NUMBER SYSTEMS

NUMBER SYSTEMS

BRIEF SYNOPSIS

·         Numbers that can be expressed in the form  p/q , where q is a non- zero integer and p is any integer are called rational numbers.

·         Numbers that cannot be put in the form p/q , where q≠0, p , q € are called irrational numbers.

·         Every integer is a rational number but a rational number need not be an integer.

·         Every fraction is a rational number but a fraction need not be a rational number.

·         A rational number p/q is said to be in the standard form if q is positive integer and the integer and the integers p and q have no common divisor other than 1.

·         Between two rational numbers x and y , there is a rational number  x + y .

                                                                                                              2

·         We can find as many rational numbers between x and y as we want.

·         If x and y are any two rational numbers, then:

                                 i.            x+ y is a rational number.

                                ii.            x-y is a rational number.

                              iii.            x× y is a rational number.

                              iv.            x÷ y is a rational number (y≠0)

·         Every rational number can be expressed as a decimal.

·         The decimal representation of a rational number is either terminating  or non terminating.

·         If the denominator of a rational number written in standard form contains no prime factors other than 2 or 5 or both, then it can be represented as a terminating decimal.

·         If the denominator of a rational number written in standard form has prime factors other than 2 or 5 or , then it cannot be represented as a terminating decimal.

·          A number is the  terminating decimal form can be converted to one in the rational form by summing the place values of all the digits.

·         A number in the recurring decimal can be converted to one in the rational form by multiplying the decimal by suitable power of 10 and eliminating the decimal point.

·         A  useful result to convert a number in the recurring decimal form to the rational form:

If all the digits on the right side of the decimal part are repeated , then

Given decimal =  Integral part of the decimal number+ Number formed by the digits in decimal part

                                                                                       

                                                                                      Number formed by the same number of 9's as the                  

                                                                                      number of digits in the decimal part

For example :0.7  = 0+ 7  = 7   , 2.49  =2+49 , 5.04 =5+4    etc.

                                       9     9                     99                 99

                                      

·         A number can have decimal representation in one of the following forms:

                     i.          terminating

                   ii.          non- terminating but repeating( recurring)

                  iii.          non-terminating and non -repeating.

The numbers of the type (i) and (ii) are rational numbers where  as of type(iii) are known as irrational.

·         For positive real numbers a and b, the following identities hold:

                                 i.            √ab = √a √b

                                ii.            a

b

                              iii.            (√a+√b) (√a-√b) =a-b

                              iv.            (a+√b)(a-√b)=a²-b

                               v.            (√a+√b)²=a +2√ab+b

·         To rationalise the denominator of        1      , we multiply this by √a-b , where a and b are integers and

                                                       √a +b                                        √a-b                                                                          

·         √a +b≠0 .

·         Let  a> 0 be a real number and p and q be rational numbers .Then

a)     ap