““The appropriate numbering system The appropriate numbering system to represent a value depends on the to represent a value depends on the application. There are advantages application. There are advantages

and disadvantages to each numbering and disadvantages to each numbering system”system”

Numeration System Numeration System Past and Present ..Past and Present ..The symbols for writing numbers are called numerals and the methods for calculating are called algorithms. Taken together, any particular system of numerals and algorithms is called a numeration system. The earliest means of recording numbers consisted of a set of tallies – marks on stone, stones in a bag, notches in a stick – one-for-one each item being counted. Example: Often such a stick was split in half with one half going to the debtor and the other half to the creditor.

It is still common practice today to keep count by making tallies or marks with the minor but useful refinement of marking off tallies in group of five. Thus,

IIII IIII IIII IIII III is much easier to read as twenty-three

than IIIIIIIIIIIIIIIIIIIIIIIIBut such system for recording numbers were much too simplistic for large numbers and for calculating. Numbers actually originated in the history of the Hindus of India. They have changed greatly over the centuries, passing first to the Arabs of the Middle East and finally to Europe in the Middle Ages.

As early as 3400 B.C., the Egyptians developed a system for recording numbers on stone tablets using hieroglyphics. Used number systems other than base 10, like the duodecimal (base 12) in which they counted fingers-joints instead of fingers. Divided their days into twenty-four periods, which is, in turn, why we have twenty-four hours in a day.The Egyptians had symbols for one and the first few powers of ten, and then combined symbols to represent other numbers.

The Ancient Egyptians ..

The Ancient Babylonians ..

Famous for their astrological observations and calculationsUsed a sexagesimal (base 60) numbering systemMade use of six and ten as sub-bases.First to develop the concept of cipher position or place value, in representing larger numbers

The Babylonians Numbers..

Where then is 60?

Even the Chinese ..

One 一 (Yī)

Two 二 (Èr)

Three 三(Sān)

Four 四 (Sì)

Five 五(Wǔ)

Six 六 (Liù)

Seven 七 (Qī)

Eight 八 (Bā)

Nine 九(Jiǔ)

Ten 十 (Shí)

Hundred 百 (Bǎi)

Thousand 千 (Qiān)

10 Thousands 万 (Wàn)

Million 百万100 Millions 亿 (Yì)

Billion 万亿

The Aztecs, Eskimos

and Indian Merchants .. Aztecs - Developed vigesimal (base 20) systems because they counted using both fingers and toes.The Ainu of Japan and the Eskimos of Greenland are the two people who make use of vigesimal systems to the present day.Relatively easy to understand is quinary (base 5) which uses five digits: 0,1,2,3,4. This system is particularly interesting, in that a quinary finger counting scheme is still in use today by Indian merchants near Bombay. This allows them to perform calculations on one hand while serving their customers with the other.

The Greeks and the Hebrews ..

Used letters for numbers.Every letter in the Greek or Hebrew alphabet corresponded to a different number.For example, in Greek, the letter Alpha corresponded to 1 and the letter Theta corresponded to 9.

The Greek Numbers..

The Romans ..

Devised a system that was an important improvement over hash marks Roman Numeral SystemAlso used letters to represent numbers

DISADVANTAGES: Very difficult to apply in large numbers; there is no provision for representing the number zero or negative numbers which are very important concepts in mathematics.

The Arabs..Transmitted to us the Hindu Numeral SystemThe use of Arabic Numerals in Europe is attributed to the Italian mathematician Fibonacci.In 1202, he published a book called Liber Acci, which taught Arabic Numerals and Algebra and strongly advocated the use of Arabic numerals in society.The use of the Hindu-Arabic numerals is now the prevalent number system throughout the world. With the development of the printing press in the 16th century, the numerals have become standardized, and this only increased with the development of computers.

Decimal NumerationSystem ..Used on daily basis and has 10 digitsLatin decem “ten”Symbols used to represent these digits arrived in Europe around the 13th century from the Arabs.Said to be base 10 or radix 10 where the term radix comes from the Latin word meaning “root”

Decimals also refer to decimal fractions, either separately or in contrast to vulgar fractionsIt has been also adopted almost universally = due to the fact that we happen to have 10 fingersA place value system – the value of a particular digit depends both on the digit itself and on its position within the number

Almost any other base would be as good as or better than base 10. This is because, for many arithmetic operations, the use of a base that is wholly divisible by many numbers especially the smaller values , conveys certain advantages.Some mathematician would ideally prefer a system with a prime number as a base: for example, seven or eleven

Nature of ten fingers, place value system, good choice for interfacing with non-engineers

ADVANTAGES:

DISADVANTAGES:

Mathematical terminology and concept use is identical to the base that is use for counting everyday numbersValues for currency, demography, economics, for quantitative backup in science, etc.

