Consider Y = x2.25 Y = x2.25 we raise the equation to the power 4 (y + iz)4 = x9 (y2 z2 + 2yzi)2 = x9 – so (y2 z2)2 4y2z2 + i4yz(y2 z2) = x9 – – – The x values are real numbers < 0 so i4yz(y2 z2) = 0 – Which implies that either z = 0 or y = 0 or y = ± z If z = 0, the equation just becomes the original y4 = x9 or y = x2.25 If y = 0, the equation becomes z4 = x9 or z = x2.25 If y = ± z, the equation becomes -4z4 = x9 so x = -4(1/9) z(4/9) Consider Y = x2.75 Similarly for Y = x2.75 we raise the equation to the power 4 (y + iz)4 = x11 (y2 z2 + 2yzi)2 = x11 – so (y2 z2)2 4y2z2 + i4yz(y2 z2) = x11 – – – The x values are real numbers < 0 so i4yz(y2 z2) = 0 – Which implies that either z = 0 or y = 0 or y = ± z (N.B. In this case I chose y = z and so y = t because (-1)2.25 = 0.7 0 .7i – – – and squaring the equation introduces extra solutions.) If z = 0, the equation just becomes the original y4 = x11 or y = x2.25 If y = 0, the equation becomes z4 = x11 so x = z(4/11) If y = ± z, the equation becomes -4z4 = x11so x = -4(1/11) z(4/11) in the correct sequence: Y = x2.25 Y = x2.5 Y = x2.75 Y = x3
Consider Y = x2.25 Y = x2.25 we raise the equation to the power 4 (y + iz)4 = x9 (y2 – z2 + 2yzi)2 = x9 so (y2 – z2)2 – 4y2z2 + i4yz(y2 – z2) = x9 The x values are real numbers < 0 so i4yz(y2 – z2) = 0 Which implies that either z = 0 or y = 0 or y = ± z If z = 0, the equation just becomes the original y4 = x9 or y = x2.25 If y = 0, the equation becomes z4 = x9 or z = x2.25 If y = ± z, the equation becomes -4z4 = x9 so x = -4(1/9) z(4/9) Consider Y = x2.75 Similarly for Y = x2.75 we raise the equation to the power 4 (y + iz)4 = x11 (y2 – z2 + 2yzi)2 = x11 so (y2 – z2)2 – 4y2z2 + i4yz(y2 – z2) = x11 The x values are real numbers < 0 so i4yz(y2 – z2) = 0 Which implies that either z = 0 or y = 0 or y = ± z (N.B. In this case I chose y = – z and so y = – t because (-1)2.25 = 0.7 – 0 .7i and squaring the equation introduces extra solutions.) If z = 0, the equation just becomes the original y4 = x11 or y = x2.25 If y = 0, the equation becomes z4 = x11 so x = z(4/11) If y = ± z, the equation becomes -4z4 = x11so x = -4(1/11) z(4/11) in the correct sequence: Y = x2.25 Y = x2.5 Y = x2.75 Y = x3
0_the_theory_of_phantoms_of_y_=_x^n.docx