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Dot 084: √-1 = i, √i = ?

11/17/2018

 
  • Q: What is √i ?
    • Know: √-1 = i
    • Suppose: √i = a + bi
    • Answer: 
      • (√i)^2 = (a + bi)^2
      • i = a^2 + 2abi + (bi)^2
      • i = a^2 + b^2*(i)^2 + 2abi
      • i = a^2 + (-1)*b^2 + 2abi
      • i = (a^2 - b^2) + 2abi
      • 0 + i = (a^2 - b^2) + 2abi
      • So, 
        • a^2 - b^2 = 0, 
        • 1 = 2ab, 
          • ——> b = 1/2a ——> a^2 - (1/2a)^2 = 0
          • ——> (4a^4 - 1) / 4a^2 = 0
      • So,
        • a = ± 1 / √2
        • b = ± 1/√2
        • √i =  1/√2 + 1/√2i or √i = -1/√2 - 1/√2 i 











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