The writing of numbers above 2,880,000 up to and including 12,489,781 (the highest number found in the codices) involves the use of six places, or terms—kins, uinals, tuns, katuns, cycles, and great cycles—the last of which (the sixth place) has the numerical value 2,880,000. It will be remembered that some have held that the sixth place in the inscriptions contained only 13 units of the fifth place, or 1,872,000 units of the first place. In the codices, however, there are numerous calendric checks which prove conclusively that in so far as the codices are concerned the sixth place was composed of 20 units of the fifth place. For example, the number 5,832,060 is expressed as in figure [63], j. The 0 in the first place equals 0 (0×1); the 3 in the second place, 60 (3×20); the 0 in the third place, 0 (0×360); the 10 in the fourth place, 72,000 (10×7,200); the 0 in the fifth place, 0 (0×144,000); and the 2 in the sixth place, 5,760,000 (2×2,880,000). The sum of these six terms equals 5,832,060 (0+60+0+72,000+0+5,760,000). The highest number in the codices, as explained above, is 12,489,781, which is recorded on page 61 of the Dresden Codex. This number is expressed as in figure [63], k. The 1 in the first place equals 1 (1×1); the 15 in the second place, 300 (15×20); the 13 in the third place, 4,680 (13×360); the 14 in the fourth place, 100,800 (14×7,200); the 6 in the fifth place, 864,000 (6×144,000); and the 4 in the sixth place, 11,520,000 (4×2,880,000). The sum of these six products equals 12,489,781 (1+300+4,680+100,800+864,000+11,520,000).
It is clear that in numeration by position the order of the units could not be reversed as in the first method without seriously affecting their numerical values. This must be true, since in the second method the numerical values of the numerals depend entirely on their position—that is, on their distance above the bottom or first term. In the first method, the multiplicands—the period glyphs, each of which had a fixed numerical value—are always expressed[[86]] with their corresponding multipliers—the numerals 0 to 19, inclusive; in other words, the period glyphs themselves show whether the series is an ascending or a descending one. But in the second method the multiplicands are not expressed. Consequently, since there is nothing about a column of bar and dot numerals which in itself indicates whether the series is an ascending or a descending one, and since in numeration by position a fixed starting point is absolutely essential, in their second method the Maya were obliged not only to fix arbitrarily the direction of reading, as from bottom to top, but also to confine themselves exclusively to the presentation of one kind of series only—that is, ascending series. Only by means of these two arbitrary rules was confusion obviated in numeration by position.
However dissimilar these two methods of representing the numbers may appear at first sight, fundamentally they are the same, since both have as their basis the same vigesimal system of numeration. Indeed, it can not be too strongly emphasized that throughout the range of the Maya writings, codices, inscriptions, or Books of Chilam Balam[[87]] the several methods of counting time and recording events found in each are all derived from the same source, and all are expressions of the same numerical system.
That the student may better grasp the points of difference between the two methods they are here contrasted:
Table XII. COMPARISON OF THE TWO METHODS OF NUMERATION
| FIRST METHOD | SECOND METHOD |
| 1. Use confined almost exclusively to the inscriptions. | 1. Use confined exclusively to the codices. |
| 2. Numerals represented by both normal forms and head variants. | 2. Numerals represented by normal forms exclusively. |
| 3. Numbers expressed by using the numerals 0 to 19, inclusive, as multipliers with the period glyphs as multiplicands. | 3. Numbers expressed by using the numerals 0 to 19, inclusive, as multipliers in certain positions the fixed numerical values of which served as multiplicands. |
| 4. Numbers presented as ascending or descending series. | 4. Numbers presented as ascending series exclusively. |
| 5. Direction of reading either from bottom to top, or vice versa. | 5. Direction of reading from bottom to top exclusively. |
We have seen in the foregoing pages (1) how the Maya wrote their 20
numerals, and (2) how these numerals were used to express the higher numbers. The next question which concerns us is, How did they use these numbers in their calculations; or in other words, how was their arithmetic applied to their calendar? It may be said at the very outset in answer to this question, that in so far as known, numbers appear to have had but one use throughout the Maya texts, namely, to express the time elapsing between dates.[[88]] In the codices and the inscriptions alike all the numbers whose use is understood have been found to deal exclusively with the counting of time.