Number Systems

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Step 1: Divide 7 by 8 using long division: 0.875 8)7.000 64 --- 60 56 --- 40 40 --- 0 Step 2: Since remainder becomes 0, it's a terminating decimal. Answer: 7/8 = 0.875 (terminating…

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Step 1: Let x = 0.333... Step 2: Multiply by 10: 10x = 3.333... Step 3: Subtract original equation: 10x - x = 3.333... - 0.333... Step 4: Simplify: 9x = 3 Step 5: Solve for x: x = 3/9 = 1/3 Answer: 0…

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Step 1: Find the value of √25 √25 = 5 (since 5² = 25) Step 2: Check if 5 can be written as p/q 5 = 5/1, where p = 5, q = 1 (both integers, q ≠ 0) Step 3: Conclusion Since √25 = 5 can be expressed as…

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Method: Convert to common denominator Step 1: LCM of 3 and 2 is 6 1/3 = 2/6 and 1/2 = 3/6 Step 2: Since we need 5 numbers, multiply by 10 1/3 = 20/60 and 1/2 = 30/60 Step 3: Find rational numbers be…

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Proof by contradiction: Step 1: Assume 2 + √3 is rational Then 2 + √3 = p/q where p, q are integers, q ≠ 0 Step 2: Rearrange √3 = p/q - 2 = (p - 2q)/q Step 3: Analyze Since p, q are integers, (p - 2…

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Step 1: Apply the identity (a + b)(a - b) = a² - b² Here, a = √5 and b = √2 Step 2: Substitute (√5 + √2)(√5 - √2) = (√5)² - (√2)² Step 3: Calculate = 5 - 2 = 3 Answer: 3…

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Step 1: Multiply numerator and denominator by conjugate (3 - √2) 1/(3 + √2) × (3 - √2)/(3 - √2) Step 2: Apply (a + b)(a - b) = a² - b² in denominator Numerator: 1 × (3 - √2) = 3 - √2 Denominator: (3 …

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Step 1: Let x = 2.373737... Step 2: Since 2 digits repeat, multiply by 100 100x = 237.373737... Step 3: Subtract to eliminate repeating part 100x - x = 237.373737... - 2.373737... 99x = 235 Step 4:…