What does the rupee symbol stand for?
Why do most popular currency symbols have strikeouts in their symbols?
It is clear that the designers of recent currency symbols have their own reasons for including the slashes or "strikeouts" in the symbol. It is also clear that these elements evolved naturally in older currency symbols through the use of the abbreviation and shorthand.
It is more than likely that modern currency symbols use this older convention, originally developed naturally, which is abbreviated by a symbol to indicate the currency. This makes it much easier to distinguish between normal characters. As you said in your question, the practice has arguably become a pseudo-standard.
The euro is a relatively new currency and the symbol was designed and presented in 1996. The declaration of the design by the European Commission (focus on mine):
The inspiration for the € symbol itself came from the Greek epsilon (Є) - a reference to the cradle of European civilization - and the first letter of the word Europe, which was crossed by two parallel lines to "certify" the stability of the euro .
A number of explanations for the crossbeams are contained in Udaya Kumar's draft proposal for the Indian rupee sign:
The use of Shiro Rekha (the horizontal top line) in Devanagari script is unique in India. The Devanagari script is the only script that has letters on the top line rather than on a baseline. The symbol retains this unique and essential feature of our Indian script that cannot be found in any other script in the world.
The two horizontal lines with an equal negative space (imaginary space) between them create a foreground and background effect of three stripes (tricolor). The stripes subtly represent the tricolor of our Indian national flag waving above.
The horizontal lines also indicate the arithmetic sign "equal". [...] The arithmetic sign denotes the ratio of the comparison of currency values. The equal sign also means a balanced economy, our economy should be secure and stable forever.
The origins of the dollar symbol are a little more controversial than newer symbols. It originated when English Americans were trading with Spanish Americans in the 1770s.
The most credible theory is that it is derived from the abbreviation 'ps' for the peso. Another theory is that the symbol originated as '8' with a slash denoting eight parts or the Spanish dollar.
Theories about the making of the dollar using two vertical lines include the idea that it originated as the abbreviation "US" or as a representation of the Spanish coat of arms showing the pillars of Hercules with a banner wound between them.
In The American Accomptant, published 1979 (a great find from @Yorik!), You can clearly see the notation used for 'Federal Money'. 1 cent is expressed as. 1 cent superimposed on an over and 1 dollar superimposed on a double crossed over. This prompts one to question the other theories about the origin of the dollar symbols. However, the document does not include an explanation of the notation (as far as I can tell - I haven't read all 320 pages!)
The pound symbol is more obviously derived from an italic capital letter 'L', which represents 'scales', the basic unit of weight in the Roman Empire.
The slashes through the pound come from writing abbreviations that were in use in the Roman Empire, and the use of shorthand became common around this time.
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