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Complex Matters
(Alex and Charlie are reading a newspaper)
Alex: Look! The Great Whodunni, human calculator and mathemagician, is coming!
Charlie: I've heard he can find the 13th root of a 100-digit number in a minute!
Alex: I wish I was as good as him. I have to settle for finding the square root of 4-digit numbers.
Charlie: OK then. 4316.
Alex: 66. Now you try this. 49.
Charlie: That's obvious! It's 7! Give me a harder one.
Alex: Once you've solved that one.
Charlie: What do you mean? Don't tell me it's -7!
Alex: Yes it is. But if 72 = 49 and (-7)2 = 49 then what is the square root of -49? Or just what is the square root of -1, for that matter?
Charlie: Check it up on Castleran. It seems to know everything.
(They plug in Castleran)
Alex: WHAT IS THE SQUARE ROOT OF -1?
Castleran: Loading search features... The square root of -1 is an imaginary number commonly known as i. The number system containing i is known as complex numbers. It is possible to find the square roots of all complex numbers. There is no definite way to order numbers, such that for every distinct pair of numbers, one of them is greater. Multiplication is commutative, meaning that x times y is always the same as y times x.
Charlie: Interesting... But what is the root of 2i, for example?
Alex: I think it's i + 1. It seems to work. So the root of i is (i + 1) / √2. But then there is that rule that multiplication is commutative. What happens if I break that rule?
Alex: WHAT IF MULTIPLICATION IS NOT COMMUTATIVE?
Castleran: You must then use quaternions, also called Hamilton algebra since they were discovered by William Rowan Hamilton. It involves 1, i, j, and k. i2 = j2 = k2 = ijk = -1, as carved on Brougham Bridge by Hamilton when he discovered quaternions. ij = k, jk = i, and ki = j. However, since multiplication is not commutative, ji = -k, kj = -i, and ik = -j. Let H be a quaternion. You can also define a quaternion as a point in 4-space: H = (a, b, c, d) where it is equal to a + bi + cj + dk. The norm of a quaternion is √a2+b2+c2+d2 = ||H||. The conjugate of H is H* = a - bi - cj - dk. The reciprocal H-1 is then H*/ (||H||)2. Then division of quaternions X and Y can be defined. Left-division gives the value as Y-1X, whereas right-division gives it as XY-1. Thus you must always be clear about whether left- or right-division is used.
Charlie: HOW DO YOU EXTEND QUATERNIONS?
Castleran: For that, you will need octonions, discovered by Arthur Cayley. These use 1, i, j, k, l, m, n, and o. The quaternion rules apply but on = i, mo = j, nm = k, mi = nj = ok = l, il = kn = oj = m, jl = io = mk = n, and kl = jm = ni = o. Of course, inverting one of these pairs makes the product negative. This makes the multiplication table easy to memorise. Multiplication is not associative, so abc does not have to equal a(bc). As an example, jli = ni = o but j(li) = j(-m) = -jm = -o. However, it is alternative, meaning that any subalgebra of order 2 is associative. In other words, aab = a(ab) and abb = a(bb). It has no zero divisors, which means that if ab = 0, a = 0 or b = 0. If it is extended further, the sedenions are not alternative and can have zero divisors. The octonions are the most complex division algebra with the reals as a proper subalgebra.
Alex: THANKS, CASTLERAN!
(They turn off Castleran and hurry to the theatre)
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• Quaternions