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4) Substituting i 2 = j 2 = −1 into Eq. 5) which leaves the terms ij and ji undeﬁned. These stumped Hamilton for many years, but his tenacity won the day, and he eventually came up with an incredible idea which involved extending the triple into a 4-tuple: z = a + ib + jc + kd. 7) 40 Geometric algebra for computer graphics which expands to z1 z2 = a1 a2 + ia1 b2 + ja1 c2 + ka1 d2 + ib1 a2 + i 2 b1 b2 + ijb1 c2 + ikb1 d2 + jc1 a2 + jic1 b2 + j 2 c1 c2 + jkc1 d2 + kd1 a2 + kid1 b2 + kjd1 c2 + k 2 d1 d2 .
Paul Nahin’s book An Imaginary Tale: The Story of −1  records that the Norwegian surveyor, Caspar Wessel [1745–1818], was another person to discover the geometric interpretation of complex numbers. His paper entitled “On the Analytic Representation of Direction: An Attempt ” was announced in 1797 and in 1799 was published in a local Danish journal with a small international circulation, which ensured that his idea remained hidden for almost 100 years! Wessel’s paper was discovered in 1895 by an antiquarian and its importance recognized by the Danish mathematician, Christian Juel [1855–1935], but it was too late — the Swiss-born writer, Jean-Robert Argand [1768–1822], had also thought of the same idea in 1806, and it is Argand’s name that is associated with the complex plane rather than Wessel’s.
But what happens if we make x a complex number in Eq. 53)? e ix = lim n→∞ 1+ ix i2x 2 i3x 3 inx n + + + ... + 1! 2! 3! n! 56) 20 Geometric algebra for computer graphics which simpliﬁes to ix x2 ix 3 x4 ix 5 − − + + .... 1! 2! 3! 4! 5! Collecting up real and imaginary terms: e ix = 1 + e ix = 1 − x4 x6 x3 x5 x2 x7 + − ... + i x − + − ... 2! 4! 6! 3! 5! 7! 58) which we recognize as the sin and cos functions, therefore, e ix = cos x + i sin x. 60) which is the beautiful relationship discovered by Euler.