Real Numbers

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A rational number is any number that can be expressed in the form p/q, where p and q are integers and q ≠ 0. Examples: 3/4, -7/5, 0.25. Set notation: Q = {p/q | p, q ∈ I, q ≠ 0}…

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1. Terminating decimals (e.g., 2/5 = 0.4, 7/8 = 0.875) 2. Non-terminating recurring decimals (e.g., 1/3 = 0.333..., 22/7 = 3.142857142857...) Note: The bar notation is used for recurring decimals like…

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1. Commutative: a + b = b + a (e.g., 2/3 + 1/4 = 1/4 + 2/3) 2. Associative: (a + b) + c = a + (b + c) 3. Identity: a + 0 = 0 + a = a (0 is additive identity) 4. Inverse: a + (-a) = 0 (-a is additive i…

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1. Commutative: a × b = b × a 2. Associative: a × (b × c) = (a × b) × c 3. Identity: a × 1 = 1 × a = a (1 is multiplicative identity) 4. Inverse: a × (1/a) = 1 for a ≠ 0 (1/a is multiplicative inverse…

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Look at the prime factors of the denominator (in lowest terms): - If only 2 and/or 5 are prime factors → Terminating decimal - If any other prime factors are present → Non-terminating recurring decima…

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Irrational numbers are real numbers that cannot be expressed as p/q where p and q are integers. Their decimal expansion is non-terminating and non-recurring. Examples: √2 = 1.414213..., √3 = 1.732050.

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Assume √2 is rational, so √2 = p/q (in lowest terms). ∴ 2 = p²/q² → 2q² = p² ∴ p² is even → p is even → p = 2k ∴ 2q² = 4k² → q² = 2k² → q is even Both p and q are even, contradicting that p/q is in lo…

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π ≈ 22/7 ≈ 3.14159... (the ratio of circumference to diameter of any circle) The great Indian mathematician Aryabhata calculated π = 62832/20000 = 3.1416 in 499 CE. π is an irrational number with non-…