




employs ten
numerals (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) and is said to operate in "base
ten." In a baseten number, each position has a value ten times that of
the position to its immediate right; for example, in the number 23 the
numeral 3 represents three units (ones), and the numeral 2 represents
two tens. Decimal numbers do not necessarily contain a decimal point;
563, 5.63, and –563 are all decimal numbers. 








Decimal numbers
may be thought of as written under column headings based on the number
ten. For example, the number 2,567 stands for 2 thousands, 5 hundreds, 6
tens, and 7 ones. Large decimal numbers may also be expressed in
floatingpoint notation. 








Account of how
numbers developed from their ancient Indian origins to the Modern Arabic
numerals that are generally used today. There is a brief description of
numbering systems used by the ancient Egyptians, Greeks, and Romans, and
some of the problems these systems presented. 








binary
number system Binary numbers were first developed by the German
mathematician Gottfried Leibniz in the late 17th century. The binary
system is a system of numbers to base two, using combinations of the
digits 1 and 0. 








The value of any
position in a binary number increases by powers of 2 (doubles) with each
move from right to left (1, 2, 4, 8, 16, and so on). For example, 1011
in the binary number system represents (1 × 8) + (0 × 4) + (1 × 2) + (1
× 1), which adds up to 11 in the decimal system. 








The value of any
position in a normal decimal, or baseten, number increases by powers of
10 with each move from right to left (1, 10, 100, 1,000, 10,000, and so
on). For example, the decimal number 2,567 stands for: 








(2 × 1,000) + (5
× 100) + (6 × 10) + (7 × 1) 








Codes based on
binary numbers are used to represent instructions and data in all modern
digital computers, the values of the binary digits (contracted to
"bits") being stored or transmitted as, for example, open/closed
switches, magnetized/unmagnetized disks and tapes, and high/low voltages
in circuits. Because the main operations of subtraction, multiplication,
and division can be reduced mathematically to addition, digital
computers carry out calculations by adding, usually in binary numbers in
which the numerals 0 and 1 can be represented by off and on pulses of
electric current. 








Numbers can be
categorized broadly as real and complex. Concepts such as negative
numbers, rational numbers, and irrational numbers can be rigorously and
precisely defined in terms of the natural numbers. There remains then
the problem of defining the natural numbers. A modern approach defines
the natural numbers in terms of sets (see below). Zero is defined to be
the empty set: 0 = ř (i.e. the set with no elements). Then 1 is defined
to be the union of 0 and the set that consists of 0 (which is a set with
1 element, zero). Now we can define 2 as the union of 1 and the set
containing 1 (which is a set containing 2 elements, zero and one), and
so on. 








An alternative
procedure for constructing a number system is to define the real numbers
in terms of their algebraic and analytical properties. 








binary number system The capital
letter A represented in binary form. 








real
numbers Real numbers include all rational numbers (integers, or
whole numbers, and fractions) and irrational numbers (those not
expressible as fractions). Rational numbers are whole numbers, or
integers, and fractions. The whole numbers are represented by the
natural numbers, 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9, which give a counting
system that, in the decimal system, continues 10, 11, 12, 13, and so on.
Fractions of these numbers are represented as, for example,
,
,
,
or as decimal fractions (0.25, 0.5, 0.75). 








Irrational
numbers cannot be represented in this way and require symbols, such
as Ö2,
p,
and e. They can be expressed numerically only as the (inexact)
approximations 1.414, 3.142, and 2.718 (to three places of decimals)
respectively. The symbols p
and e are also examples of transcendental numbers, because they
(unlike Ö2)
cannot be derived by solving a polynomial equation (an equation with one
variable quantity) with rational coefficients (multiplying factors). 








complex
numbers Complex numbers include the real numbers described above
and imaginary numbers, 



