Let the i-j plane be the complex plane, with basic operations ij=−i ji=−j i^2 =−1 j^2 =−1 z=r+xi+yj w=s+pi+qj zw=rs+pri+qrj+sxi−px−qxi +syj−pyj−qy = (rs−px−qy)+(pr+sx−qx)i +(qr+sy−py)j wz=rs+sxi+syj+pri−px−pyi +qrj−qxj−qy = (rs−px−qy)+(pr+sx−py)i +(qr+sy−qx)j (little difference..) ⇒ zw−wz= (py−qx)(i−j) = 0 if py−qx= 0 z^2 = (r^2 −x^2 −y^2 )+(2rx−xy)i +(2ry−xy)j And if y=0 ⇒ z=r+xi z^2 =(r^2 −x^2 )+2rxi either way! (z^2 )z=r(r^2 −x^2 )+2r^2 xi +x(r^2 −x^2 )i−2rx^2 ⇒ (z^2 )z= r(r^2 −3x^2 )+x(3r^2 −x^2 )i z(z^2 )= r(r^2 −x^2 )+x(r^2 −x^2 )i +2r^2 xi−2rx^2 = r(r^2 −3x^2 )+x(3r^2 −x^2 )i so z(z^2 )=(z^2 )z = z^3 (so far so good) .....


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Question Number 140784 by ajfour last updated on 12/May/21

$L e t t h e i - j p l a n e b e t h e c o m p l e x$ $p l a n e, w i t h b a s i c o p e r a t i o n s$ $i j = - i$ $j i = - j$ $i^{2} = - 1$ $j^{2} = - 1$ $z = r + x i + y j w = s + p i + q j$ $z w = r s + p r i + q r j + s x i - p x - q x i$ $+ s y j - p y j - q y$ $= (r s - p x - q y) + (p r + s x - q x) i$ $+ (q r + s y - p y) j$ $w z = r s + s x i + s y j + p r i - p x - p y i$ $+ q r j - q x j - q y$ $= (r s - p x - q y) + (p r + s x - p y) i$ $+ (q r + s y - q x) j$ $(l i t t l e d i f f e r e n c e . .)$ $\Rightarrow z w - w z = (p y - q x) (i - j)$ $= 0 i f p y - q x = 0$ $z^{2} = (r^{2} - x^{2} - y^{2}) + (2 r x - x y) i$ $+ (2 r y - x y) j$ $A n d i f y = 0 \Rightarrow z = r + x i$ $z^{2} = (r^{2} - x^{2}) + 2 r x i$ $e i t h e r w a y!$ $(z^{2}) z = r (r^{2} - x^{2}) + 2 r^{2} x i$ $+ x (r^{2} - x^{2}) i - 2 {r x}^{2}$ $\Rightarrow$ $(z^{2}) z = r (r^{2} - 3 x^{2}) + x (3 r^{2} - x^{2}) i$ $z (z^{2}) = r (r^{2} - x^{2}) + x (r^{2} - x^{2}) i$ $+ 2 r^{2} x i - 2 {r x}^{2}$ $= r (r^{2} - 3 x^{2}) + x (3 r^{2} - x^{2}) i$ $s o z (z^{2}) = (z^{2}) z = z^{3} (s o f a r s o g o o d)$ $. . . . .$
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