Informe número 44 - Numeral Systems

Numeral Systems

Various notational systems (number systems) have been or are being used in mathematics to represent the abstract quantities called numbers.

A number system is defined by the base it uses, the base being the number of different symbols required by the system to represent any of the infinite series of numbers. Thus, the decimal system in universal use today (except for computer application) requires ten different symbols, or digits, to represent numbers and is therefore a base-10 system.

Throughout history, many different number systems have been used; in fact, any whole number greater than 1 can be used as a base. Some cultures have used systems based on the numbers 3, 4, or 5.

The Babylonians used the sexagesimal system, based on the number 60, and the Romans used (for some purposes) the duodecimal system, based on the number 12.

The Mayas used the vigesimal system, based on the number 20.

The binary system, based on the number 2, was used by some tribes and, together with the system based on 8, is used today in computer systems.

The earliest forms of number notation were simply groups of straight lines, either vertical or horizontal, each line corresponding to the number. Such a system is inconvenient when dealing with large numbers, and as early as 3400 BC in Egypt and 3000 BC in Mesopotamia a special symbol was adopted for the number 10.

The addition of this second number symbol made it possible to express the number 11 with 2 instead of 11 individual symbols and the number 99 with 18 instead of 99 individual symbols. Later numeral systems introduced extra symbols for a number between 1 and 10, usually either 4 or 5, and additional symbols for numbers greater than 10.

In Babylonian cuneiform notation the numeral used for 1 was also used for 60 and for powers of 60; the value of the numeral was indicated by its context. The Egyptian hieroglyphic system used special symbols for 10, 100, 1000, and 10,000.

The ancient Greeks had two parallel systems of numerals. The earlier of these was based on the initial letters of the names of numbers: The number 5 was indicated by the letter Pi; 10 by the letter Delta; 100 by the antique form of the letter H; 1000 by the letter Chi; and 10,000 by the letter Mu.

The later system, which was first introduced about the 3rd century BC, employed all the letters of the Greek alphabet plus three letters borrowed from the Phoenician alphabet as number symbols. The first nine letters of the alphabet were used for the numbers 1 to 9, the second nine letters for the tens from 10 to 90, and the last nine letters for the hundreds from 100 to 900. Thousands were indicated by placing a bar to the left of the appropriate numeral, and tens of thousands by placing the appropriate letter over the letter M.

The late Greek system had the advantage that large numbers could be expressed with a minimum of symbols, but it had the disadvantage of requiring the user to memorize a total of 27 symbols.

The decimal system is believed to have originated in counting on the fingers, using both hands as the most convenient method. Arabic numerals are 1, 2, 3, 4, etc.

Roman numerals are I, II, III, IV, etc. Both the Arabic and the Roman symbols are believed to be related to this method: 1 or I is one finger, 2 or II is two fingers, and 3 or III is three fingers. The word "digit" is from the Latin digitus, meaning "finger".

Some of the symbols are less easily explained, but V seems to be the open hand, and X seems to be two open hands. The Roman system has no symbol for zero, and in the Arabic system zero is much more recent than the other symbols. The Maya, who were skilled in mathematics, had a symbol for zero.

Some languages show traces of reckoning by units other than ten, e.g., 8, 12, 20. In the case of 20, the toes as well as the fingers may have been used in counting.

Informe número 43 - Hebrew Numerals

Hebrew Numerals

The Hebrew Alphabet:

The Hebrew use a different alphabet than English. The Hebrew alphabet is often called the 'alefbet', because of its first two letters. Note that Hebrew is written from right to left, rather than left to right as in English, so "Alef" is the first letter of the 'alefbet' and "Tav" is the last. The picture below illustrates the 'alefbet', in Hebrew alphabetical order:

If you are familiar with Greek, you will no doubt notice substantial similarities in letter names and in the order of the alphabet.
Note that there are two versions of some letters. "Khaf", "Mem", "Nun", "Pe" and "Tzade" all are written differently when they appear at the end of a word than when they appear in the beginning or middle of the word. The version used at the end of a word is referred to as Final "Khaf", Final "Mem", etc. The version of the letter on the left is the final version. In all cases except Final "Mem", the final version has a long tail. The "Kh" and the "Ch" are pronounced as in German or Scottish, a throat clearing noise, not as the "ch" in "chair".

Numerical Values:

Each letter in the 'alefbet' has a numerical value. These values can be used to write numbers, as the Romans used some of their letters (I, V, X, L, C, M) to represent numbers.

"Alef" through "Yod" have the values 1 through 10. "Yod" through "Qof" have the values 10 through 100, counting by 10s. "Qof" through "Tav" have the values 100 through 400, counting by 100s. Final letters have the same value as their non-final counterparts.
The number 11 would be rendered "Yod-Alef", the number 12 would be "Yod-Bet", the number 21 would be "Kaf-Alef", the word Torah (Tav-Vav-Resh-He) has the numerical value 611, etc.
The only significant oddity in this pattern is the number 15, which if rendered as 10+5 would be a name of God, so it is normally written "Tet-Vav" (9+6).

The order of the letters is irrelevant to their value; letters are simply added to determine the total numerical value. The number 11 could be written as "Yod-Alef", "Alef-Yod", "Heh-Vav", "Dalet-Dalet-Gimel" or many other combinations of letters.

