Class IX Topic : NUMBER SYSTEM KEY POINTS Natural Numbers ( N) : Whole Numbers (W) : Integers ( Z) : Rational Numbers ( Q) : A number is called Rational Number, if it can be written in the form —where p and q are integers and q 0. Examples , etc Irrational Numbers . 2'9'1' 3 1 A number is called Irrational Number, if it cannot be written where p and q are integers and q * 0. Examples MO, VS, IT, 0.1011011... in the form — q Real Numbers ( R): Collection of all Rational numbers and Irrational numbers together are called Real Numbers. REAL NUMBERS AND THEIR DECIMAL EXPANSION • A number whose decimal expansion is either terminating or non — terminating recurring is a Rational Number. Examples are 3.375, 0.25, 0.23535353535 1.27272727 A number whose decimal expansion is non — terminating non - recurring is a Irrational Number. Examples are 1.414213562 3.1415926 OPERATION ON REAL NUMBERS If we ADD, SUBTRACT, MULTIPLY or DIVIDE two Rational Numbers we get a Rational Number. The SUM or DIFFERENCE of a Rational Number and an Irrational Number is Irrational. The PRODUCT or QUOTIENT of a non-zero Rational Number with an Irrational Number is Irrational. If we ADD, SUBTRACT, MULTIPLY or DIVIDE two Irrational Numbers, the result may be Rational or Irrational. IDENTITIES RELATING TO SQUARE ROOTS : ( Here a and b are positive Real Numbers ) a b
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INTEGRAL EXPONENTS OF REAL NUMBERS a-n=l/an m+n am xa am / an m-n ( am ) n amn = ( an ) m (ab = an. Ion • (a / b) n = an / b n ao=l ( Here a and b are Real Number, m and n are positive integers ) RATIONAL EXPONENTS OF REAL NUMBERS . 1. 2. Principal nt root of a positive real number is denoted by al/ n or 8/7. Here a is a positive real number and n is a positive integer. If a and b are positive Real Numbers and m, n are Rational Numbers, then m l/n l/n m m+n am xa • (am P = mn a am / an m-n • (ab = an. bn = an / bn RATIONALISATION OF DENOMINATORS : • Rationalisation Factor of 1 / y/ä is • Rationalisation Factor of 1/ 6/7 ) is 6/7 ) Rationalisation Factor of 1/ (MG - A/T ) is (N/F + N/b) • Rationalisation Factor of 1/ (a + VB) is (a - A/T ) • Rationalisation Factor of 1/ (a-vfi) is (a + N[B) OTHER IMPORTANT POINTS : If n is a natural number other than a perfect square, than is an Irrational Number. • If a and b are two distinct positive Rational Numbers then, v/7b is an Irrational Number between a and b provided ab is not a perfect square.
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Every Real Number can be represented on the Number Line i.e. every point on the Number Line represents a Real Number. SOME IMPORTANT ALGEBRIC IDENTITIES : (a + b = a2+ 2ab + b2 • (a-b = a2 - 2ab+ b2 a2—b2=(a+b).(a-b) = a2 + b2 + c2 + 2ab + 2bC + 2ca = + b3+ 3ab( a + b) (a-b = a3 - b3- 3ab( a -b) a2—ab+b2) a3 b3 =(a-b).(a2+ab+b2) 3— a2 + b2 + c2 - ab 2 O, then a3 + b3 + c3 = 3abc If a + b + c = -bc -ca )
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