Mathematics in India until 650 CE

2018 ◽  
pp. 82-94 ◽  
Author(s):  
Toke Lindegaard Knudsen
Keyword(s):  
Pythagorean Theorem ◽  
Place Value ◽  
Square Root ◽  
Approximate Methods ◽  
Decimal System ◽  
Ancient India ◽  
Squaring The Circle ◽  
Great Size ◽  
Mathematical Astronomy ◽  
Solar Eclipses

The chapter studies the mathematics of ancient India, from the Vedic (Indo-European) period, ca1500–500 bce, and later. They used a decimal system to express numbers, often of great size. The texts called Śulba-sūtras (Rules of the Cord, ca 800–200 bce) prescribe detailed rituals involving geometrical arrangements of bricks forming altars, using pegs and ropes. These texts present a system of mathematics involving the full application of the Pythagorean theorem, a rather precise approximation to the square root of 2, and approximate methods for squaring the circle and circling the square. Around 500 bce, the place-value decimal system was created, including the zero. Moreover, the bhūta-saṃkhyā system allowed place-value numbers to be represented using a sequence of fixed words, e.g., “eyes” always meant “two.” The analysis of possible metrical forms led to the development of simple combinatorics, including a form of what we would call Pascal’s triangle. Jain mathematics speculated about types of infinity. Mathematical astronomy, from ca 400 ce, included computation of mean and true planetary positions, and computation of lunar and solar eclipses. The chapter concludes with brief surveys of notable Indian mathematicians.

In the classroom: Binary can be F-U-N

1963 ◽  
Vol 10 (6) ◽  
pp. 354-355
Author(s):  
Marion E. Ochsenhirt ◽  
Mary M. Wedemeyer
Keyword(s):  
Binary System ◽  
Place Value ◽  
Seventh Grade ◽  
Educational Value ◽  
Decimal System ◽  
The North ◽  
New Situation ◽  
Modern Mathematics

The seventh-grade students of the North Hills Joint Schools have found that there is fun as well as educational value in using the binary system. As all teachers of modern mathematics know, one of the main reasons for teaching the binary system is that the pattern for place value in this system is identical to that of the traditional decimal system. Developing the pattern in an entirely new situation gives the student a better understanding of the decimal system.

The Effects of Studying Decimal and Nondecimal Numeration Systems on Mathematical Understanding, Retention, and Transfer

1970 ◽  
Vol 1 (3) ◽  
pp. 162-172
Author(s):  
Richard C. Diedrich ◽  
Vincent J. Glennon
Keyword(s):  
Multivariate Analysis ◽  
Mathematical Understanding ◽  
Analysis Of Covariance ◽  
Value Systems ◽  
Place Value ◽  
Decimal System ◽  
Statistical Procedures ◽  
System 2 ◽  
Grade Groups ◽  
Numeration Systems

What are the effects of studying 1, 3, or 5 place-value systems of numeration? Each of 3 experimental 4th grade groups received 30 minutes of instruction for 9 consecutive school days. A 4th group served as control. Statistical procedures included multivariate analysis of covariance. The results suggest: for the group under consideration (1) a study of the decimal system alone tends to be just as effective as a study of 3 or 5 systems in promoting understanding of the decimal system; (2) a study of either 3 or 5 place-value systems of numeration appears to be more effective than a study of the decimal system alone in promoting understanding of the general principles of place-value systems of numeration.

Arithmetic in Maya Numerals

1971 ◽  
Vol 36 (1) ◽  
pp. 54-63 ◽  
Author(s):  
W. French Anderson
Keyword(s):  
Evolutionary Development ◽  
Place Value ◽  
Square Root ◽  
Advantages And Disadvantages ◽  
Root Extraction ◽  
Two Systems

AbstractArithmetical procedures, including addition, subtraction, multiplication, division, and square root extraction, are demonstrated using the Maya numerals. All procedures can be carried out efficiently. The Maya system is relatively unique in that it combines properties of both place-value and non-place-value numerical systems. The Babylonian system, discussed briefly, also utilizes a mixture of properties from the two systems. In order to take into account the unique hybrid characteristics of these two systems, as well as the subtractive principle of the Roman numerals, we here define a third category of numerical systems designated as mixed-place-value in type. The three types of numerical systems are compared and the advantages and disadvantages of each are mentioned. The evolutionary development of numerical systems in relation to the mathematical needs of societies is discussed.

The Explosion

2016 ◽  
Author(s):  
Joseph Mazur
Keyword(s):  
Coordinate System ◽  
Rene Descartes ◽  
Pythagorean Theorem ◽  
Square Root ◽  
Cartesian Coordinate System ◽  
René Descartes ◽  
Algebraic Operations ◽  
Standard Rule ◽  
Cartesian Coordinate ◽  
And Algebra

This chapter focuses on the Cartesian coordinate system and how it provides a link between geometry and algebra. In Geometria, René Descartes introduced a new idea for notation, a rule: beginning letters of the alphabet were to be reserved for fixed known quantities and letters from p onward were to represent variables or unknowns that could take on a succession of values. To this day, this division of the alphabet at p remains the loose standard rule. The chapter also considers the Pythagorean theorem as a way of finding the distance between any two points in space; how the algebraic operations of addition, subtraction, multiplication, division, and extraction of a square root could actually be performed; and who came up with the idea of the vinculum.

