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lines rather than just one. Now we can go not only from the real line to anyone of the other two imaginary lines, but we can also go from one of the imaginary lines to the other imaginary line. The former is solved by our straightforward extension of the old idea of mUltiplying by i: To get from the real line to the new imaginary line, we just multiply by the new imaginary number j. The latter, however, is still unresolved. There are two key questions that we must answer in order to complete our three-dimensional generalization of the connection in two dimensions between multiplication of complex numbers and rotation: 1. By what number can we multiply a vector that lies on the i-imaginary line to rotate it to the j-imaginary line 2. Analogously, by what number can we multiply a vector that lies on the jimaginary line to rotate it to the i-imaginary line Let us concentrate on question 1 first. Clearly, this number cannot be real, for it would only change the length or invert the direction of the vector without making it leave the i-imaginary line. Neither can it be i nor j. It cannot be i, because ii = -1, which takes the vector to the real axis. It cannot be j because ji = j would imply that i = 1. So we are forced to assume the existence of a/ourth axis with a new unit number k, whose multiplication takes a vector on the i-imaginary line onto the j-imaginary line; that is, (C.7) ki = j. Being a fourth dimension, this new axis is at 90 degrees to the other axes. Then multiplication by k twice must mean a rotation of 180 degrees, which implies that

this forth axis is also an imaginary axis; that is,

Unbilled Costs by Job: This report is not really an A/R report so much as it is an invoicing reminder. It lists all expenses you have incurred on speci c jobs for which you have not charged the customer. Transaction List by Customer: This report lists all customers and their transactions with your company during the relevant time period identi ed on the report.

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The EFPIA also contends that revisions to the EU regulations are necessary for the pharmaceutical industry to remain viable in Europe The EFPIA suggested revisions, referred to as the G10 Medicines Recom mendations for Action, include lifting the price controls for over-thecounter medicines that are not reimbursed by national health systems Another recommendation of the EFPIA is to allow pharmaceutical rms to provide information on drugs directly to patients, sidestepping the ban on drug advertisement Europe s infrastructure for protecting intellectual property, although fundamentally sound, provides less protection to biotech innovations than does the United States For example, unlike United States patent law, Eu ropean patent law doesn t explicitly allow patenting of genes, cells, plants, or animals Furthermore, intellectual property protection agreements ex tended to pharmaceuticals and agricultural biotech products have been only sporadically enforced in Europe since the late 1990s.

(C.8)

Notice that in order for this scheme to make any sense, multiplication must be noncommutative8 ; otherwise, squaring (e.7), we would find k 2 = 1 instead of (C.8). At this point, the reader might be wondering whether we will find out next that in order to have a consistent scheme, we require yet another fifth, sixth, seventh dimension, and so on, never stopping. Well, part of the beauty of this is that four dimensions is just enough to make the entire scheme self-consistent, as we will show now. Multiplying (e.7) by k from the left and using (e.8), we find that kj = -i. (C.9)

On almost all QuickBooks reports, whenever the mouse pointer is a magnifying glass with a Z in the middle, you can use QuickZoom to zoom in on a detail. QuickZoom opens a separate report with more information about the one line or total that you clicked on.

So we have answered question (2) as well: Multiplication by minus this third imaginary number k takes a vector on the j-imaginary line to the i-imaginary line. But what about the new problem introduced by this fourth axis of rotating a vector that lies on one of the old imaginary lines to this new imaginary line By what number can we multiply such vector to achieve this rotation To answer these questions we first multiply (C.9) from the right by j; we find that

For example, the Trade Related Intellectual Property Agreement (TRIPS) agreement re.

(e.lO)

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In other words, multiplication by i takes a vector on the j-imaginary line to the new k-imaginary line. Next, we multiply (C.7) by i from the right and find that

(e.ll)

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This means that multiplication by - j takes a vector on the i-imaginary line to the k-imaginary line. Thus, with a real line and three orthogonal imaginary lines, we have a closed algebraic system whose multiplication rules can be summarized by the simple expression that Hamilton carved on the stone of Brougham Bridge9 :

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