8 ms·
i, j, and k come from Hamilton's Quaternions. Euler picked i for the imaginary unit. Hamilton needed two more square roots of -1: ii = jj = kk = ijk = -1
by mnemonicsloth 16y ago
i, j, and k come from Hamilton's Quaternions.
Euler picked i for the imaginary unit.
Hamilton needed two more square roots of -1:
ii = jj = kk = ijk = -1
Hamilton was really influential, and quaternions were the standard way to do 3D analysis before 1900. By then, mathematicians were used to thinking of (ijk) as a matched set.
Vector calculus replaced quaternionic analysis in the 1890s because it was a better way to write Maxwell's equations. But people tended to write vector quantities as like this: (3i-2j+k) instead of: (3,-2,1). So (ijk) became the standard basis vectors in R^3.
Finally, physicists started using group theory to describe symmetries in systems of differential equations. So (ijk) started to connote "vectors that get swapped around by permutation groups," then drifted towards "index-like things that take on all possible values in some specified set," which is basically what they mean in a for loop.
- ced 16y agoWhat variable did mathematicians use for summation before physicists started using group theory to describe symmetries in systems of differential equations? Isn't that a fairly recent development? My search for an old paper with a bloody summation symbol found nothing. I did, however, find this OT gem about Newton, written in 1848: From the thick darkness of the middle ages man's struggling spirit emerged as in new birth; breaking out of the iron control of that period; growing strong and confident in the tug and din of succeeding conflict and revolution it bounded forwards and upwards with resistless vigour to the investigation of physical and moral truth; ascending height after height; sweeping afar over the earth, penetrating afar up into the heavens; increasing in endeavour, enlarging in endowment; every where boldly, earnestly out stretching, till, in the Author of the Principia, one arose, who, grasping the master key of the universe and treading its celestial paths, opened up to the human intellect the stupendous realities of the material world, and in the unrolling of its harmonies, gave to the human heart a new song to the goodness, wisdom, and majesty of the all creating, all sustaining, all perfect God.
- J3L2404 16y agoDo you mean to imply that the convention for loop variables came from on high?
- robinhouston 16y agoIf you want an old paper with a summation symbol, you could do worse than the book where the symbol was first used, by Euler: http://www.math.dartmouth.edu/~euler/docs/originals/E212sec1ch1.pdf http://www.math.dartmouth.edu/~euler/docs/originals/E212sec1... (He uses x as the index variable there.)
- gruseom 16y agoWhat a great comment. So little historical knowledge circulates online.