Tuesday, 27 December 2016

Number Systems for Competitive Examinations

Number Systems

Number Systems for Competitive Examinations

Since ancient civilizations numbers have been used to count and measure items. Different symbols or numerals were used to denote the same number.

Figures 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 are called digits. The digit 0 is called insignificant while digits 1 to 9 are called significant. Decimal system (base 10) is used world wide as cited by Aristotle that we are born with ten fingers and ten toes.

Old civilizations like Maya civilization used base 20 or vigesimal system.

Numerals: All Arabic (digits 0 to 9) numbers that we use today are international numerals. There are ideograms created by Abu Ja'far Muhammad ibn Musa al-Khowarizmi (c.778 - c.850).

Natural Numbers:
Also called Counting Numbers.  Denoted by symbol N i.e. N = {1, 2, 3, 4,...}. It starts from one.

Whole Numbers:
It includes number '0' plus natural numbers. i.e.
W = {0, 1, 2, 3, ...}

BAKHSHÂLÎ MANUSCRIPT (400AD) shows that decimal sub-system have been in use before 400 AD

Integers (Z): By adjoining negatives of natural number to a set of whole numbers, constitute Integers.
i.e. Z = {..., -3, -2, -1, 0, 1, 2, 3 ...}

Rational Numbers (Q): When a number can be represented in the form of a/b, where a and b are integers and b ≠ 0

Irrational numbers (I): A number which can't be represented in the form of a/b is called irrational numbers. e.g. √, π (pi) are examples of irrational numbers.

Real Numbers: A set of numbers comprising both rational and irrational numbers is called real numbers (R). It includes all types of decimal.

Prime Numbers:
A number other than 1 is called a prime number number, if it is divisble by 1 and by itself only.
The prime numbers up to 100 are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.

Composite Numbers: A number other than one, which is not prime is called composite number.

Co-Prime Numbers: Those numbers which are prime to each other i.e. HCF of these two numbers is 1, are called co-prime number numbers. A pair of co-prime numbers can be prime or composite numbers.
e.g. 23 & 24, 16 & 49 are co-prime numbers.

Twin Primes: Any two consecutive prime numbers that have a difference of two are called twin primes. e.g. 3 & 5, 11 & 13.

Perfect Number: If the sum of factors of a number is the number it self, then the number is called perfect number. e.g. Factors of 28 are: 1, 2, 4, 7 and 14  then 1 + 2 + 4 + 7 + 14 = 28. Thus 28 is a perfect number.
"Without mathematics, there's nothing you can do. Everything around you is mathematics. Everything around you is numbers.
Shakuntla Devi

Prime Triplet: A set of three successive prime numbers differing by 2 is called a prime triplet. The only example of prime triplet is {3,5,7}

Abstract Number: It is a number which does not refer to any particular unit. Two, Five, seven are abstract numbers.

Concrete Number: A number which refers to a particular unit as three pens, seven girls, twenty one rupees are examples of concrete numbers.

Even Number: A number divisible by an even number is called even number. e.g. 2, 4, 6, 8, 10...

Odd Number: A number not divisible by 2 is called an odd number. e.g. 1, 5, 7, 11, 13, 121...

Hardy–Ramanujan number: 1729 is called Hardy–Ramanujan number, it is the smallest number expressible as the sum of two cubes in two different ways.

Googol: Incredibly large number i.e. 10^100 (1 followed by 100 zeroes)

The largest Number: According to the Guiness Book of Records, is Graham's number.

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