In the United States, we are used to a decimal monetary system, the $/¢ system, in which the base unit (the dollar, $) is divided into 100 subunits (the cent, ¢). Canadian monetary history closely follows the United States for economic reasons, and many other nations have had decimal currencies for over a century. However, one currency that was famous for not having decimal subdivisions was the United Kingdom's pound. The pound was subdivided into the shilling, which was further subdivided into the penny.
In the £/s/d system, there are 12 pence (d) to a shilling (s), and 20 shillings to the pound (£). There have at times been ½d (the "halfpenny", pronounced "ha'p'ny") and even ¼d (the "farthing") subdivisions of the penny. These are nondecimal subdivisions of the pound; there were 960 farthings, 240 pence, or 20 shillings to the pound (none of these are powers of 10).
Several nations used the £/s/d monetary system, generally inherited from the United Kingdom but with their own local version of the pound. Throughout the 1950s and 1960s, especially as the use of computers for financial transactions increased, there was a movement among these nations to create a decimal currency, i.e., a currency with subdivisions that are powers of 10.
There were two basic implementations of this decimalization:
South Africa created the Rand (R) in 1961, New Zealand the Dollar ($) in 1967, and Australia the Dollar ($) in 1966. All of these followed the first method above, where a new subdivision (the "cent", c) has 100 to the new base unit, and a shilling was worth 10 cents. This made the new base unit worth half a pound.
South Africa created ½c coins during the transition, as well as 2½c coins (worth exactly 3d). There was no replacement for existing ¼d coins, nor for a coin worth 2/6 (exactly 25c).
In Australia, the minimum subdivision in decimal currency was 1c.
In the years since decimalization, some countries have stopped minting or officially withdrawn low-denomination coins; South Africa has ceased minting all coins smaller than 10c, and Australia has withdrawn all coins smaller than 5c. This does, however, allow for accounting (i.e., non-cash) in smaller subdivsions.
The United Kingdom and Ireland, in 1971, followed the second route, mostly because the United Kingdom did not want to eliminate the pound, as it was a well-respected currency. They then created a new subdivision, the "new penny" (p), with 100 new pence to the pound (and five new pence to the shilling).
The United Kingdom also created ½p coins during the transition. Like South Africa, they eliminated the 2/6 coin (exactly 12½p) as part of the transition; the farthing was eliminated a few years prior.
There had also been proposals in the UK, which were implemented elsewhere, to subdivide the pound into 1000 mils, or "new farthings". By 1971, the farthing had been eliminated in the UK, so the mil would have been regarded as insignificantly small.
Two key factors in the currency replacement were public-relations campaigns to educate the public and the continued acceptance of £/s/d currency. Both of these factors were crucial in not undermining the new currencies in the eyes of the public. Common refrains from government PR campaigns emphasized that only the names of the money were changing, not their values, and that computations would be much easier in the new system.
What I have done below is created a form to both convert among the £/s/d and decimal systems and perform addition and subtraction in £/s/d and decimal systems.
For each currency, there is a user-selectable minimum subdivision, so that any computations are rounded to the nearest minimum. For the decimal currencies, this could be zero (meaning arbitrary precision is allowed), ⅒ (using the mil model), ½, 1, 5, or 10. For £/s/d, since the values are not floating-point numbers, arbitrary precision is not possible.
South Africa had set rates for converting (whole) pence to cents, rounding to the nearest cent if a conversion wasn't exact (i.e., 1d = 1c, 2d = 2c, 4d = 3c, 5d = 4c, but 3d = 2½c). I have not, as yet, implemented this quirk of rounding.
Australia has a specific example in their advertisements of adding in £/s/d vs. adding in dollars and cents. That example is loaded above with the button.
Although it would have been much easier to implement addition and subtraction on £/s/d with floating-point arithmetic (which is easily simulated by converting to £/p with arbitrary precision, doing the arithmetic, and converting back), in the spirit of pre-decimal currency, I've implemented them strictly using integers and variable place value. Yes, it was a pain, but that's really the whole point of not using that system any longer.