The document provides an overview of number systems used by different civilizations and an introduction to basic number concepts: - It discusses ancient number systems including the Egyptian base-12 and Babylonian base-60 systems, as well as modern systems like binary and decimal. - Basic number types are defined such as integers, rational numbers, irrational numbers, and real numbers. Fractions and decimal expansions are also introduced. - Famous mathematicians who contributed to the study and development of number systems throughout history are acknowledged.

1. NUMBER SYSTEM The mysterious world of numbers… 3 1 0 2

2. Contents Acknowledgment Introduction Brief Introduction about numbers History of Number System Number System according to different civilizations Types of Numbers Decimal Expansion Of Number System Scientists related to Number System What is a number line? What is the difference between numeral and number? Word Alternatives

3. ACKNOWLEDGEMENT We would like to thank Teema madam for giving us an opportunity to express ourselves via mathematical projects. We are also thankful to Bharti madam, our computer teacher for letting us use the school computers for presentation and providing us with an e-mail ID. Also, we thank our friends for their ideas and co-operation they provided to us. We are grateful to all of them.

4. Introduction A number system defines a set of values used to represent a quantity. We talk about the number of people attending school, number of modules taken per student etc. Quantifying items and values in relation to each other is helpful for us to make sense of our environment. The study of numbers is not only related to computers. We apply numbers everyday, and knowing how numbers work, will give us an insight of how computers manipulate and store numbers.

5. A number is a mathematical object used in counting and measuring. It is used in counting and measuring. Numerals are often used for labels, for ordering serial numbers, and for codes like ISBNs. In mathematics, the definition of number has been extended over the years to include such numbers as zero, negative numbers , rational numbers, irrational numbers, and complex numbers .

6. Numbers were probably first used many thousands of years ago in commerce, and initially only whole numbers and perhaps rational numbers were needed. But already in Babylonian times, practical problems of geometry began to require square roots. Certain procedures which take one or more numbers as input and produce a number as output are called numerical operation.

7. The History Of Number System The number system with which we are most familiar is the decimal (base-10) system , but over time our ancestor have experimented with a wide range of alternatives, including duo-decimal (base-12) , vigesimal (base-20) , and sexagesimal (base-60) …

8. The Ancient Egyptians The Ancient Egyptians experimented with duo-decimal (base-12) system in which they counted finger-joints instead of finger . Each of our finger has three joints. In addition to their base-twelve system, the Egyptians also experimented with a sort –of-base-ten system. In this system , the number 1 through 9 were drawn using the appropriate number of vertical lines. A human hand palm was the way of counting used by the Egyptians…

9. The Ancient Babylonians Babylonians, were famous for their astrological observations and calculations, and used a sexagesimal (base-60) numbering system. In addition to using base sixty, the babylonians also made use of six and ten as sub-bases. The babylonians sexagesimal system which first appeared around 1900 to 1800 BC, is also credited with being the first known place-value of a particular digit depends on both the digit itself and its position within the number . This as an extremely important development, because – prior to place-value system – people were obliged to use different symbol to represent different power of a base.

10. Aztecs, Eskimos, And Indian Merchants. Other cultures such as the Aztecs, developed vigesimal (base-20) systems because they counted using both finger and toes. The Ainu of Japan and the Eskimos of Greenland are two of the peoples who make use of vigesimal systems of present day . Another system that is relatively easy to understand is quinary (base-5), which uses five digit : 0, 1, 2, 3, and 4. The system is particularly interesting , in that a quinary finger-counting scheme is still in use today by Indian merchant near Bombay . This allow them to perform calculations on one hand while serving their customers with the other. Aztecs were the ethnic group of Mexico

11. Number System according to different civilizations…

12. THE DECIMAL NUMBER SYSTEM The number system we use on day-to-day basis in the decimal system , which is based on ten digits: zero through nine. As the decimal system is based on ten digits, it is said to be base -10 or radix-10. Outside of specialized requirement such as computing , base-10 numbering system have been adopted almost universally. The decimal system with which we are fated is a place-value system, which means that the value of a particular digit depends both on the itself and on its position within the number.

13. MAYAN NUMBER SYSTEM This system is unique to our current decimal system, as our current decimal system uses base -10 whereas, the Mayan Number System uses base- 20. The Mayan system used a combination of two symbols. A dot (.) was used to represent the units and a dash (-) was used to represent five. The Mayan's wrote their numbers vertically as opposed to horizontally with the lowest denomination on the bottom. Several numbers according to Mayan Number System

14. BINARY NUMBER SYSTEM The binary numeral system , or base-2 number system , represents numeric values using two symbols, 0 and 1. More specifically, the usual base-2 system is a positional notation with a radix of 2. Owing to its straight forward implementation in digital electronic circuitry using logic gates, the binary system is used internally by all modern computers. Counting in binary is similar to counting in any other number system. Beginning with a single digit, counting proceeds through each symbol, in increasing order. Decimal counting uses the symbols 0 through 9 , while binary only uses the symbols 0 and 1 .

15. FRACTIONS AND ANCIENT EGYPT Ancient Egyptians had an understanding of fractions, however they did not write simple fractions as 3/5 or 4/9 because of restrictions in notation. The Egyptian scribe wrote fractions with the numerator of 1. They used the hieroglyph “an open mouth" above the number to indicate its reciprocal. The number 5, written , as a fraction 1/5 would be written as . . . There are some exceptions. There was a special hieroglyph for 2/3, , and some evidence that 3/4 also had a special hieroglyph. All other fractions were written as the sum of unit fractions. For example 3/8 was written as 1/4 + 1/8.

