The algebra we use commonly today has been in existence since the 800s when the first book on the topic was published in Persia. It arrived in Europe in the 1200s. But before this, the famous mathematician Diophantus (200 - 284 AD) had developed his own system. Previously, unknown quantities could only be represented if they had geometric solutions. Diophantus worked on a system of algebra to solve problems that did not need to be drawn geometrically to solve. Indeed, Diophantus once had his life represented in an algebraic problem in 500AD:
... his boyhood lasted 1 / 6 th of his life; he married after 1 / 7 th more; his beard grew after 1 / 12 th more, and his son was born 5 years later; the son lived to half his father's age, and the father died 4 years after the son.
Diophantus worked on algebraic problems for which there were integer solutions. Many of these problems are popular today because they make such fun problems you can try with your friends. The following are three ‘magician' type algebra problems. See if you can figure out how each of them work.
Magician Problem 1
Get someone to pick a number [don't let them tell you]
Get them to:
From that you will know their original number. In fact their original number will always be 4 less than the number they gave you. Why does this work?
Interesting Cat Problem
The Egyptians loved cats. In fact they worshipped the cat and one of their goddesses, Bast, had the head of a cat. Killing a cat in ancient Egypt was a crime punishable by death. Often cats were mummified upon death and eventually buried with their owners. It is not clear where the legend that cats had nine lives came from but the following problem comes from an ancient Egyptian puzzle. A mother cat has already spent seven of her nine lives. She had a number of kittens. Some had spent six and some have spent four of their lives. Together, mother and kittens have a total of 25 lives left. Can you tell with certainty how many kittens there are? In other words, how many answers to this problem are there?
Eratosthenes (276 - 194 BC) had a very diverse set of interests ranging from mapping through to cataloguing stars. He was able to make a surprisingly accurate measurement of the circumference of the Earth. He compared the noon shadow at midsummer between Syene (now Aswan in Egypt) and Alexandria. He assumed that the sun was so far away that its rays were essentially parallel, and then with a knowledge of the distance between Syene and Alexandria, he gave the length of the circumference of the Earth as 250,000 stadia. Eratosthenes also worked on prime numbers. He is remembered for his prime number sieve, the 'Sieve of Eratosthenes' which is still an important tool in number theory research.
The Sieve of Eratosthenes
A prime number is a number greater than 1 where the only factors of that number are itself and 1. Numbers that have additional factors are called composite. There are actually an infinite amount of prime numbers. In the early days of number theory, mathematicians were very interested in finding prime numbers easily. Eratosthenes developed a ‘sieve' to identify all primes up to a given maximum.
Here is how it works: list all the integers from 2 to your maximum. Let us use 50 as our maximum.
The first number on our list is a prime – 2. Highlight it. All the numbers that are a multiple of 2 are clearly not prime as they have an addition factor of
two. Cross out all the multiples of 2.
The first number after 2 is 3, which is not a multiple of 2 and so it too must be prime. Highlight 3 and cross out multiples of 3. [Remember, a number is a multiple of 3 if the digits in that number sum to 3!]
We have eliminated all the multiples of 2 and 3. The next number we see is 5. It is not a multiple of 2 or 3 and so must be prime. We highlight it and cross out the remaining multiples of 5.
Then we do the same for 7, 11, and 13 and so on until we are left with:
Using this method, can you determine how many prime numbers there are between 2 and 200? Use the following applet to help you. If you make a mistake, you must reset.
Use the following program to determine how many prime numbers there are between 2 and 200.