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More about Numbers

Overview

iPhone calculator application

A number is an abstract mathematical concept representing a quantity. It is used in counting. Numbers have been used from ancient times, first in the form of tally marks — scratches on wood or bone, and then as more abstract systems. There are several ways of expressing numbers in numeric systems. Some of them are not in use today.

Different Ways of Representing Numbers

It is believed by some researchers that the concept of number was created independently in different regions. The originally written representations of numbers through symbols evolved independently, but once trade across countries and continents became widespread, people learned and borrowed from each other and the number systems currently in use were created through collective knowledge.

Hindu-Arabic Numerals

The Hindu-Arabic numeral system is one of the most widely used in the world today. It was originally developed in India and improved by the Persian and Arab mathematicians. In the Middle Ages, it spread to the Western world through commerce, to replace the Roman numeral system. It was further modified and widely adopted around the world because of European trade and colonization. It is a base-10 system, meaning that it is based on multiples of ten and that it uses ten symbols to represent all numbers.

Ten is a common number to use for counting because people have ten fingers, and body parts were often used for counting historically. Even today people learning to count or who want to illustrate a point about counting in conversation often use fingers. Some cultures also used toes, spaces between fingers, and knuckles for counting. Curiously, numbers are represented by “digits,” the same word that is used to refer to fingers and toes in English and many other languages.

An inscription in Latin and with Roman numerals on Admiralty Arch in London. It reads ANNO : DECIMO : EDWARDI : SEPTIMI : REGIS : VICTORIÆ : REGINÆ : CIVES : GRATISSIMI : MDCCCCX : (In the tenth year of King Edward VII, to Queen Victoria, from most grateful citizens, 1910).

Roman

Roman numerals were used in the Roman Empire and Europe until the 14th century. They are still used today in some contexts, for example on clocks, to represent the hours. Roman numerals are based on seven numbers written with the letters of the Latin alphabet:

I 1 V 5 X 10 L 50 C 100 D 500 M 1000

The order is important in the Roman system because a greater number followed by the smaller means that the two need to be added, but a smaller number in front of the larger one means that the smaller number is subtracted from the larger. For example, XI is 11, but IX is 9. The subtraction rule is not universal, it only works for these numbers: IV, IX, XL, XC, CD, and CM. In some cases the subtraction rules are not used, and numerals are written in succession instead.

Systems in Other Cultures

People in many geographic areas had systems of representing numbers, similar to the Roman or the Hindu-Arabic ones. For example, some Slavic people used the Cyrillic alphabet to represent numbers such as 1 to 9, multiples of 10, and multiples of 100, with special symbols for greater numbers, as well as symbols to differentiate the numerals from the letters. The Hebrew number system uses the Hebrew alphabet to represent numbers from one to ten, multiples of ten, 100, 200, 300, and 400. The rest of the numbers are represented as multiples or sums. The Greek number system is also similar.

Some cultures use simpler representations, like the Babylonian system, which has only two cuneiform symbols, for one (somewhat resembling the letter “T”) and for ten (slightly similar to the letter “C”). So for example 32 would be written (using the proper symbols) as CCCTT. The Egyptian system was very similar, except that there were additional symbols for zero, one hundred, one thousand, ten thousand, one hundred thousand, and one million, as well as special notations for fractions. Numbers in the Mayan culture had symbols for zero, one, and five, with special notation for numbers above nineteen.

Unary numeral system. Tally marks in various cultures

Unary

The unary system represents each number with the same number of symbols as its value. These symbols are usually the same, therefore if 1 is represented with A, then 5 would be represented as AAAAA. When children learn to count, their teachers often use this system to help create a link between a concrete, easy to understand the system, and a more abstract representation of numbers. This system is also sometimes used in games and other simple calculations. Different countries may use different types of representation for this. For example, when keeping score of the winning teams or counting items or days, people in the Western world and some other regions would often write four vertical lines, then cross them with a fifth horizontal line, and repeat the process. For example, in part A) in the picture the person counting reached four, crossed it out, then reached four again, crossed it out, and continued to write tally marks until they added up to twelve. People who use or have historically used Chinese characters in their writing systems, for example in China, Japan, and Korea use a certain Chinese character with five strokes to do the same. In part B) in the picture the person counts to five, completing the character, and then starts a new character, continuing the count to seven. The stroke order is pre-determined, as shown in the picture. The unary system is also used in computer science.

