Category: Number Theory

Ordinals, Cardinals, Representation of Numbers in Language.

Quoted from:  Pi in the Sky, by John Barrow. Oxford University Press, 1992. p. 37-38 “English: one/first ; two/second ; three/third ; four/fourth French: un/premier ; deux/second or deuxième ; trois/troisième ; quatre/quatrième German: ein/erste ; zwei/ander or zweite ; drei/dritter ; vier/vierte Italian: uno/primo ; due/secondo ; tre/terzo ; quattro/quarto In each of these four languages the words for ‘one’ and ‘first’ are quite distinct in form and emphasize the distinction between solitariness (one) and priority (being first). In Italian and the more old-fashioned German and French usage of ander and second, there is also a clear difference between...

Fibonacci Sequence

Definition A Fibonacci sequence is easily constructed:  Start with 0 and 1, and for each following number, add the previous two: 0, 1, 0+1=1, 1+1=2, 1+2=3, and so on. Here is the beginning of the sequence:  0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, … The Fibonacci sequence is named after Leonardo of Pisa, who is also known as Fibonacci. In 1202 he wrote a book entitled “Liber Abaci“,  which introduces the sequence to Western European mathematics. It had been described earlier, however, in Indian mathematics.  Fibonacci number sequences are everywhere in nature, and the math can quickly get...

Prime Numbers

Prime numbers are the basic units of the system of numbers since every natural number is either a prime or can be expressed as a composite of prime numbers. Here are some excerpts from the Wikipedia article about Prime numbers: Definition “A prime number (or a prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example, 5 is prime, as only 1 and 5 divide it, whereas 6 is composite since it has...

Perfect Numbers

Perfect numbers are gateways to the wonders of the mathematical world. Contemplating them, one realizes how small our human minds really are compared to the reality that surrounds us, and creates us. Definition A perfect number is a positive integer that is equal to the sum of its positive divisors excluding itself.  The first perfect number is six. Six is the number of sides to each cell in the bee’s honeycomb,  or the number of points of all snowflakes.  ALL SNOWFLEKES HAVE SIX CORNERS, OR POINTS. Amazing, isn’t it? There is an infinite number of snowflakes, each different from every...

The Number Pi

From Wikipedia: “π (sometimes written pi) is a mathematical constant that is the ratio of any circle’s circumference to its diameter. π is approximately equal to 3.14 in the usual decimal notation. Many formulae in mathematics, science, and engineering involve π, which makes it one of the most important mathematical constants.For instance, the area of a circle is equal to π times the square of the radius of the circle. π is an irrational number, which means that its value cannot be expressed exactly as a fraction having integers in both the numerator and denominator. Consequently, its decimal representation never...

Fundamental Theorem of Arithmetic

The Fundamental Theorem of Arithmetic states that every positive integer can be written as a product of prime numbers in a unique way. The following description of the theorem and its proof is quoted from the Wikipedia encyclopedia: Definition “In mathematics, and in particular number theory, the fundamental theorem of arithmetic is the statement that every positive integer can be written as a product of prime numbers in a unique way. For instance, we can write 6936 = 23 · 3 · 172   or   1200 = 24 · 3 · 52 and there are no other such factorizations...

Number Theory

The concept of “numbers” is itself very perplexing, and shows the evolution of mathematics from simple arithmetic,  to very complex operations.  Here is a brief overview of the types of numbers: Each type of numbers in the list includes all the types listed above it. N = Natural Numbers = {0, 1, 2, …}. These are also known as the “counting numbers”. Sometimes zero is excluded from this set, but included in the “Whole” numbers. Z = Integers = {…, -2, -1, 0, 1, 2, …}. This set is the union of Natural Numbers and their counterparts, negative numbers. Q...