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## Using Your Hands to Count to 60 - Babylonian Counting System

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Origin[ edit ] It is possible for people to count on their fingers to 12 using one hand only, with the thumb pointing to each finger bone on the four fingers in turn. A traditional counting system still in use in many regions of Asia works in this way, and could help to explain the occurrence of numeral systems based on 12 and 60 besides those based on 10, 20 and 5.

In this system, one hand usually right counts repeatedly to 12, displaying the number of iterations on the other usually leftuntil five dozens, i. Throughout their many centuries of use, which continues today for specialized topics such as time, angles, and astronomical coordinate systems, sexagesimal notations have always contained a strong undercurrent of decimal notation, such as in how sexagesimal digits are written.

Their use has also always included and continues to include inconsistencies in where and how various bases are to represent numbers even within a single text.

In ancient texts this shows up in the fact that sexagesimal is used most uniformly and consistently in mathematical tables of data. The early shekel in particular was one-sixtieth of a mana, [3] though the Greeks later coerced this relationship into the more base compatible ratio of a shekel being one-fiftieth of a mina.

Apart from mathematical tables, the inconsistencies in how numbers were represented within most texts extended all the way down to the most basic Cuneiform symbols used to represent numeric quantities.

But within the same texts in which these symbols were used, the number 10 was represented as a circle made by applying the round end of the style perpendicular to the clay, and a larger circle or "big 10" was used to represent Such multi-base numeric quantity symbols could be mixed with each other and with abbreviations, even within a single number.

The details and even the magnitudes implied since zero was not used consistently were idiomatic to the particular time periods, cultures, and quantities or concepts being represented.

While such context-dependent representations of numeric quantities are easy to critique in retrospect, in modern time we still have "dozens" of regularly used examples some quite "gross" of topic-dependent base mixing, including the particularly ironic recent innovation of adding decimal fractions to sexagesimal astronomical coordinates.

Instead, the cuneiform digits used ten as a sub-base in the fashion of a sign-value notation: The value of the digit was the sum of the values of its component parts: Numbers larger than 59 were indicated by multiple symbol blocks of this form in place value notation. Because there was no symbol for zero in Sumerian or early Babylonian numbering systems, it is not always immediately obvious how a number should be interpreted, and its true value must sometimes have been determined by its context.

Without context, this system was fairly ambiguous. For example, the symbols for 1 and 60 are identical. The same stem and branch repeat every 60 steps through this cycle.

This number has the particularly simple sexagesimal representation 1,0,0,0,0. Later scholars have invoked both Babylonian mathematics and music theory in an attempt to explain this passage. The sexagesimal number system continued to be frequently used by European astronomers for performing calculations as late as The Greeks limited their use of sexagesimal numbers to the fractional part of a number and employed a variety of markers to indicate a zero. By the seventeenth century it became common to denote the integer part of sexagesimal numbers by a superscripted zero, and the various fractional parts by one or more accent marks.

The same minute and second nomenclature is also used for units of time, and the modern notation for time with hours, minutes, and seconds written in decimal and separated from each other by colons may be interpreted as a form of sexagesimal notation.

In modern studies of ancient mathematics and astronomy it is customary to write sexagesimal numbers with each sexagesimal digit represented in standard decimal notation as a number from 0 to 59, and with each digit separated by a comma. When appropriate, the fractional part of the sexagesimal number is separated from the whole number part by a semicolon rather than a comma, although in many cases this distinction may not appear in the original historical document and must be taken as an interpretation of the text.

Sexagesimal degree Unlike most other numeral systems, sexagesimal is not used so much in modern times as a means for general computations, or in logic, but rather, it is used in measuring anglesgeographic coordinates, electronic navigation, and time. One hour of time is divided into 60 minutesand one minute is divided into 60 seconds. Thus, a measurement of time such as 3: Similarly, the practical unit of angular measure is the degreeof which there are six sixties in a circle.

There are 60 minutes of arc in a degree, and 60 arcseconds in a minute. In some usage systems, each position past the sexagesimal point was numbered, using Latin or French roots: To this day we call the second-order part of an hour or of a degree a "second".

However, the representation of these fractions as sexagesimal numbers does not depend on such an interpretation.

However, it does not provide support for numbers expressed in the sexagesimal 60 base. This lack of support is surprising given that the sexagesimal base is the foundation of modern time and angular measures. Within the sexagesimal base, time is simply expressed as hours with a radix point. The "minute" is a "minute hour" or a 60th of an hour.

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The "second" is a "second minute hour", a 60th of a minute hour, or a th of an hour. Likewise, the angular degree is so divided and named.

## How to Add Angles and Time Measures (Ice Cream Mathematics)

More pragmatically, the time and angles can be counted as integral "seconds". The higher order components, minutes and hours, will be read directly from the higher order digits. With sexagesimal numbers, the representation of time needs only three digits, one for the hour, one for the minute, and one for the second.

Furthermore, no punctuation is necessary. This economy contrasts markedly with conventional time notation, which requires six digits and two punctuation marks. Likewise, the representation of angles needs only four digits, two for the degree, one for the minute, and one for the second.

Again, no punctuation is necessary, and its economy contrasts with seven digits and three punctuation markers in conventional angular notation. Solution The straightforward approach to supporting an additional number base is to extend the existing mechanism for indicating hexadecimal numbers.

Unfortunately, the number of sexagesimal digits exceeds the number of decimal digits and lower-case letters. The obvious workaround is to have upper-case and lower-case letters represent different digits. While this workaround does meet the need for digits, it is inconsistent with the syntax for hexadecimal numbers. Additionally, the explicit base marker reduces the advantage in economy of sexagesimal representations. More importantly, mapping between digit value and the letter forms is less than obvious, entailing a significant cognitive burden.

So, extending the hexadecimal approach is not useful. The solution to this notational problem was well-solved by the Old Babylonian period, the first half of the second millennium BC [1] [2] [3] [4].

Rather than adopt an untested and invented number representation, we propose to represent sexagesimal numbers with historically proven notation, Babylonian cuneiform numbers.

With cuneiform digits, the base of the number is obviously sexagesimal, and additional marking of the base is unnecessary. This lack of a base marker preserves the economy of representation described above. Proposal We propose to use the ISO [6] [7] [8] standard to represent cuneiform numbers. ISO provide characters to represent cuneiform digits [9] [10] [11] [12] [13]. However, rather than provide one character per digit, it provides to characters per digit.

The first character encodes the number of tens .

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