10th Class Mathematics Number System and its Operations Number System

      Introduction

Every number we have studied so far are real numbers. The real numbers are divided into two categories as rational and irrational numbers. All the positive counting numbers are called the natural numbers. It starts from 1 till infinity. The positive numbers which starts from zero are called whole numbers. The collections of natural numbers, their negatives along with the number zero are called integers the rational numbers are the numbers in the form \[\frac{p}{q},q\ne 0\] where p and q are integers.  

       Decimal Expansion of Rational Numbers

There are rational numbers which can be expressed as terminating decimals or non-terminating decimals. The non-terminating decimals may be repeating or non-repeating.. The rational number whose denominator has factor 2 or 5 are terminating and rest are non-terminating.  

       Tests of Divisibility

•   A number is divisible by 2, if its units place have the digit 1,2,4,6 and 8.
•   A number is divisible by 3, if sum of all the digits in the given number is a multiple of 3.
•   A number is divisible by 4, if its last two digits are divisible by 4.
•   A number is divisible by 5, if the units place of the given number is either 0 or 5.
•   A number is divisible 6, if it is divisible by both 2 and 3.
•   A number is divisible by 8, if its last three digits are divisible by 8.
•   A number is divisible by 9, if sum of digits is divisible by 9.
•   A number is divisible by 10, if its units place have the digit 0.
•   A number is divisible by 11 if the difference between the sum of the digits at odd and even places is either 0 or a multiple of 11.  

       Some Important Results

•   \[1+2+3+\,-------\,+n=\frac{n\left( n+1 \right)}{2}\]
•   \[{{1}^{2}}+{{2}^{2}}+{{3}^{2}}+\,-------\,+{{n}^{2}}=\frac{n\left( n+1 \right)\left( 2n+1 \right)}{6}\]
•   \[{{1}^{3}}+{{2}^{3}}+{{3}^{3}}+\,-------\,+{{n}^{3}}={{\left[ \frac{n\left( n+1 \right)}{2} \right]}^{-2}}\]
•   \[{{x}^{n}}+{{y}^{n}}\] in divisible by \[\left( x+y \right)\] for all values of add
•   \[{{x}^{n}}-{{y}^{n}}\] in divisible by \[\left( x-y \right)\] for all values of n
•   If a number is divisible by m and n, then it is always divisible by the LCM of m and n.
•   \[{{x}^{n}}-{{y}^{n}}\] IS divisible by \[\left( x+y \right)\] if in even