Tag Archives: Euler

Pi day Countdown #12 – Pi and the Transcendental Number

Leonhard Euler, considered to be one of the best mathematicians of all time, had the honor of having a few Transcendental numbers named after him. The Euler’s number or 2.718281828459… is not only a beautiful and a powerful number, it also has a very interesting connection with \pi.

The Euler’s number e can be expressed in the Euler’s Identity as:

e^{ix} = \sin(x) + i \cos(x)

and with x = \pi the identity becomes

e^{i \pi} = \sin(\pi) + i \cos(\pi) = 1 or e^{i \pi} -1 = 0

Swiss Mathematician, Leonhard Euler (15 April 1707 – 18 September 1783), was famous for many discoveries in mathematics such as the Euler's number and graph theory
Swiss Mathematician, Leonhard Euler (15 April 1707 – 18 September 1783), was famous for many discoveries in mathematics such as the Euler’s number and graph theory

Which unites some of the most famous mathematical constants, not only just \pi. There is also i the number that was so controversial that it was named imaginary as a joke. The number 0 which represented the concept of nothingness, a concept that wasn’t accepted as a number for millennia. The number 1 that started it all, and not to forget \pi one of the most famous numbers in mathematics.

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Fun with Numbers #17 – Perfect numbers

A perfect number is a number n where if all the factors of n were summed up and divided by 2 then the result is n. An example of this is the first perfect number 6, the factors of 6 are 1, 2, 3 and 6: (1+2+3+6/2) = 6. The first four perfect numbers 6, 28, 486 and 8128 were known and discovered by the classical Greeks, the 5th number was discovered in the middle ages the and the 8th was discovered by Euler.

Euler also strengthened an observation that Euclid made about the Perfect numbers. Euclid had noticed that given a perfect number, it can be expressed as p(p+1) where p is a Mersenne Prime (a prime number that can be expressed as 2^n+1). Euler expanded on this by showing that p(p+1), where p is a Mersenne prime will always result in a perfect number, thus each perfect number will correspond to a Mersenne prime. 6 and 3, 28 and 7, 486 and 31 and so on.

As a result there are as many Perfect numbers as there are Mersenne Primes, however whether or not there is a finite number of Perfect numbers is still unknown.

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Fun with Numbers #15 – Amicable Numbers

In the ninth century Iraqi mathematician Thabit Ibn Qurra discovers a way to generate a specific pair of numbers, the amicable numbers. Amicable numbers are pairs of numbers where the sum of the factors of the first number is the second and the sum of the factors of the second number is the first number. For example the numbers 220 and 284 are amicable numbers. The factors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 55, and 110 which the sum is 284. The factors of 284 are 1, 2, 4, 71 and 142 which the sum is 220.

The amicable numbers were known to the Pythagoreans who believed they had mystical powers. They were also well studied by Arab mathematicians. Later on Swiss mathematician, Leonhard Euler would generalize Thabit Ibn Qurra’s rules allowing the formula to find more amicable numbers.

Fun fact the smallest pair of amicable numbers is 220 and 284 and the largest currently known pair discovered in 2005 is individually 24073 digits long. At the moment it is not known if there is a limited finite number of amicable numbers or an infinite list of them.

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