Rational and Irrational Numbers

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A rational number is a number that can be expressed as a/b, where 'a' and 'b' are both integers and 'b' ≠ 0. The set of rational numbers is denoted by Q. Examples: 3/4, -5/7, 0, 2 (which is 2/1).

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(i) b ≠ 0 (ii) a and b have no common factor other than 1 (they are co-primes) (iii) b is usually positive, while a may be positive, negative, or zero.

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Step 1: Find d = (y-x)/(n+1). Step 2: The n rational numbers are: x+d, x+2d, x+3d, ..., x+nd. Example: To insert 2 numbers between 1 and 4: d = (4-1)/(2+1) = 1, so numbers are 2 and 3.

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A terminating decimal is a decimal representation that ends after a finite number of digits. Examples: 1/8 = 0.125, 1/25 = 0.04, 3.4. These occur when division gives an exact result with no remainder.

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A recurring decimal is a non-terminating decimal where a digit or set of digits repeats continuously. Notation: A bar or dot is placed over the repeating part. Examples: 4/9 = 0.4̅ = 0.4444..., 4/7 =

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A rational number a/b has a terminating decimal if and only if the denominator b can be expressed as 2ᵐ × 5ⁿ, where m and n are whole numbers. Examples: 17/50 terminates because 50 = 2¹ × 5², but 23/7

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Let x = 0.47̅ = 0.474747... Since 2 digits repeat, multiply by 100: 100x = 47.474747... Subtract: 100x - x = 47.474747... - 0.474747... → 99x = 47 → x = 47/99. Therefore, 0.47̅ = 47/99.

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An irrational number is a real number that cannot be expressed as a/b where a and b are integers and b ≠ 0. It has a non-terminating, non-recurring decimal representation. Examples: √2 = 1.414213...,