Imaginary Power

This blog has included many articles on imperial economic and political power, a reality that must be understood in order to make sense of the world. As something to consider as a contrast over the holiday period, following is a note on imaginary power taken from the work of Swiss genius, Leonhard Euler.

Firstly, start with Euler's famous formula:

e^ix = cosx + i sinx

For x = π, this produces the even more striking result

e^iπ + 1 = 0

(Nobody has constructed a simpler or more concise - some would say beautiful - equation, one that includes so many of the fundamental operations and core elements of mathematics: unity, zero, e, π, i, equality, addition, exponentiation and multiplication)

But, for x = π/2, Euler's formula can be used to answer the question: what is i, the so-called imaginary unit (the square root of -1), to the power of i?

In other words, what is iⁱ? This is a particularly weird question if translated as 'an imaginary number to the power of an imaginary number', stranger perhaps than asking: what colour is 5.6 and is it different from the flavour of 6.5?

Using x = π/2, Euler's formula becomes

e^iπ/2    = cos π/2 + i sin π/2

          = 0 + i

so

e ^iπ/2 = i

in which case, raising both sides of the equation to the power i gives

iⁱ        = (e^iπ/2)ⁱ

          = e^{i.i. π/2}

          = e^-π/2

So, it turns out that

          iⁱ = 0.2079 (to 4 decimal places)

and it is a real (and a positive) number.

But that is not the end of the process, because i = e^iπ/2 is just one of an infinite number of values for the equation

e^ix = cosx + i sinx

since sin π/2 , sin 5π/2 , sin 9π/2 , etc, are also equal to 1, while the respective cosx values are all equal to 0 (note these are measured in terms of radians, and that π/2 radians is 90 degrees).

So the general equation for i is:

i = e^i(4n+1)π/2

For example, when

          n= 0,             iⁱ = 0.2079…

          n = -1           iⁱ = 111.3178…

          n = 1            iⁱ = 3.3882… x 10^-4

etc, for positive and negative integer values of n

This odd result follows from the nature of i. I think that it is best to think of i as a mathematical operator that, when applied to itself squared, as  (i x i) or i², results in minus 1, rather than in the unnecessarily mysterious form of an 'imaginary' unit or number.

Perhaps surprisingly, Euler has not been considered to be one of the founders of Rastafarianism, despite his understanding of i to the power of i.

Tony Norfield, 14 December 2013


Note added on 4 July 2014: there are several ways of deriving the above results, some of which can be found on Wikipedia and in other Internet sources. However, the method used above was taken from a book by Y E O Adrian, The Pleasures of Pi, e and Other Interesting Numbers, World Scientific Publishing, 2006, pp. 205-207.

Sitting on the Dock of the Bay

(This is a guest article)

That millions of workers in Asia on minimal wages produce a huge amount of consumer goods for the West is such a well-established and undisputed fact that it does not require much further comment. These goods are often so cheap that their price astonishes us. Of course, once we consider the economics of the lives of the people who produce these goods, there is no mystery in this. Yet we rarely ponder such issues for long, because the inevitable conclusion can only be that living standards in the West are supported by the toil and sweat of millions of others.

But the systematic exploitation of what used to be called the ‘Third World’ - and is now fast becoming the First World in terms of industrial organisation and manufacturing competence - is not restricted to production. Every aspect of this production and trade is parasitical and hugely exploitative. Consider, for example, maritime shipping - the main way these goods get from the hands of distant toiling masses into the hands of consumers in the rich countries.

Almost all goods produced in Asia for the West are transported in large container ships. Airfreight accounts for less than 7% of the total. Despite the West’s clear technical superiority, not a single developed western nation builds container ships. They are all built in Asia, mainly in South Korea. So, it is not only the goods, but also the ships they travel in that are produced in Asia. What little shipbuilding of any kind remains in the West survives only because of the most stringent protective barriers or due to social policy protecting employment (the disparity in wages is so great that a global free market in shipbuilding would wipe out what is left of this protected industry).

The exploitative and parasitical nature of Western consumption even determines the design of container ships because of the unequal loading on the forward and return journeys. Ships stacked up with containers on the outward East-West journey can be the equivalent of a 10-storey building above the water line. A ship is stable when a proportion of it is below the water line, but a ship built to handle such huge capacities would be unstable in rough seas when unladen. Because we give Asia practically no goods in return, container ships have to return empty. So, to maintain stability the ships have to be built with huge ballast tanks to take on seawater. The ships are designed on the assumption that the West takes but does not give and that this will continue to be the case throughout the working life of the vessel!

A large container ship has a crew of around 30. The captain is almost always a very-well-paid European. The crew is invariably staffed by ratings from extremely poor countries that command extremely poor wages (mostly from the Philippines, Bangladesh and Malaysia). Were merchant seamen paid decent wages these would be reflected in a higher price for the goods transported.

Considering that 80% of world trade is from ‘East to West’, and that all container ships are built in the Far East, it would not be unreasonable to expect Far Eastern operators to dominate world maritime business. Not a bit of it. For 120 years very powerful Western companies, backed by monopoly practices of linked banks and insurance companies, and supported by port authority regulations, ensure that a whopping 90% of world shipping is controlled by a dozen Western cartels. Only 8% of shipping is in the hands of Far Eastern operators, the people who build the ships, who sail them, who make the goods transported in them, and who dispose of the ships at the end of their working life. Cartel shipping fees represent another transfer of income from Asia to the West.

A container ship has a working life of around 20 years. The cost of disposal is also a cost that must be reflected in the price of goods transported. Ship breaking is a very labour intensive and extremely dangerous activity. There are no breaker’s yards catering for large ships in the West. They are all located in countries where wages are extremely low (Bangladesh, Pakistan), where health and safety legislation is non-existent or not enforced, and where the compensation for death and injury at work is a pittance. Another sign of how cheap goods are bought on the exploitation of others.

O Redding, 13 December 2013