APPLICATIONS

The decimal numeration system uses ten ciphers,

and place-weights that are multiples of ten.

What if we made a numeration What if we made a numeration system with the same strategy system with the same strategy

of weighted places, except with of weighted places, except with fewer or more ciphers? fewer or more ciphers?

The binary numeration system is such a system ..

Instead of ten different cipher symbols, with each weight constant being ten times the one before it, we only have two cipher symbols, and each weight constant is twice as much as the one before it. The two allowable cipher symbols for the binary system numeration are “1” and “0”, and these ciphers are arranged right=to=left in doubling values of weight.Binary is referred to as “base 2” numeration. Each cipher position in binary is called bit

Why use Binary System ?

The primary reason that the binary numeration system is used in modern electronic computers is because of the ease of representing two cipher states electronically.Binary numeration also lends itself well to the storage and retrieval of numerical information: on magnetic tape, optical disks, or a variety of other media types.

The Real Number System ..

The real number system evolved over time by expanding the notion of

what we mean by the word “number.”

Natural Numbers ..

At first, “number” meant something you could count, like how many sheep a farmer owns. These are called the natural numbers, or sometimes the counting numbers.1,2,3,4,5… The use of three dots at the end of the list is a common mathematical notation to indicate that the list keeps going forever.

Whole Numbers ..

At some point, the idea of “zero” came to be considered as a number. If the farmer does not have any sheep, then the number of sheep that the farmer owns is zero. We call the set of natural numbers plus the number zero the whole numbers.0,1,2,3,4,5…

About ZERO .. After more than a, 500 years of potentially inaccurate calculations, the Babylonians finally began the special sign for zero.Many historians believe that this sign, which first around 300 BC, was one of the most significant inventions in the history of mathematics.However, the Babylonians only used their symbol as place holder and they didn’t have the concept of zero as an actual value.The concept of zero first appeared in India around 600 AD.

Integers ..

Even more abstract than zero is the idea of negative numbers. If, in addition to not having any sheep, the farmer owes someone 3 sheep, you could say that the number of sheep that the farmer owns is negative 3. It took longer for the idea of negative numbers to be accepted, but eventually they came to be seen as something we could call “numbers.” The expanded set of numbers that we get by including negative versions of the counting numbers is called the integers.Whole numbers plus negative numbers

About Negative Numbers ..

First appeared in India around 600 ADIn 18th century, the great Swiss mathematician Leonhard Euler believed that negative numbers were greater than infinity, and it was common practice to ignore any negative results returned by equations on the assumptions that they were meaningless

Rational Numbers ..While it is unlikely that a farmer owns a fractional number of sheep, many other things in real life are measured in fractions, like a half-cup of sugar. If we add fractions to the set of integers, we get the set of rational numbers. All numbers of the form , where a and b are integers (but b cannot be zero)Rational numbers include what we usually call fractions Notice that the word “rational” contains the word “ratio,” which should remind you of fractions.

Irrational Numbers ..Now it might seem as though the set of rational numbers would cover every possible case, but that is not so. There are numbers that cannot be expressed as a fraction, and these numbers are called irrational because they are not rational.Cannot be expressed as a ratio of integers. As decimals they never repeat or terminate (rationals always do one or the other)

Real Number ..

Rationals + Irrationals All points on the number line Or all possible distances on the number line

An Ordered Set ..

The real numbers have the property that they are ordered, which means that given any two different numbers we can always say that one is greater or less than the other. A more formal way of saying this is: For any two real numbers a and b, one and only one of the following three statements is true:

1.     a is less than b, (expressed as a < b)2.     a is equal to b, (expressed as a = b)3.     a is greater than b, (expressed as a > b)

The Number Line .. Every real number corresponds to a distance on the number line, starting at the center (zero). Negative numbers represent distances to the left of zero, and positive numbers are distances to the right. The arrows on the end indicate that it keeps going forever in both directions.

Absolute Value .. When we want to talk about how “large” a number is without regard as to whether it is positive or negative, we use the absolute value function. The absolute value of a number is the distance from that number to the origin (zero) on the number line. That distance is always given as a non-negative number.In short:If a number is positive (or zero), the absolute value function does nothing to it If a number is negative, the absolute value function makes it positive

Group 3

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