Dot 084: √-1 = i, √i = ?

• 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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