I knew you were eager to read about this. 
Let us keep to the following basic hypothesis:
The Arabic numerals do not catch on in Europe for general usage.
At least not, say, until during the 16th century.
The are known as an academic and exotic curiosity, but never become standard in either mathematical theory or commercial practice.
(More precisely, I consider that Fibonacci (or an alternate of him) does not obtain a version of Al-Chowarismi's book, or does not take it as important as IOTL. In particular, he still writes a book on the resolution of linear and quadratic equations and the abacus, but not numerals and symbolic computation. In order to do what Fibonacci has done I think it suffices to have access to the scriptures of Euclid and Diophantos.)
So around 1250, we have an Occident where some educated people know how to solve quadratic equations; but to perform concrete numerical computations, they have to resort to their abacuses.
What are the consequences?
Most obviously, computing is harder, and it is likely that fewer people will learn it, or people will learn it to a lesser extent than IOTL.
But I have an additional suspect:
I suppose that the development of the balance sheet and of double-entry accounting (which roughly took place in the 14th and 15th century) would be seriously harmed.
While writing up inventory lists of goods is practically as old as scripture itself,
and the step towards inventories of debts and claims is a minor effort for the human desire for abstraction,
I think that the structure of accounting will not take the form it actually has since these days.
The basic target of traditional accounting is: make it easier to double-check - usually one checks by finding identical items in several places, or by adding items and checking sums.
But if you express everything in roman numerals, you cannot check sums in writing anyway!
So there is no need to develop that form in the first place.
Of course a 15th century balance, translated into Roman numerals, sheet would provide useful information to a merchant from my ATL.
To check or extract information, he would have to put the numbers into his abacus, compute there, and get back to the account; similar to what an accountant did in the 1970s and 1980s with an electronic calculator.
But I think for all these circumstances, the users of Roman numerals would develop only rudimentary forms of bookkeeping.
What's your opinion on that?
Let us keep to the following basic hypothesis:
The Arabic numerals do not catch on in Europe for general usage.
At least not, say, until during the 16th century.
The are known as an academic and exotic curiosity, but never become standard in either mathematical theory or commercial practice.
(More precisely, I consider that Fibonacci (or an alternate of him) does not obtain a version of Al-Chowarismi's book, or does not take it as important as IOTL. In particular, he still writes a book on the resolution of linear and quadratic equations and the abacus, but not numerals and symbolic computation. In order to do what Fibonacci has done I think it suffices to have access to the scriptures of Euclid and Diophantos.)
So around 1250, we have an Occident where some educated people know how to solve quadratic equations; but to perform concrete numerical computations, they have to resort to their abacuses.
What are the consequences?
Most obviously, computing is harder, and it is likely that fewer people will learn it, or people will learn it to a lesser extent than IOTL.
But I have an additional suspect:
I suppose that the development of the balance sheet and of double-entry accounting (which roughly took place in the 14th and 15th century) would be seriously harmed.
While writing up inventory lists of goods is practically as old as scripture itself,
and the step towards inventories of debts and claims is a minor effort for the human desire for abstraction,
I think that the structure of accounting will not take the form it actually has since these days.
The basic target of traditional accounting is: make it easier to double-check - usually one checks by finding identical items in several places, or by adding items and checking sums.
But if you express everything in roman numerals, you cannot check sums in writing anyway!
So there is no need to develop that form in the first place.
Of course a 15th century balance, translated into Roman numerals, sheet would provide useful information to a merchant from my ATL.
To check or extract information, he would have to put the numbers into his abacus, compute there, and get back to the account; similar to what an accountant did in the 1970s and 1980s with an electronic calculator.
But I think for all these circumstances, the users of Roman numerals would develop only rudimentary forms of bookkeeping.
What's your opinion on that?
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