Numbers and Their Principles

 

Numbers and their Principles

 

NUMBERS AND THEIR STRUCTURE

Numbers are the most important tool in our daily life for counting, measuring, labeling, sequencing and coding.

Numbers are represented by using the symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 in Hindu-Arabic System.

These symbols are called digits. 0 is called insignificant digit, whereas others are called significant digits.

 

Face Value and Place Value of a Digit Face Value

It is the value of the digit itself e.g., in '2563', face value of 5 five and face value of 3 is three.

 

Place Value

It is the face value of the digit multiplied by the place value of which it is situated e.g., in '8963', place value of 6 is

\[\text{6}\times \text{1}{{\text{0}}^{1}}=60\]and place value of 8 is \[8\times {{10}^{3}}=8000.\]

 

Notation and Numeration

Notation Representing a number in figures is known as notation e.g., 250, 300, 7200, 255.

 

Numeration Representing a number in words as numeration e.g., Numeration of 36 is thirty six.

 

 To write a number, we put digits from right to left at the places designated as units, tens, hundreds, thousands, ten thousands, lakhs, ten lakhs, crores, ten crores etc. Let us see how the number 53461902 is denoted.

 It is read as 'five crore thirty four lakh sixty one thousand nine hundred and two'.

 

Type of Numbers

Real Numbers

Real numbers include rational and irrational numbers both.

e.g.,      \[\frac{7}{9},\]\[\sqrt{2},\]\[\sqrt{5},\]\[\pi ,\frac{8}{9}\] etc.

 

Rational Numbers

A number that can be expressed as \[\frac{p}{q}\] is called a rational number, where p and q are integers and \[q\ne 0.\]

e.g.,      \[2,\]\[-\,\,5,\]\[\frac{3}{5},\]\[\frac{7}{9},\]\[\frac{8}{9},\]\[\frac{13}{15}\]etc.

Irrational Numbers

The numbers that cannot be expressed in the form of p/q are called irrational numbers, where p and q are integers and \[q\ne 0.\]

e.g.,      \[\sqrt{2},\] \[\sqrt{3},\]\[\sqrt{7},\]\[\sqrt{11}\]etc.

\[\pi \] is an irrational number as \[\frac{22}{7}\] is not the actual value of\[\pi \] but it is its nearest value.

 

Natural Numbers

Natural numbers are counting numbers.

e.g.,      \[1,\]\[2,\]\[3,\]\[4,\]\[5,....\infty .\]

·         Zero is not a natural number.

·         All natural numbers are positive. Therefore, 1 is the smallest natural number.

 

Whole Numbers

All natural numbers and zero form the set of whole numbers, e.g., 0, 1, 2, 3, 4, 5,...

 

Integers

Whole numbers and negative form the set of integers.

e.g.,      \[-\,4,\]\[-3,\]\[-\,\,2,\]\[-1,\]\[0,\]\[1,\]\[2,\]\[3,\]\[4\]

·         Integers are denoted by I.

·         0 is neither \[(+)\] ve nor \[(-)\] ve.

 

Even Numbers

A counting number which is divisible by 2 is called an even number.

e.g., 2, 4, 6, 8, 10, 12,...... etc.

 

Odd Numbers

A counting number which is not divisible by 2 is known as an odd number.

e.g., 1, 3, 5, 7, 9, 11, 13, 15, 17, 19,...... etc.

 

Prime Numbers

A counting number is called a prime number when it has exactly two factors, 1 and itself.

e.g., 2, 3, 5, 7, 11, 13 etc.

·         2 is the only even number which is prime.

·         A prime number is always greater than 1.

 

Co-primes

Two natural numbers are said to be co-primes if their HCF is 1.

e.g., (7, 9), (15, 16)

Co-prime numbers may or may not be prime.

 

Composite Numbers

Composite numbers are non-prime natural numbers. They must have atleast one factor apart from 1 and itself.

e.g., 4, 6, 8, 9 etc.

·         Composite numbers can be both odd and even.

·         1 is neither prime nor composite.

 

To Find the Unit's Place Digit in the Product of Numbers

To find the unit's place digit in the product of number, we take units place digits of each number. Now, find the product of these digits. If there is any number (more than 9) in the product, then we proceeds as above i.e., we again take unit's place digits and find the product, e.g., Unit's place digit in

\[569\times 369\times 477\times 444\]= Unit's place digit in \[9\times 9\times 7\times 4\]

= Unit's place digit in \[81\times 28\]

= Unit's place digit in \[1\times 8=8\]

 

To Find the Unit's Place Digit in a Number of the Form \[{{\mathbf{N}}^{\mathbf{n}}}\]

·         If unit's place digit of number is 0, 1, 5 or 6, then the unit's place digit in \[{{N}^{n}}\]remains unaltered. e.g., Unit's place digit in \[{{(576)}^{1151}}=6\]

·         If unit's place digit of a number is 2, then divide the index of number by 4, then write in form of \[{{2}^{4}}\] and solve. (Unit's place digit in\[{{2}^{4}}=6\])

e.g., Unit's place digit in \[{{(572)}^{443}}\]

= Unit's place digit in \[{{({{2}^{4}})}^{110}}\times {{2}^{3}}\]

= Unit's place digit in \[6\times 8\]

= Unit's place digit in 48 = 8

\[\therefore \] Unit's place digit in \[{{(572)}^{443}}=8\]

·         Similarly, we find the unit's place digit of a given number which has unit's place digit either 4 or 8.

