Educator

Complex Numbers

Main definition and formulas:

• Complex numbers are numbers of the form x + yi, where x and y are real numbers, and i satisfies i² = − 1.
• The powers of i satisfy the following pattern:
  

  i0 = 1 i1 = i i2 = − 1 i3 = − i i4 = 1 i5 = i i6 = − 1 i7 = − i i8 = 1 :

• Operations:
  

  Addition: (x + yi) + (a+bi) = (x +a) + (y + b)i Subtraction: (x + yi) − (a+bi) = (x − a) + (y − b)i Multiplication: (x + yi)(a+bi) = (xa − yb) + (xb + ya)i Conjugation: x + yi   = x − yi x − yi   = x + yi (x+yi)(x− yi) = x2 + y2

• Division:
  

  x+yi a+bi = x+yi a+bi · - a+bi - a+bi = x+yi a+bi · a− bi a− bi = (xa +yb) + (ya − xb)i a2 +b2 = ( xa +yb a2 +b2 ) + ( ya − xb a2 +b2 ) i