Congruences : Multiplicative Inverse

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  d) From the same multiplicative table we see that 2*8 =6,4*4 = 6 and 6*6 =6 So 8 is the inverse of 2 (and vice versa), 4 is the inverse of 4 ,and 6 is the inverse of 6.

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  (b) The statement is false, as the zero complex number 0+i0 does not possess the multiplicative inverse. For the complex number (x, y) the inverse is given by ((x/x^2 + y^2) - (y/x^2 + y^2)) which requires x^2 + y^2 ≠ 0.

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  Thus, is the left and right multiplicative inverse of . (c) We wish to show that the map given by is a ring monomorphism (1-1 homomorphism). To see this, we first note that for all nonzero quaternions and such that .