Numbers

A number is an value assigned to quantity; that is, it is a mental concept.   A numeral is a symbol for a number.   The numerals 0, 1, 2, 3, ... are called arabic numerals   Examples of other numerals comparable to arabic numerals are shown below:

^{Nelson 302}

Numbers are classified as follows:

Natural numbers: 1, 2, 3, ... .   Zero is not included.   These are also called "counting numbers".

Whole numbers: 0, 1, 2, 3, ...   Zero is included.

Integers: Negative and positive natural numbers, and zero:   -6, - 2, 0, +4, +10, ... .   Zero cannot be positive or negative.   An even integer is an integer that when divided by 2 results in another integer; that is, it has no remainder.   An odd integer is an integer that when divided by 2 results in a non-integer; that is, it has a remainder of 1   Zero is an even integer, since it has no remainder when divided by 2.

Rational numbers: numbers that can be expressed as a ratio of an integer and a natural number: 1/2, 7/9, ... .   Note that dividing by zero results in an undefined number; that is, no number.   Rational numbers are represented by numerals called fractions.   Integers also are rational numbers since they can be expressed as a ratio of an integer and 1:   1/1, 5/1, 56/1, ... .

Irrational numbers: numbers that cannot be expressed as a ratio of two integers:   square root of 2, "pi" (= circle circumference / diameter), etc.

Real numbers are all rational and irrational numbers.

An imaginary number is a number of the form ix, where x is a real number and i is the positive square root of -1.   In spite of the unfortunate name, imaginary numbers have applications in the "real" world; viz., physics, electrical engineering, etc.

A complex number is a number of the form v + iw, where v and w are real numbers, and i represents the positive square root of -1.   Note that if w = 0, then the complex number becomes a real number.   Therefore, all real numbers are a subset of complex numbers.   Like imaginery numbers, complex numbers also have useful applications in the real world of science and technology.

A prime number is a natural or prime number greater than 1, whose only two whole-number factors are 1 and itself.   2, 3, 5, 7, 11, 13, 17, etc. are prime whole numbers.   Numbers that are not prime numbers are called composite numbers.   Every composite number can be expressed as a product to two or more prime numbers.   All even numbers are the sum of two prime numbers. (This is Goldbach's unproven conjecture.)

Fractions can be represented by decimals when the numerator is divided by the denominator:   3/4 = .75, 2 3/4 = 2.75 and square root of 2 = 1.414213562... .

Number concepts, arithmetic and rudimentary mathematics and geometry are extremely old, judging from the monuments, irrigation, and property demarcations that were made in ancient times, which must have required calculations.   Irrational numbers (numbers that cannot be expressed as a fraction, such as pi and the square root of 2), were known to Greeks by 520 BCE, but they tried to avoid them, since they made calculations difficult.   The origin of numerals to represent numbers is lost in antiquity, so we can only speculate on how and when numerals came into use:   Hash marks for numeral ones - / / / / / - could be conveniently summed into simpler numerals, such as 5 = 1 + 1 + 1 + 1 +1, 2 5's could be summarized into 10, etc.   Roman numerals (see above chart) illustrate this primitive system very well, suggesting it is very old.   In 250 CE, Diophantus, a Greek, solved some arithmetic problems by algebraic methods.   Around 500 CE, Hindu mathematicians used the zero ( 0 ) as a positional symbol.   Thus, 67, 607 and 670 represent different numerals.   The Arabs are thought to have learned this system from the Hindus around 700 CE during their Asian conquests.   The Arab mathematician, Al-Khwarizmi, wrote about these "arabic" numerals and their associated number system around 810 CE. They gradually spread to Europe, where Roman numerals then held sway.   The Italian mathematician, Leonardo Fibonacci, who traveled in North Africa with his merchant father, learned the system and introduced it into Italy in 1202 in his book, Liber Abaci, but it would not completely replace the Roman numerals and their associated number system until around 1500.   (It's not easy to change habit and vested interests.)   Negative numbers came into use in 1545 by Cardano, an Italian mathematician.   Trigonometric tables were invented in 1551 by the German mathematician, George Joachim Iserin von Lauchen.   In 1586, Simon Stevin, a Dutch mathematician, showed that fractions could be indicated by positional notation.   Thus, 1 1/4 = 1.25, etc.   This system was useful for many applications, although it presented rounding problems for irrational numbers, such as 1 1/3 = 1.333333...., ad infinitum. ^{Asimov 43-169}

The invention of numerals and arithmetic operations (addition, subtraction, multiplication, and division) based on them can be considered one of mankind's greatest achievements.   Can you imagine a world without counting and calculating?   Numbers are presented here as an introduction to number systems, which in turn leads to the study of and logarithms, which are important to scientific and engineering principles and their history, as will be shown.