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Misc 20 - If (1 + i/1 - i)m then find least integral value of m - Proof- Solving

  1. Chapter 5 Class 11 Complex Numbers
  2. Serial order wise
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Misc, 20 If ((1 + ๐‘–)/(1 โˆ’ ๐‘–))^๐‘š = 1, then find the least positive integral value of m. We need to find minimum value of m which is positive as well as integer. Lets first find the value of ((1 + ๐‘–)/(1 โˆ’ ๐‘–)) (1 + ๐‘–)/(1 โˆ’ ๐‘–) Rationalizing = (1 + ๐‘–)/(1 โˆ’ ๐‘–) ร— (1 + ๐‘–)/(1 + ๐‘–) = ((1 + ๐‘–) (1 + ๐‘–))/((1 โˆ’ ๐‘–)(1 + ๐‘–)) = (1 + ๐‘– )2/((1)2 โˆ’ (๐‘–)2) = (1 + (๐‘–)2 + 2 ร— 1 ร— ๐‘–)/(1 โˆ’ ๐‘–2) = (1 + ๐‘–2 + 2๐‘–)/(1 โˆ’ ๐‘–2) Putting i2 = โˆ’1 = (1+ (โˆ’1) + 2๐‘–)/(1 โˆ’(โˆ’1) ) = (1 โˆ’ 1+ 2๐‘–)/(1+1) = (0 + 2๐‘–)/2 = 2๐‘–/2 = ๐‘– ย  Hence, (1 + ๐‘–)/(1 โˆ’ ๐‘–) = ๐‘– Given ((1 + ๐‘–)/(1 โˆ’ ๐‘–))^๐‘š = 1 (๐‘–)๐‘š = 1 We know that ๐‘–2 = โˆ’1 Squaring both sides (๐‘–2)2 = (โˆ’1)2 ๐‘–4 = 1 Hence the minimum value of m which satisfies the equation is 4 ย 

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