This site is about some extraordinary numbers in the history of mathematics.
Some numbers are very commonly known like 0, 1, 1.618, 2, 2.71828, 3, or 3.14159, others are little known like 2.54, i, 1.414, .618, 6.67384 × 10-11, or 10^100. Each of these has a significant place in mathematics both historically and currently.
Below I discuss a few features of some of these amazing numbers. I have included 32 significant places or more of many of the long decimals. This is not the entire number of course. The irrational numbers never end so it would be impossible for me to include them in their entirety.
Some important numbers:
Zero was the last digit to be introduced to the field of mathematics. When it was, however, it opened up a whole new way to write and manipulate numbers. Before zero was introduced there was no place value. In the Roman Numeral system I=1, V=5, X=10, etc… and XI:11 had a different meaning then IX:9
It is the most commonly used number anywhere. It is also the only number which you can multiply anything by and get that same number back.
This is (5^.5+1)/2. It is commonly known as the golden ratio because it appears repeatedly in mathematics and nature. In the rectangle on the right the small rectangle is similar to the total rectangle (the small rectangle plus the square) and both have a golden ratio between their length and their height. We can also get the golden ratio from the Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89,… where the first two terms are one and each following term is the sum of the two proceeding ones. if we divide any number by the proceeding one we get a ratio that approaches the golden ratio the higher we go.
Two is the smallest prime and the only even prime. Every prime following it is odd 3,5,7,11,13,17,19,… it is also the only prime that is right next to the following prime (3). two is also the base of choice for programming and computer calculations. two bits 0 and 1 can reflect a current or lack of one. Example base two numbers are: 101011=43, 110010=50, 111111=63.
The number e comes up repeatedly in mathematics. One of it’s most famous uses is in Euler’s formula e^i*pi+1=0 but it also shows up in business where it is used to compute continuously compounded interest A=Pe^(rt). It’s applications to calculus include the fact that the function y=e^x is it’s own integral and derivative, also that the inverse of this function y=ln(x) is the integral of y=1/x.
Three is the smallest odd prime. It is also the smallest nontrivial triangular number.
Pi is probably the most famous number in the history of mathematics. Defined as the ratio of the diameter of a circle to it’s circumference, it appears everywhere we look. It is implied implicitly in every circle but it also appears in relation to trigonometric functions, in relation to e, and as a limit to many mathematical series.
Important as a link between the English and the metric system of measurement it is the number of centimeters in an inch.
8 is the largest cube that is one less than a square number. (2^3+1=3^2)
The only other case is the trivial one 0. (0^3+1=1^2)
This is the result of Catalan’s Conjecture proven in 2002 by Preda Mihailescu.
8 is also the sum of all the powers of 2 less than itself. 8=4+2+1+1/2+1/4+1/8… to 1/infinity.
8 is the largest cube in the Fibonacci Sequence: 1,1,2,3,5,8,13,21,34,55,….
Reversing the digits gives you 96.
The difference of the digits 6 and 9 is 3 and the difference of 69 and 96 is 3^3.
69 is the only number x for which x^2 and x^3 together contain all the digits from 0 to 9 exactly once.
Since 12=1 and (-1)2=1 what is the sqrt(-1)? It is not anywhere on the real number line. Since it cannot be placed into the real numbers we call it imaginary. We can still graph it, however, we just place it beside the real number line on a plane. Multiples of i added to the real numbers then create the complex plane. The complex plane is very similar to the Cartesian coordinate system with the real components graphed on the X axis or real axis and the imaginary components graphed on the Y axis or imaginary axis.
The square root of two is irrational, in fact it was the first irrational number to be discovered. If a right triangle is created with the sides closest to the right angle each being one unit long then the length of the remaining side will be the square root of two. There are many other irrational numbers including the non-integer roots of integers, pi, and e.