Coins dating and the conversion from hebrew calendar to christian calendar:

Modern Israel's coins carry hebrew dating formed from a combination of the 22 consonant letters of the hebrew alphabet and read from rigth to left. The hebrew new year falls in september or october by christian calendar reckoning.
The hebrew calendar is 3,760 years greater than the christian dating. For example, 1992 is 5752 according to the jewish calendar. So... the hebrew calendar dates back more that five milleniums (5,000).

In the israeli coins from 1981 full dates appear on the coins, but sometimes are assumed in the dating of coins, and in general as well. Thus, the year 5735 (1975) for example, apears as 735 (and not 5,735) with the two first caracters from the rigth indicating the number of years in hundreds ("tav"=400 + "shin"=300), The next is "lamed" (= 30) followed by a separation mark which has the appearance of double quotation marks, then "he" (= 5).

In total: 400+300+30+5=735. Very simple...:)

The separation mark, generally similar to a double quotation marks, serves the purpose of indicating that the letters form a number, not a hebrew word.

Informe número 42 - Arabic Numerals

Arabic Numerals

The common system of number notation in use in most parts of the world today is the Arabic system. This system was first developed by the Hindus and was in use in India in the 3rd century BC. At that time the numerals 1, 4, and 6 were written in substantially the same form used today. The Hindu numeral system was probably introduced into the Arab world about the 7th or 8th century AD. The first recorded use of the system in Europe was in AD 976.

The important innovation in the Arabic system was the use of positional notation, in which individual number symbols assume different values according to their position in the written numeral. Positional notation is made possible by the use of a symbol for zero. The symbol "0" makes it possible to differentiate between 11, 101, and 1001 without the use of additional symbols, and all numbers can be expressed in terms of ten symbols, the numerals from 1 to 9 plus 0. Positional notation also greatly simplifies all forms of written numerical calculation.

Countries in the Arabic sphere generallly date their coins to the Hejira date (AH=Anno Hegirae). Hejira is the name of the Mohammedan era, which commenced on July 16, 622 of the Christian era (AD=Anno Domini) when the prophet Mohammed fled from Mecca, escaping to Medina to avoid persecution from the Koreish tribesmen. Based on a lunar year, is 11 days shorter.

Follow the instructions and you'll see how easy is to indentify your coins, even if you don't know a single arabic word. You just have to memorize the following number and everything will be ok.

While Arabic letters are written from right-to-left, numbers in Arabic are written from left-to-right. For example: the following number is "120", not "021".

In the Arabic world the Hejira date may be indicated in Arabic numerals and the Christian date in Western numerals, or both dates represented in either form. Some coins carry dates according to both locally observed and Christian eras.

Some countries, such as Morocco, Algeria, and Tunisia, use the Western "Arabic" numerals, for example, 1984 instead of:

Informe número 41 - Dating modern Japanese coins

Dating modern Japanese coins

The Japanese counting system and number-related expressions are one of the difficult parts of learning Japanese. Numbers in Japanese can either be written in the usual "European" fashion (0...9), or using kanji. 0...9 is more often used when writing horizontally, kanji when writing vertically, although there are plenty of exceptions to both. When using kanji to write numbers, a positional system is employed. 25 is "two ten five", 1246 is "thousand two hundred four ten six", etc. Note that there is a character and word for 10,000 (man), but not for a million. To say a million, you have to say "one hundred ten-thousand".

The following number are the basic numbers used in dating Japanese coins:

Modern, post-WWII coins are quite easy to date. Japan does not use the calendar year of minting, but rather the year of reign of the emperor. To begin with, we must look at the emperors of this time frame, and understand their years of reign.

Hirohito, who used the regal name Showa, ruled from 1926 until 1989.
Akihito, who used the regal name Heisei, has ruled since 1989.

1926 would be Showa yr. 1,
1927 would be Showa yr. 2, etc.
1989 was Showa yr. 64.
1989 was also Heisei yr. 1, as he ascended the throne in that year.
1990 would be Heisei yr. 2, etc.

To read the date:

Find the legend (usually on the lower side of the reverse, under the English value on all except the 5 yen, which does not have an English value). This legend will contain either 4,5, or 6 Japanese characters. Reading the normal western style of left to right, the first 2 characters are the regal name of the emperor, either Showa or Heisei. The last character on the right is irrelevant (it means year). The characters in the middle are the important ones for dating.

Read the numbers left to right:

If there is only a single number, (between 2 and 10*), then that is the actual year of reign, and would mean that the coin is a Heisei issue. There is a special character for yr 1, which means "first".

If there are 2 numbers, and the first number is 10, then add the second number to it to get the year, such as 10 1 would be 11 (2001 is Heisei yr 13).

If there are 2 numbers, and the second number is 10, then multiply the first number by 10, such as 4 10 would be 40.

If there are 3 numbers, the second number should always be 10. Multiply the first number by 10, and add the third number, such as 4 10 5 would be 45 (4 x 10 + 5).

To determine the actual year of minting, it must first be determined under which emperor the coin was minted. Currently, that is not a problem since the years do not overlap. Current Japanese coins were begun in 1948, Showa yr 23.

If the coin is dated above 23, then it is a Showa issue. If the coin is a Showa issue, add the year of reign to 1926, and subtract 1. For example, Showa year 55 would be 1980: 1926 + 55 -1.

If it is dated under 11, then it is a Heisei issue. If the coin is a Heisei issue, add the year of reign to 1989, and subtract 1. For example, Heisei year 8 would be 1996: 1989 + 8 - 1.