = 15 = 9 = 16 = 10 = 18 = 12 = 20 = 14 = Any integer above 1 is either a Prime Number, or can be made by multiplying Prime Numbers together, also known as Prime Number Decomposition
Prime Number Decomposition Some Prime Number decompositions are easy: For example, which prime numbers multiply together to make 21? Others are a bit more difficult. For example, which prime numbers multiply together to make 60? We can use factor trees to help us work it out.
Prime Factor Trees 60 12 5 Can you write your answer in a different way? 3 4 2 2 2 2 3 5 x x x
On your whiteboard: What are the missing values?
On your whiteboard: What are the missing values? 42 21
On your whiteboard: 72 Write 72 as a product of prime factors Can you write your answer in a different way? 9 8 2 3 4 3 2 2 x x x x
On your whiteboard: In how many different ways can you find the prime number decomposition of 90 How do you know you have got all of the different ways?
On your whiteboard: In how many different ways can you find the prime number decomposition of 360 How do you know you have got all of the different ways?
Title: Prime Factor Decomposition Express the following as a product of prime factors 1. 300 2. 600 3. 150 4. 450 5. 4500 6. 45
Mark and correct your work 1. 300 =2 2 3 5 5 2. 600 = 2 2 2 3 5 5 3. 150 = 2 3 5 5 4. 450 = 2 3 3 5 5 5. 4500 = 2 2 3 5 5 5 6. 45 = 3 3 5
Further Practice Each number is made of 4 prime factors. Write the prime factors of the number in the 4 white surrounding rectangles. This prime factor would have to be a prime factor of both 24 & 40 This prime factor would have to be a prime factor of both 2440, 132 and 16! 2 3 2 2
If you have the CASIO calculator that we recommend, you can actually check your answers on your calculator using the FACT button. Can you find it? Press your number, then the = button. Now press SHIFT then FACT. Use it to check your answers for question 3.
