Page 1 : Introduction:, Number system is divided into two types of numbers:, (a), , Imaginary numbers, , (b), , Real numbers, , Square root of every negative number is an, imaginary number. For example: √−𝑥𝑥 , √−5 , √−3, , If the numbers are not imaginary. For example: √3 , √4, or 2., Real Numbers (R), , Rational numbers (Q), , Irrational numbers (Q), , 𝑎𝑎, A no. that can be expressed in , where ‘a’ and ‘b’ both are integers and b ≠ 0. For, 𝑏𝑏, 5 2, example: {-6, 0, 5, , , etc.}, 3 7, , Irrational numbers (Q): The square roots, cube roots, etc. of natural numbers are, irrational numbers. Here, even the numbers who’s exact, square and cube root does not exist comes in irrational, numbers. For example: √3, √7, √−5, 2, 5√2, etc., How to find a rational number in between two rational numbers a and b., Basic formula to find a rational number in between two rational numbers a and b is, , 𝑎𝑎+𝑏𝑏, 2, , . Thus, if a > b = a >, , 𝑎𝑎+𝑏𝑏, 2, , >b, , and, , if a < b = a <, , 𝑎𝑎+𝑏𝑏, 2, , <b, , Terminating Decimals, , : The division is exact i.e. no remainder is left., , Non-Terminating Decimals, , : The division never ends no matter how long it is., , Recurring or Periodic Decimals, , : In which a digit or set of digits repeats, continually., , Non recurring or terminating, , : In which a digit or set of digits does not repeats, continually.

Page 2 : How to find an irrational number between two rational numbers a and b., Basic formula to find an irrational number between two rational numbers a and b is, √𝑎𝑎𝑎𝑎, SURDS (Radicals) :, , If x is a positive rational number and n is a positive integer, 𝑛𝑛, , such that x1/n i.e., √𝑥𝑥 is irrational, then x1/n is called a surd, 4, , 3, , or a radical. For example : √3, √8, √20, Positive integer, , surd of order n, , Rationalization, , 3, , √20, , Positive rational number, , The process of rationalizing a surd by multiplying it with its rationalising factor is called, rationalization. For example :, 5√2 × 3√2 = 15 × 2 = 30, Rationalizing factors, Q. How to find the least rationalizing factor of the following :, (a), , √125, , √5 × 5 × 5 = 5√5, , Now, multiply √5 with 5√5 as √5 becomes the rationalizing factor of √125, 5√5 × √5 = 5 × 5 = 25, (b), , 5, , 2√2, 5, , 2√2, , 5√2, , ×, , 2×2, 5√2, 4, , √2, √2, , Ans., , (as √2 becomes the rationalizing factor of 2√2)