An arithmometer that uses the decimal system and a microprocessor chip that uses the binary system

Positional System

Positional systems work with a base. For example, in base-10 we have the following:

The first position is for numbers from zero to nine, that is, the number in the first position has to be multiplied by ten to the power of zero. The number in the second position is multiplied by ten to the power of one. The number in the third position is multiplied by ten to the power of two, and so on, until the numbers in all positions are exhausted.

To arrive at the final value of the number represented one needs to add all the values at each position. This is a convenient way of representing numbers because it allows one to work with numbers relatively large in value, without using large space to write them down.

Example: 3102 = 3 × 10³ + 1 × 10² + 0 × 10¹ + 2 × 10⁰

Binary

The binary numeral system is widely used in mathematics and computer science. It is based on two characters, “0” and “1” to represent all possible numbers. In other words, it is a base-2 system. Numbers are represented as follows: 0=0, 1=1, and from 2 the principle of addition is used. Addition in base-2 is similar to addition in base-10. To increment a number by one:

An artistic representation of binary numbers

If the number ends in a zero, the last zero is replaced by one: e.g. 100 (4) + 1 (1) = 101 (5). Here the base-10 numbers are used in brackets for comparison.

If the number ends with a one but is not all ones, the first zero from the right is replaced by one, while all the ones following it on the right become zeros: 1011 (11) + 1 (1) = 1100.

If the original number is all ones, then they are all changed to zeros, and a one is added at the front: 111 (7) + 1 (1) = 1000 (8).

To add two numbers, they are aligned under each other, and for each place, 0+0 produces 0, 1+0 produces 1, and 1+1 produces 10, where 0 is put in that position, and the 1 is carried over to the next position. For example:

11111 (31)

+1011 (11)

———————————

101010 (42)

In this case, working from right to left:

1+1 produces 0, with one carried over

1+1+1 produces 1, with one carried over

1+1 produces 0, with one carried over

1+1+1 produces 1, with one carried over

1+1 produces 10

So, putting this together, we get 101010.

Subtraction works using the same principle, except instead of carrying over ones, we “borrow” ones. Multiplication is also similar to base-10 multiplication. Multiplying by 0 results in a 0, while multiplying 1 by 1 is 1. So, for example:

101 (5)

×10 (2)

———————————

000

101

———————————

1010 (10)

Division and calculation of square roots are also very similar to base-10.

Number Classification

All numbers can be divided into subsets. Some of the subsets below partly overlap.

Debt is a negative number

Negative Numbers

Negative numbers are numbers that represent a negative value. A minus sign is placed in front of them. For example, if person A has no money and owes 5 dollars to person B, then person A has −5 dollars. Here –5 is a negative number.

Rational Numbers

Rational numbers are numbers that can be expressed as fractions where a denominator is a natural number that is not zero, and the numerator is an integer. For example, both 3/4 and −10/5 (the same as −2) are rational numbers.

Natural Numbers

Natural numbers are those that are positive (including 0), and are not fractions, for example 7 or 86,766,575,675,456.

Integers

Integers include zero, negative, and positive numbers that are not fractions. Examples include −65 and 11,223.

Complex Numbers

Complex numbers are all numbers that are a sum of one real number and a product of another real number and the square root of a negative one.

Prime Numbers

Prime numbers are natural numbers greater than one that produce an integer only when divided by one or by itself. Some examples are 3, 5, and 11. 257,885,161−1 is the largest known prime number as of winter 2013. It contains 17,425,170 digits. Prime numbers are used in public-key cryptography, a system of encoding data, often used in online secure data exchange, such as in online banking.