18. The real numbers include all of the measuring numbers . Real numbers are usually written using decimal numerals , in which a decimal point is placed to the right of the digit with place value one. It includes all types of numbers such as Integers, Whole numbers, Natural numbers, Rational number, Irrational numbers and etc… Let us see them in detail…

19. A rational number is a number that can be expressed as a fraction with an integer numerator and a non-zero natural number denominator. The symbol of the rational number is ‘Q’. It includes all types of numbers other than irrational numbers, i.e. it includes integers, whole number, natural numbers etc…

20. This is a type of a rational number. Fractions are written as two numbers, the numerator and the denominator ,with a dividing bar between them. In the fraction m/n ‘m’ represents equal parts, where ‘n’ equal parts of that size make up one whole. If the absolute value of m is greater than n ,then the absolute value of the fraction is greater than 1.Fractions can be greater than ,less than ,or equal to1 and can also be positive ,negative , or zero.

21. If a real number cannot be written as a fraction of two integers, i.e. it is not rational, it is called irrational numbers . A decimal that can be written as a fraction either ends(terminates)or forever repeats about which we will see in detail further. Real number pi (π) is an example of irrational. π=3.14159365358979……the number neither start repeating themselves or come in a specific pattern.

22. Integers are the number which includes positive and negative numbers. Negative numbers are numbers that are less than zero. They are opposite of positive numbers . Negative numbers are usually written with a negative sign(also called a minus sign)in front of the number they are opposite of .When the set of negative numbers is combined with the natural numbers zero, the result is the set of integer numbers , also called ‘Z’.

23. The most familiar numbers are the natural numbers or counting numbers: One, Two, Three and so on…. Traditionally, the sequence of natural numbers started with 1.However in the 19 th century, mathematicians started including 0 in the set of natural numbers. The mathematical symbol for the set of all natural numbers is ‘N’.

24. Moving to a greater level of abstraction, the real numbers can be extended to the complex numbers. This set of number arose historically, from trying to find closed formulas for the roots of cubic and quadratic polynomials. This led to expressions involving the square roots of negative numbers, eventually to the definition of a new number: the square root of negative one denoted by “I”. The complex numbers consist of all numbers of the form (a+bi) ; Where a and b are real numbers.

25. Other Types There are different kind of other numbers too. It includes hyper-real numbers, hyper-complex numbers, p-adic numbers, surreal numbers etc. These numbers are rarely used in our day-to-day life. Therefore, we need not know about them in detail.

26. Decimal Expansion of Numbers A decimal expansion of a number can be either, Terminating Non-terminating, non recurring Non terminating, recurring Let us see each of the following briefly…

27. Terminating decimal A decimal expansion in which the remainder becomes zero. For example, 54 9 = Terminating decimal is always a rational number. It can be written in p/q form. 54 9 6 54 0 As the remainder is zero, this is a terminating decimal

28. Non terminating non recurring “ Recurring” means “repeating”. In this form, when we divide a number by another, remainder never becomes zero, and also the number does not repeat themselves in any specific pattern. If a number is non terminating and non repeating, they are always classified as irrational number. For example, 0.10100100010000100000100.... does have a pattern, but it is not a fixed-length recurring pattern, so the number is irrational.

29. Non terminating, recurring In this form, when a number is divided by the other, the remainder never becomes zero, instead the numbers of the quotient start repeating themselves. Such numbers are classified as rational numbers. For example, 3.7250725072507250… In this example, “7250” have started repeating itself. Hence, it is a rational number. It can be expressed in p/q form.

30. Mathematicians related to Number System Euclid : Euclid was an ancient mathematician from Alexandria, who is best known for his major work, Elements. He told about the division lemma, according to which, A prime number that divides a product of two integers must divide one of the two integer. Euclid – The father of geometry

31. Mathematicians related to Number System R. Dedekind And G. Cantor : In 1870s two German mathematicians; Cantor and Dedekind, showed that : Corresponding to every real number, there is a point on the number line, and corresponding to every point on the number line, there exists a unique real number. R. Dedekind G. Cantor

32. Archimedes : He was a Greek mathematician. He was the first to compute the digits in the decimal expansion of π (pi). He showed that - 3.140845 < π < 3.142857 Mathematicians related to Number System Archimedes

33. A number line is a line with marks on it that are placed at equal distance apart. One mark on the number line is usually labeled zero and then each successive mark to the left or to the write of the zero represents a particular unit such as 1, or 0.5. It is a picture of a straight line. A number line

34. No, number and numerals are not same. Numerals are used to make numbers. It is a symbol used to represent a number. For example, the NUMERAL 4 is the name of NUMBER four. Numeral “7” Number 7

35. Word Alternatives Some numbers traditionally have words to express them, including the following: Pair, couple, brace: 2 Dozen: 12 Bakers dozen: 13 Score: 20 Gross: 144 Ream(new measure): 500 Great gross: 1728

37. Project made and Compiled by ~ Samarth Agrawal Yogesh Surve Arnab Das Arijit Sharma Ankita Sinha Ayushi Sur Nimisha Singh