The world of Numbers
Those ten simple symbols, digits, or numbers that we all learn early in life that influence our lives in far more ways than we could ever imagine. Have you ever wondered what our lives would be like without these 10 elegant digits and the infinite array of other numbers that they can create? Birthdays, ages, height, weight, dimensions, addresses, telephone numbers, license plate numbers, credit card numbers, PIN numbers, bank account numbers, radio/TV station numbers, time, dates, years, directions, wake up times, sports scores, prices, accounting, sequences/series of numbers, magic squares, polygonal numbers, factors, squares, cubes, Fibonacci numbers, perfect, deficient, and abundant numbers, and the list goes on ad infinitum. Engineers, accountants, store clerks, manufacturers, cashiers, bankers, stock brokers, carpenters, mathematicians, scientists, and so on, could not survive without them. In a sense, it could easily be concluded that we would not be able to live without them. Surprisingly, there exists an almost immeasurable variety of hidden wonders surrounding or emanating from these familiar symbols that we use every day, the natural numbers.
Over time, many of the infinite arrays, or patterns, of numbers derivable from the basic ten digits have been categorized or classified into a variety of number types according to some purpose that they serve, fundamental rule that they follow, or property that they possess. Many, if not all, are marvelously unique and serve to illustrate the extreme natural beauty and wonder of our numbers as used in both classical and recreational mathematics.
In the interest of stimulating a broader interest in number theory and recreational mathematics, this collection will endeavor to present basic definitions and brief descriptions for several of the number types so often encountered in the broad field of recreational mathematics. The number type descriptions that follow will not be exhaustive in detail as space is limited and some would take volumes to cover in detail. A list of excellent reading references is provided for those who wish to learn more about any specific number type or explore others not included. It is sincerely hoped that the material contained herein will stimulate you to read and explore further. I also hope that after reading, digesting, and understanding the material offered herein, that you will have enjoyed the experience and that you will never utter those terrible, unforgettable words, "I hate math."
Some basic definitions of terms normally encountered in the classroom are given
The different Numbers
Integers
Any of the positive and negative whole numbers, ..., -3, -2, -1, 0, +1, +2, +3, ... The positive integers, 1, 2, 3..., are called the natural numbers or counting numbers. The set of all integers is usually denoted by Z or Z+
Digits
The 10 symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9, used to create numbers in the base 10 decimal number system.
Numerals
The symbols used to denote the natural numbers. The Arabic numerals 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 are those used in the Hindu-Arabic number system to define numbers.
Natural Numbers
The set of numbers, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17,....., that we see and use every day. The natural numbers are often referred to as the counting numbers and the positive integers.
Whole Numbers
The natural numbers plus the zero.
Rational Numbers
Any number that is either an integer "a" or is expressible as the ratio of two integers, a/b. The numerator, "a", may be any whole number, and the denominator, "b", may be any positive whole number greater than zero. If the denominator happens to be unity, b = 1, the ratio is an integer. If "b" is other than 1, a/b is a fraction.
Fractional Numbers
Any number expressible by the quotient of two numbers as in a/b, "b" greater than 1, where "a" is called the numerator and "b" is called the denominator. If "a" is smaller than "b" it is a proper fraction. If "a" is greater than "b" it is an improper fraction which can be broken up into an integer and a proper fraction.
Irrational Numbers
Any number that cannot be expressed by an integer or the ratio of two integers. Irrational numbers are expressible only as decimal fractions where the digits continue forever with no repeating pattern. Some examples of irrational numbers are
Transcendental Numbers
Any number that cannot be the root of a polynomial equation with rational coefficients. They are a subset of irrational numbers examples of which are Pi = 3.14159... and e = 2.7182818..., the base of the natural logarithms.
Real Numbers
The set of real numbers including all the rational and irrational numbers.
Rational numbers include the whole numbers (0, 1, 2, 3, ...), the integers (..., - 2, - 1, 0, 1, 2, ...), fractions, and repeating and terminating decimals.
Draw a line.
Put on it all the whole numbers 1,2,3,4,5,6,7.....etc
then put 0
then put all the negatives of the whole numbers to the left of 0
........-10,-9,-8,-7,-6,-5,-4,-3,-2,-1,0
Then put in all of the fractions .
Then put in all of the decimals [ some decimals aren't fractions ]
Now you have what is called the "real number line"
The number types currently entered, and/or planned to be entered, are listed below and will be updated as new entries are made in the future. When appropriate, and time permitting, some of the number definitions/descriptions will be expanded further to provide additional information.
Abundant, Algebraic, Amicable, Arrangement, Automorphic, Binary, Cardinal, Catalan, Complex, Composite, Congruent, Counting, Cubic, Decimal, Deficient, Even, Factor, Factorial, Fermat, Fibonacci, Figurate, Fractional, Friendly, Generating, Gnomon, Golden, Gyrating, Happy, Hardy-Ramanujan, Heronian, Imaginary, Infinite, Integers, Irrational, Mersenne, Monodigit, Narcissistic, Natural, Oblong, Octahedral, Odd, Ordinal, Parasite, Pell, Pentatope, Perfect , Persistent, Polygonal, Pronic, Pyramidal , Pythagorean, Quasiperfect, Random, Rational, Real, Rectangular, Relatively Prime, Semi-perfect, Sequence, Sociable, Square, Superabundant, Tag, Tetrahedral, Transcendental, Triangular, Unit Fraction, Whole.
Several of the numbers form unique patterns that are often used in the solution of mathematical problems. When distinct patterns are applicable, the first ten numbers of the patterns will be given along with specific relationships, or equations, that will enable you to find any number in the pattern.
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