column. For the sake of simplicity, the example chosen did not have farthings. Operations in practice During the 19th and 20th centuries various aids were developed to aid the manipulation of compound units, particularly in commercial applications. The most common aids were mechanical tills which were adapted in countries such as the United Kingdom to accommodate pounds, shillings, pence and farthings, and ready reckoners, which are books aimed at traders that catalogued the results of various routine calculations such as the percentages or multiples of various sums of money. One typical booklet that ran to 150 pages tabulated multiples "from one to ten thousand at the various prices from one farthing to one pound". The cumbersome nature of compound unit arithmetic has been recognized for many years—in 1586, the Flemish mathematician Simon Stevin published a small pamphlet called De Thiende ("the tenth") in which he declared the universal introduction of decimal coinage, measures, and weights to be merely a question of time. In the modern era, many conversion programs, such as that included in the Microsoft Windows 7 operating system calculator, display compound units in a reduced decimal format rather than using an expanded format (e.g., "2.5 ft" is displayed rather than ). Number theory Until the 19th century, number theory was a synonym of "arithmetic". The addressed problems were directly related to the basic operations and concerned primality, divisibility, and the solution of equations in integers, such as Fermat's Last Theorem. It appeared that most of these problems, although very elementary to state, are very difficult and may not be solved without very deep mathematics involving concepts and methods from many other branches of mathematics. This led to new branches of number theory such as analytic number theory, algebraic number theory, Diophantine geometry and arithmetic algebraic geometry. Wiles' proof of Fermat's Last Theorem is a typical example of the necessity of sophisticated methods, which go far beyond the classical methods of arithmetic, for solving problems that can be stated in elementary arithmetic. Arithmetic in education Primary education in mathematics often places a strong focus on algorithms for the arithmetic of natural numbers, integers, fractions, and decimals (using the decimal place-value system). This study is sometimes known as algorism. The difficulty and unmotivated appearance of these algorithms has long led educators to question this curriculum, advocating the early teaching of more central and intuitive mathematical ideas. One notable movement in