(Unit's place digit in \[{{4}^{4}}=\]Unit's place digit is\[{{8}^{4}}=6\])

·         If there is unit's place digit either 3 or 7 in given number, then we solve the number as above, but unit's place digit in

\[{{3}^{4}}=\]Unit's place digit in \[{{7}^{4}}=1.\]

·         If number contains 9 as unit's place digit and index is odd, then the required unit's place digit in \[{{N}^{n}}\] is 9 and if index is even, then the required unit's place digit in \[{{N}^{n}}\] is 1.

e.g., Unit's place digit in \[{{(539)}^{140}}=1\]

and unit's place digit in \[{{(539)}^{141}}=9\]

            Quicker One

Ø  \[\frac{{{(a+1)}^{n}}}{a}\] gives always remainder 1.

Ø  \[\frac{{{a}^{n}}}{(a+1)}\]  gives remainder 1, when n is even and gives the remainder a itself, when n is odd, where a is any integer and n being a positive integer.

Ø  Every prime number greater than 3 can be written in the form of \[(6k+1)\] or \[(6k-1),\] where k is an integer.

Ø  There are 15 prime numbers between 1 and 50 and 10 prime numbers between 50 and 100.

Ø  The product of three consecutive natural numbers is always divisible by 6.

Ø  \[({{x}^{m}}-{{a}^{m}})\] is divisible by \[(x-a)\] for all values of m.

Ø  \[({{x}^{m}}-{{a}^{m}})\] is divisible by \[(x+a)\] for even values of m.

Ø  \[({{x}^{m}}+{{a}^{m}})\] is divisible by  for odd values of m.

 

DIVISIBILITY

When D and d are two numbers, then \[\frac{D}{d}\] is called the operation of division, where D is the dividend and d is the divisor. A number which tells how many times a divisor (d) exists in dividend (D) is called the quotient (Q). If dividend (D) is not a multiple of divisor (d) then D is not exactly divisible by d and in this case a remainder (R) is obtained.

Dividend = (Divisor \[\times \] Quotient) + Remainder

 

Divisibility Test

 

by 3     When the sum of the digits of a number is divisible by 3, then the number is divisible by 3. e.g.,

§  \[\mathbf{1233}\to 1+2+3+3=9,\] which is divisible by 3, so 1233 must be divisible by 3.

§  \[\mathbf{156}\to 1+5+6=12,\] which is divisible by 3, so 156 must be divisible by 3.

 

by 6     When a number is divisible by both 3 and 2, then that particular number is divisible by 6 also.

e.g., 18, 36, 720, 1440 etc. are divisible by 6 as they are divisible by both 3 and 2.

 

by 7     A number is divisible by 7 when the difference between twice the digit at ones place and the number formed by other digits is either zero or a multiple of 7.

e.g., 658 is divisible by 7 because \[65-2\times 8=65-16=49.\] As, 49 is divisible by 7, the number 658 is also divisible by 7.

 

by 8     When the number made by last three digits of a number is divisible by 8, then the number is also divisible by 8. Apart from this, if the last three or more digits of a number are zeros, then the number is, divisible by 8. e.g.,

§  \[\mathbf{2256}\to \] As, 256 (the last three digits of 2256) is divisible by 8, therefore 2256 is also divisible by 8.

§  \[\mathbf{4362000}\to \] As, 4362000 has three zeros at the end, therefore it will be divisible by 8.

 

by 9     When the sum of all the digits of a number is divisible by 9, then the number is also divisible by 9. e.g.,

§  \[\mathbf{936819}\to 9+3+6+8+1+9=36,\] which is divisible by 9. Therefore, 936819 is also divisible by 9.

§  \[\mathbf{4356}\to 4+3+5+6=18,\] which is divisible by 9.

Therefore, 4356 is also divisible by 9.

 

by 11   When the sums of digits at odd and even places are equal or differ by a number divisible by 11, then the number is also divisible by 11. e.g.,

§  \[\mathbf{2865423}\to \]Let us see

Sum of digits at odd places\[(A)=2+6+4+3=15\]

Sum of digits at even places\[(-\,\,B)=8+5+2=15\]                                          \[A=B\]

§  Hence, 2865423 is divisible by 11.

§  \[217382\to \] Let us see

Sum of digits at odd places\[(A)=2+7+8=17\]

Sum of digits at even places\[(B)=1+3+2=6\]

\[A-B=17-6=11\]

Clearly, 217382 is divisible by 11.