Abstract
Number notations can influence the way numbers are handled in computations; however, the role of notation itself in mental processing has not been examined directly. From a mathematical point of view, it is believed that place-value number notation systems, such as the Indo-Arabic numbers, are superior to sign-value systems, such as the Roman numbers. However, sign-value notation might have sufficient efficiency; for example, sign-value notations were common in flourishing cultures, such as in ancient Egypt. Herein we compared artificial sign-value and place-value notations in simple numerical tasks. We found that, contrary to the dominant view, sign-value notation can be applied more easily than place-value notation for multi-power comparison and addition tasks. Our results are consistent with the popularity of sign-value notations that prevailed for centuries. To explain the notation effect, we propose a natural multi-power number representation based on the numerical representation of objects.
Highlights
Numbers can be represented in many ways
We found that comparison in placevalue notation was more erroneous and slower than comparison in sign-value notation; F (1,22) = 19.774, MSE = 0.001, p < 0.001 and F (1,22) = 45.393, MSE = 207,517, p < 0.001, respectively
Consistent with the comparison task, addition with place-value notation was slower than addition with sign-value notation [F (1,18) = 47.787, MSE = 23,833,720, p < 0.001], while the error rate did not differ significantly between the two notations (Figure 6)
Summary
Numbers can be represented in many ways. Indo-Arabic numbers and Roman numerals are only two well-known examples, while dozens of other notations were invented throughout the history of human culture (Ifrah, 1999; Chrisomalis, 2010). Many methods used two common structures to denote numbers: sign-value and place-value notations (Table 1). In sign-value notation, powers are denoted by symbols, and the quantity of that power is represented by repeating those symbols. In the Roman notation C, X, and I symbols represent 100, 10, and 1, respectively, and 20 is denoted with the repetition of symbol X twice: XX. In the ancient Egyptian hieroglyphic system a stroke (|) was used to denote number 1 and a heel bone (∩) to denote 10 In this notation only the powers of 10 could be symbols, and the numbers were denoted by the sum of these symbols: for example, 23 could be denoted as ∩∩|||. In a place-value system, the power is noted by the position in the string, while the quantity is represented by the symbol. In the IndoArabic 23, the tens are denoted on the second position from the right, and the quantity of tens is noted by the symbol 2
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