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Notes On Rational Number

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Published in: Mathematics
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Notes on Rational Numbers and various theroems in it it

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  1. Real numbers REAL NUMBERS CHART Real Numbers R Rational Numbers Q Irrational Numbers Non-integer Rational Numbers Integers Z Negative Numbers Whole Numbers W Zero N - Natural Numbers Countable Numbers W - Whole Numbers Natural numbers along with Zero Z - Integers Natural Numbers N Signed ( +, - ) Natural Numbers in the number line Classified as Positive and negative Integer Real Numbers The real numbers consist of all the rational numbers and all the irrational numbers. 1
  2. Real numbers Real numbers can be thought of as points on an infinitely long number line called the real line, where the points corresponding to integers are equally spaced. Rational Numbers Real Numbers Irrational Numbers Non-terminating and non-recurring Non-terminating Terminating and Recurring R or Q - Rational Number A rational number is a fraction indicating the quotient of two integers, excluding division by zero. Any number is rational , if it has the form— ( q * 0 and p, q are integers) It is a decimal with o terminating digits (Eg. 1.5) Non terminating digits or recurring decimals ( Eg. 1.325 ) Any fraction with non-zero denominators is a rational number. Hence, we can say that '0' is also a rational number P P If a rational number of the form can be expressed in the form if p q e Z and m, n e W, then it will have Terminating decimal expansion. Else it will have non- terminating and recurring decimal expansion P - Irrational Numbers Cannot be expressed as an ordinary ratio of two integers. Cannot be represented on a number line Non terminating and Recurring Denseness Property of Rational Numbers ±-2) is rational. For any two rational numbers (a,b) , their average/mid point ( 2 2 (or) a > > b 2 2
  3. Real numbers p q b Period of Decimal p+r 2 q a In the decimal expansion of the rational numbers, the number of repeating decimals is called a period of a decimal Eg) 1.325 - Period 3 1.025- Period 2 Radical Notation Let n be a positive integer, and r be a real number. If rn th n root of x, then r — x, then r is called the - nth root , radical n — index of the radical x — radicand Read as n root of x 1 can also be written as xn 1 r - base n — power x — power value Read as r power n orn power of r 1 = (xnyn th th Read as n root of m power of x Note l. Sum, Difference , Product , Division between a rational and an irrational number is always irrational 2. the sum , difference, product, quotient of any two irrational numbers could be rational or irrational Surds Irrational root of a rational number. 4/7 is a surd, provided n e N , n > l, 'a' is rational. 3
  4. Real numbers It should be irrational (non-recurring, non-terminating decimal) & is a root , Eg. = 1.4142135 VS = 1.7320508 ... All surds are irrational numbers and not vice versa Order of a Surd 8/7 — Index of the root in the surd is the order of the surd Eg ) Index 5 Classification of surds 1. 2. 3. 4. 5. 6. 7. 8. 9. Quadratic Surd — Index of the root is 2 Eg) Vis , Cubic Surd — Index of the root is 3 Surds of same order / equiradical surds - index of the root to be extracted IS same 1 Simplest form of a surd o A surd expressed as the product of a rational factor and an irrational factor. o In this form the surd has • the smallest possible index of the radical sign. • no fraction under the radical sign. • no factor is of the form an, where a is a positive integer under index n. Pure Surd — Coefficient in simplest form is l. Mixed Surd — Coefficient in simplest form is other than l. Eg) 5Nfi Simple Surd - Surd with a single term Eg) , 2Mä Compound Surd — Algebraic sum or difference of more than one surd Eg)Vä+ Binomial Surd - algebraic sum (or difference) of 2 terms both of which could be surds or one could be a rational number and another a surd. Eg)l+ 4
  5. Laws of Radicals S.N0. Radical Notation n n Real numbers xl)tl ab n Index Notation n a m mn n n mn m Operations on surds 1. Addition/ Subtration of Surds: 2. Multiplication and Division of Surds Multiplication property of surds where b,d > 0 2. Rationalisation of Surds Division property of surds (iii) (iv) where b,d > 0 Rationalising factor is a term with which a term is multiplied or divided to make the whole term rational. 3. Conjugate Surds
  6. Real numbers For, a + , the conjugate surd is a — bxfö Here , a = Rational Part bcc = Radical Part Note : Change the sign in Radical Part (Surd Part) to get the conjugate surd Conjugate surds are used for Rationalisation Methods of Rationalisation 1 ' For surds of the type, or multiply with itself For surds of the type, multiply n an m Coefficient Power 6.023 x 1023 Mantissa Base For surds of the type, a ± VB, multiply with its conjugate For surds of the type, multiply the numerator and denominator, with the denominator's conjugate Scientific Notation Expressing a number N in the form of N = ax 1011 where, 1< a
  7. Real numbers 2. Count the number of digits between the old and new decimal point. This gives 'n', the power of 10. 3. If the decimal is shifted to the left, the exponent n is positive. 4. If the decimal is shifted to the right, the exponent n is negative. Operations on numbers in scientific notation Sum and Difference Power of 10 should be same for both the notation. If not bring it to the same power and then add the coefficients , Eg)ax 10b 10 b b 10 Products Multiply the coefficients and the algebraic sum of the power is the power of 10 of the final product , 10b, c x 10 d Division —ab x 10b d Divide the coefficients. Take the reciprocal of IOS power of denominator. b ax 10 -ax 10b c x 10d Some standard Notations Notation Name Deci Centi Milli Micro Nano pico Femto d ab x 10b d S d n P f m bol Notation Name 10 10 10 10 10 10 10 Deca Hecto Kilo Mega Giga Tera Peta S mbol da h k 7

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