Interesting Facts about Numbers

Chinese anti-fraud numbers

Anti-Fraud Numerals

To prevent fraud when writing numbers in business and commerce, the Chinese language uses special complex characters that are difficult to forge by adding extra strokes. This is done because the commonly used Chinese characters for numbers are too simple and it is easy to modify their value by adding strokes.

Modern Counting in Commerce

Some languages in countries where base-10 is currently used still reflect that other number systems were common in the past. For example, English has a special word for twelve, “dozen” — currently used mainly for counting eggs, baked goods, wine, and flowers. Khmer has special words based on the ancient base-20 system, to count fruit.

Numeral Grouping

Both in China and Japan, the Hindu-Arabic numeral system is adopted, but large numbers are grouped by 10,000, and this is reflected in the language. In English, for example, there is a word for 1000, and one specifies how many thousands there are, up to 999,999. Then follows the word million, representing 1,000,000. In Japanese, there is a word for 10,000, and after that the incrementation continues to 99,999,999, followed by a special word for 100,000,000.

Unlucky Numbers

Leonardo da Vinci. The Last Supper. Church of Holy Mary of Grace (Santa Maria delle Grazie), Milan, Italy.

In the Western tradition, the number 13 is considered to be unlucky. Many believe that this is carried from the Judeo-Christian tradition, where thirteen was the number of Jesus Christ’s disciples during the last supper, after which the thirteenth disciple, Judas, betrayed Jesus. There was also a superstition among the Vikings that one at a thirteen people gathering will die the next year.

In Russia and many of the former Soviet countries, all even numbers are considered unlucky. Possibly this tradition originated from the belief that even numbers are complete, stable and static, unmoving, and thus not alive. Odd numbers, on the other hand, represent change, motion, an entity that needs completion and progression, and life. According to this belief, it is considered bad luck to give an even number of flowers to living people — these numbers are usually reserved for funerals.

In Chinese, Japanese, and Korean-speaking countries number 4 is considered to be unlucky because it is pronounced the same way as “death.” In some instances, all numbers that have a four in them are considered unlucky. For example, a building may not have floors 4, 14, and 24. In China number 7 is also unlucky because it represents the spiritual world and ghosts. The seventh month in the Chinese calendar is referred to as the “ghost month,” when the connection between the worlds of the living and the spirits is open. In Japan, the other unlucky number is 9, which has the same pronunciation as “suffering.”

In Italy 17 is an unlucky number because when its Roman representation “XVII” is rearranged, it reads VIXI or “vixi”, translated from Latin as “I have lived.” This implies that one’s life is over, and refers to death.

666 is another unlucky number, called the “Number of the Beast” in the Bible. It is sometimes believed that this number is 616, but 666 is more common. It refers to the Antichrist or Satan. Its origins are debatable but some scholars believe that 666 is the transliteration into Hebrew and 616 — into Latin of the name of Emperor Nero, who is associated with persecutions of Christians and with tyrannical and bloody reign. Nero is also thought by some to be the arsonist during the great fire in Rome, although his involvement is debated by historians.

In Afghanistan, especially in and around Kabul 39 is considered to be a cursed or a shameful number, linked with prostitution. It is connected with a story about a pimp, who had the number 39 as part of his license plate and his apartment number. Some accuse the authorities and the organized crime units of spreading this superstition to profit from buying and selling cars with the “offending” license plates. The superstition is so strong that people taunt and otherwise abuse those who have 39 in their license plate, apartment, or phone number. One such rumored instance of taunting resulted in a tragedy, when a parliamentary candidate, placed 39 on the ballot was taunted by the drivers passing by, and this caused a traffic accident. The bodyguards, fearing for his life, shot dead two of the people involved. These claims are denied by the bodyguards and the parliamentarian, and no charges have been laid, so it is unclear whether this is an urban legend or a real occurrence, but it is talked about in Kabul.

References

This article was written by Kateryna Yuri