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Numbers can be classified into
sets, called
number systems. (For different methods of expressing numbers with symbols, such as the
Roman numerals, see
numeral systems.)
[edit] Natural numbersThe most familiar numbers are the
natural numbers or counting numbers: one, two, three, ... .
In the
base ten number system, in almost universal use today for arithmetic operations, the symbols for natural numbers are written using ten
digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. In this base ten system, the rightmost digit of a natural number has a place value of one, and every other digit has a place value ten times that of the place value of the digit to its right. The symbol for the set of all natural numbers is
N, also written
.In
set theory, which is capable of acting as an axiomatic foundation for modern mathematics, natural numbers can be represented by classes of equivalent sets. For instance, the number 3 can be represented as the class of all sets that have exactly three elements. Alternatively, in
Peano Arithmetic, the number 3 is represented as sss0, where s is the "successor" function. Many different representations are possible; all that is needed to formally represent 3 is to inscribe a certain symbol or pattern of symbols 3 times.
[edit] IntegersNegative numbers are numbers that are less than zero. They are the opposite of positive numbers. For example, if a positive number indicates a bank deposit, then a negative number indicates a withdrawal of the same amount. Negative numbers are usually written by writing a negative sign (also called a minus sign) in front of the number they are the opposite of. Thus the opposite of 7 is written −7. When the
set of negative numbers is combined with the natural numbers and zero, the result is the set of integer numbers, also called
integers,
Z (German
Zahl, plural
Zahlen), also written
.[edit] Rational numbersA
rational number is a number that can be expressed as a
fraction with an integer
numerator and a non-zero natural number
denominator. The fraction
m/
n or
represents
m equal parts, where
n equal parts of that size make up one whole. Two different fractions may correspond to the same rational number; for example 1/2 and 2/4 are equal, that is:
If the
absolute value of
m is greater than
n, then the absolute value of the fraction is greater than 1. Fractions can be greater than, less than, or equal to 1 and can also be positive, negative, or zero. The set of all rational numbers includes the integers, since every integer can be written as a fraction with denominator 1. For example −7 can be written −7/1. The symbol for the rational numbers is
Q (for
quotient), also written
.[edit] Real numbersThe
real numbers include all of the measuring numbers. Real numbers are usually written using
decimal numerals, in which a decimal point is placed to the right of the digit with place value one. Each digit to the right of the decimal point has a place value one-tenth of the place value of the digit to its left. Thus
represents 1 hundred, 2 tens, 3 ones, 4 tenths, 5 hundredths, and 6 thousandths. In saying the number, the decimal is read "point", thus: "one two three point four five six ". In the US and UK and a number of other countries, the decimal point is represented by a
period, whereas in continental Europe and certain other countries the decimal point is represented by a
comma. Zero is often written as 0.0 when necessary to indicate that it is to be treated as a real number rather than as an integer. Negative real numbers are written with a preceding
minus sign:
Every rational number is also a real number. To write a fraction as a decimal, divide the numerator by the denominator. It is not the case, however, that every real number is rational. If a real number cannot be written as a fraction of two integers, it is called
irrational. A decimal that can be written as a fraction either ends (terminates) or forever repeats, because it is the answer to a problem in division. Thus the real number 0.5 can be written as 1/2 and the real number 0.333... (forever repeating threes) can be written as 1/3. On the other hand, the real number π (
pi), the ratio of the
circumference of any circle to its
diameter, is
Since the decimal neither ends nor forever repeats, it cannot be written as a fraction, and is an example of an irrational number. Other irrational numbers include
(the
square root of 2, that is, the positive number whose square is 2).
Thus 1.0 and
0.999... are two different decimal numerals representing the natural number 1. There are infinitely many other ways of representing the number 1, for example 2/2, 3/3, 1.00, 1.000, and so on.
Every real number is either rational or irrational. Every real number corresponds to a point on the
number line. The real numbers also have an important but highly technical property called the
least upper bound property. The symbol for the real numbers is
R or
.
When a real number represents a
measurement, there is always a
margin of error. This is often indicated by
rounding or
truncating a decimal, so that digits that suggest a greater accuracy than the measurement itself are removed. The remaining digits are called
significant digits. For example, measurements with a ruler can seldom be made without a margin of error of at least 0.01 meters. If the sides of a
rectangle are measured as 1.23 meters and 4.56 meters, then multiplication gives an area for the rectangle of 5.6088 square meters. Since only the first two digits after the decimal place are significant, this is usually rounded to 5.61.
In
abstract algebra, the real numbers are up to isomorphism uniquely characterized by being the only
complete ordered field. They are not, however, an
algebraically closed field.
[edit] Complex numbersMoving to a greater level of abstraction, the real numbers can be extended to the
complex numbers. This set of numbers arose, historically, from the question of whether a negative number can have a
square root. This led to the invention of a new number: the square root of negative one, denoted by
i, a symbol assigned by
Leonhard Euler, and called the
imaginary unit. The complex numbers consist of all numbers of the form
where
a and
b are real numbers. In the expression
a +
bi, the real number
a is called the
real part and
bi is called the
imaginary part. If the real part of a complex number is zero, then the number is called an
imaginary number or is referred to as
purely imaginary; if the imaginary part is zero, then the number is a real number. Thus the real numbers are a
subset of the complex numbers. If the real and imaginary parts of a complex number are both integers, then the number is called a
Gaussian integer. The symbol for the complex numbers is
C or
.
In
abstract algebra, the complex numbers are an example of an
algebraically closed field, meaning that every
polynomial with complex
coefficients can be
factored into linear factors. Like the real number system, the complex number system is a
field and is
complete, but unlike the real numbers it is not
ordered. That is, there is no meaning in saying that
i is greater than 1, nor is there any meaning in saying that that
i is less than 1. In technical terms, the complex numbers lack the
trichotomy property.
Complex numbers correspond to points on the
complex plane, sometimes called the Argand plane.
Each of the number systems mentioned above is a
proper subset of the next number system. Symbolically,
N ⊂
Z ⊂
Q ⊂
R ⊂
C.
[edit] Computable numbersMoving to problems of computation, the
computable numbers are determined in the set of the real numbers. The computable numbers, also known as the
recursive numbers or the
computable reals, are the
real numbers that can be computed to within any desired precision by a finite, terminating
algorithm. Equivalent definitions can be given using
μ-recursive functions,
Turing machines or
λ-calculus as the formal representation of algorithms. The computable numbers form a
real closed field and can be used in the place of real numbers for many, but not all, mathematical purposes.
[edit] Other typesHyperreal and hypercomplex numbers are used in
non-standard analysis. The hyperreals, or
nonstandard reals (usually denoted as *
R), denote an
ordered field which is a proper
extension of the ordered field of
real numbers R and which satisfies the
transfer principle. This principle allows true
first order statements about
R to be reinterpreted as true first order statements about *
R.
Superreal and
surreal numbers extend the real numbers by adding infinitesimally small numbers and infinitely large numbers, but still form
fields.
Tue Jan 20, 2009 12:47 pm
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