The systems discussed in the previous three chapters are primarily cumulative, repeating signs within each power of the base to indicate addition. In contrast, the next two families – the Alphabetic and South Asian systems – consist mainly of ciphered systems, which use, at most, a single sign for any power to indicate its different multiples: 1 through 9, 10 through 90, 100 through 900, and so on, in the case of decimal systems. Ciphered numeral-phrases are thus much shorter than cumulative ones, but require their users to be familiar with many more signs. Alphabetic numerical notation systems generally use phonetic script-signs, in a specified order, to express numerical values, and thus mitigate the effort needed to memorize both script-signs and numeral-signs. Despite the name, the scripts in question are not always alphabets; some, such as the Hebrew and early Arabic, are abjads or consonantaries, expressing primarily consonantal phonemes, and one, the Ethiopic Ge'ez script, is an alphasyllabary or abugida, expressing consonant + vowel clusters.
Alphabetic systems were used as far north as England, Germany, and Russia and as far south as Ethiopia, and throughout Africa and the Middle East from Morocco eastward to Iran. Their history spans over two thousand years, from the development of the Greek numerals around 600 bc to the present, but in some cases important historical questions remain unresolved. While they are mostly ciphered-additive, they are not structurally identical. We can learn much more from these structural diff erences than from the paleographic curiosities of the signs of various systems.