The Indian Gift

2016 ◽  
Author(s):  
Joseph Mazur
Keyword(s):  
Number System ◽  
Decimal Place ◽  
Place Value ◽  
Ancient India ◽  
Finger Counting ◽  
History Of ◽  
Indian Mathematics

This chapter discusses the legacy of Indian mathematics. With very few archaeological clues, the origins of the Indian numbers must rely on a small wealth of writing that survives almost exclusively in the form of stone inscriptions. Some of those stone epigraphs used decimal place-value numerals, providing some evidence that ancient India was familiar with a kind of place-value numerical system. Some letter combinations of the Sanskrit words for numbers probably contributed suggestive shapes early in the morphographic history of our current script. The chapter first considers the Brahmi number system before turning to modern Hindu-Arabic numerals. It also examines how the Western system of numerals with zero came to be by focusing on finger counting, the dust boards, and the abacus.

An additive numeral system related to place value

1965 ◽  
Vol 12 (3) ◽  
pp. 212-215
Author(s):  
Goldie E. Vitt
Keyword(s):  
Young Children ◽  
Learning Experiences ◽  
Place Value ◽  
Decimal System ◽  
Conventional Methods

The concepts relating to place value in our decimal system involve what are probably some of the most difficult learning experiences that the primary child encounters. While many young children are able to tell “how many ones,” ”how many tens,” “how many hundreds” in a numeral, the teacher sometimes has reason to suspect that this skill is a superficial one, and that many conventional methods of presenting material intended to teach place value do not promote real internalization of the concepts involved or provide realistic mean for evaluation of learnings.

Mathematics in Egypt

2018 ◽  
pp. 48-60
Author(s):  
Annette Imhausen
Keyword(s):  
Number System ◽  
Place Value ◽  
High Status ◽  
Decimal System ◽  
Unit Fractions ◽  
Essential Tool ◽  
Mathematical Texts

This chapter studies Egyptian mathematics, an essential tool to administer resources, from the invention of the script and number system, ca 3000 bce. Egyptian writing was a tool restricted to the elites. Their number system was a decimal system without positional (place-value) notation. Their concept of unit fractions, as the inverses of integers, was fundamentally different from ours; other fractional values were expressed as a sum of unit fractions. The few surviving mathematical texts contain a collection of problems or tables to aid in calculations, or actual worked calculations, or a mixture of these. Some scribes had high status, based explicitly on the numerate activity of calculating taxes owed and work produced. Later Egyptian mathematics displays Mesopotamian influences.

scholarly journals A comparative study of calculation of Solar Eclipse in ancient and present period

2017 ◽  
Vol 5 (01) ◽  
Author(s):  
Priti Bajpai
Keyword(s):  
Comparative Study ◽  
Solar Eclipse ◽  
Ancient India ◽  
Modern Times ◽  
Naked Eye ◽  
Highly Sensitive ◽  
Solar Eclipses ◽  
Advance Mathematical

The phenomenon of eclipses has been of tremendous interest to mankind. The solar eclipse, which will take place on 21st August, 2017 has kindled the curiosity towards this celestial occurrence once again. In modern times all the observations are gathered through highly sensitive telescopes. For calculations the most advance mathematical software’s and programs are used. It is of interest then, that in Ancient India, what the formulae which were used were and how were the calculations and predictions done by observing the positions of planets through naked eye and in the absence of the modern day equipments. This paper is a modest attempt to give a comparative study of the method given by Aryabhatta for calculations of Stars, Solar Eclipses and the methods used now.

The Egyptian Number System

2016 ◽  
Author(s):  
Annette Imhausen
Keyword(s):  
Number System ◽  
Decimal Place ◽  
Place Value ◽  
Decimal System ◽  
Absolute Value ◽  
Ancient Egyptian ◽  
System P ◽  
The Absolute ◽  
The Individual

This chapter describes the ancient Egyptian number system. The system can be described in modern terminology as a decimal system without positional (place-value) notation. The basis of the number system was 10 (hence decimal system), but unlike our decimal place-value notation using the ten numerics 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9, in which the absolute value is determined by its position within the number (e.g., in the number 125, the absolute value of 1 is 1 × 102, the absolute value of 2 is 2 × 101, and the absolute value of 5 is 5 × 100), the Egyptian system used individual symbols for each power of 10. Although there is no information about the choice of the individual signs for the respective values, some of them seem plausible choices. The most basic, the simple stroke to represent a unit, is used not only in Egypt but also in a variety of other cultures, possibly originating from marks on a tally stick.

Introducing the binary system in grades four to six

1973 ◽  
Vol 20 (3) ◽  
pp. 182-183
Author(s):  
Jan Unenge
Keyword(s):  
Binary System ◽  
Elementary Mathematics ◽  
Place Value ◽  
Decimal System ◽  
Mathematics Program

The teaching of place value is one of the most important—and possibly the most interesting—parts of the elementary mathematics program. You can show the children how you get another name for the number seven if you count in fives. And children will better understand what happens when you go from nine to ten and from ninetynine to one hundred in the decimal system if you discuss different bases with them. I have used the following technique for teaching the binary system in grades four to six.

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