Introduction to Numbers
“Mathematics is the queen of sciences and the theory of numbers is the
queen of mathematics”. – Gauss.
Anybody and everybody, who has undergone formal education at any level
has faced the frightening effects of draconian subject called “Mathematics”.
Although Mathematics has only 10 numerals ranging from zero to nine
(0,1,2,3,4,5,6,7,8,9), the types of problems that can trouble any individual are
magnanimous. Mathematics deals with properties and interaction of numbers and
ten characters enlisted above gave the most popular number system a name “Decimal System”.
Numbers are symbolic representations for counting objects. It brings
uniformity and makes various dimensions repeatable, independent of time, place,
person or domain. Whether it is cost of objects or number of items or weight of
an object or volume or area of a domain, numerals are frequently used to denote
quantitative parameter for any dimension. But contrary to the popular believe
that numbers in general follow a random sequence, a well recognized pattern
evolves, when one deciphers the beauty of numerals. In this first part,
introduction to numbers is presented in a systematic way.
Each of the ten symbols or characteristic, representing numbers is called
digit. But list of numbers does not
end with single digit numbers. Biggest single digit number is 9, beyond which
two-digit numbers start. Smallest two digit number is 10 and largest one is 99,
which becomes a threes digit number from 100 and so on. The smallest digit of
any number of digits contain ‘1’ at its extreme left followed by as many zeros
as required to complete number of digits. Largest number of given number of
digit contains all ‘9’s. For example largest five digit number is 99999 and
smallest five digit number is 10000. A very large number is called infinity and very small number is
replaced with zero generally. In addition to 10 symbolic representations of
numbers, they can be also expressed on a number
line, which varies from minus infinity on extreme left to plus infinity on
extreme right with centre named as origin
situated at number zero. A number lying on left side is smaller.
Numbers are classified under different heads, with certain basic
properties. The smallest such set is natural
number (denoted by N). All countable numbers from 1 to plus infinity is
included in this set (1,2,3,4,5,6,7, …….). If zero is also added to the list of
natural number, it is called whole
number (denoted by W). In the domain of whole numbers, if negative integers
are also added, Integers (denoted by
Z; German Zehlan) are formed. A
further superset of number is rational
numbers (denoted by Q: Quotient). All the numbers, which can be represented
in the fractional form p/q, where q
is not equal to zero, are included in this series. Naturally this includes all
integers. For a fractional form p/q, ‘p’ is called numerator, while ‘q’ is called denominator.
Another way of representing rational numbers is the decimal form of either terminating (0.32, 0.25 …) or recurring
(0.33333…, 0.166666….) decimal. The numbers, which are not satisfying the
fractional or decimal form requirement of rational numbers, are called irrational numbers (denoted by I). It
includes square roots (Ö2, Ö3, Ö15 etc), pi (p), base of Naperian log (e) etc. Combination of both
rational and irrational numbers forms a set of real numbers (denoted by R). The biggest set of numbers is complex number (denoted by C), which
include imaginary part (i = Ö-1) in the set of real numbers.
Based on properties of various numbers, all numbers divisible by ‘2’, are
called even numbers (2,4,6,8…….)
otherwise they are called odd numbers
(1,3,5,7,9….). A number, having only two factors namely ‘1’ and the number
itself, are called prime numbers
(2,3,5,7,11,13….), while numbers having more than two factors are called composite numbers (4,6,8,9,10,12…). If
sum of factors of a number is equal to number itself, it is called perfect number (6 = 1 + 2 + 3; 28 = 1 +
2 + 4 + 7 + 14 …). A number, which satisfies polynomial algebraic equation with
integer coefficients are called algebraic
number (Integers, square roots, cube roots, trigonometric functions of an
angle etc) else they are named transcendental
number (log 2, p,
2Ö2
etc). Reversing digits of a number is writing digits of the number in opposite turned
around way. e.g. 14537 is reverse of 73541. If a number and its reverse are
same, they are called Palindromes,
like 13631, 161 etc.
Do you know?
1.
‘1’ is neither a prime number nor a composite number.
2.
‘2’ is only even prime number.
3.
Zero is a rational number.
4.
‘p’ and ‘e’ are irrational numbers.
5.
‘p’ is approximated to 22/7 or 355/113 or 3.1415926…
6.
‘6’ is the smallest perfect number.
7.
There are infinite rational numbers between two
rational numbers.
8.
All real numbers can be represented on a number line.
9.
A rational fraction a/b in lowest terms has a
terminating decimal expansion if and only if the integer b has no prime factors
other than 2 and 5. For integers, ‘1’ remains in the denominator.
10. Complex
numbers cannot be represented on a number line. They are represented in a plane
by “Argand Diagram”.
11. All
transcendental numbers are irrational but all irrational numbers (Ö2)
are not transcendental numbers.
12. Trigonometric
functions (Sin20, Cos35, …) are algebraic irrational numbers and not
transcendental numbers.
13. Hilbert
number (2Ö2
) is transcendental irrational number.
