Advanced Mathematics - Notes on Systems of Numeration

• Textbook Information: This is a mathematics textbook for Class IX, published by the Secondary Education Department, Government of Assam.

Authors and Reviewers:

• Authors: Dr. Kailash Goswami (Units 1, 2, 5), Dibyajyoti Mahanta (Units 3, 4, 6), Bijit Kumar Dey (Units 7, 8)

• Reviewers: Dr. Prabin Das, Bijit Kumar Dey

• Coordinator: Dr. Nityajyoti Kalita (Academic Officer, SEBA)

Table of Contents:

• Chapter 1: System of Numeration (Page 1)

• Chapter 2: Sets (Page 35)

• Chapter 3: Logarithm (Page 71)

• Chapter 4: Special Product and Factorization (Page 97)

• Chapter 5: Inequalities (Page 127)

• Chapter 6: Sequence and Series (Page 179)

• Chapter 7: Plane Geometry (Page 228)

• Chapter 8: Some Special Geometrical Constructions (Page 241)

Chapter 1: System of Numeration

1.0 What are the numbers?

• Numbers are concepts arising from questions like 'how many' or 'how much'.

• Numbers are abstract but denoted by names and symbols.

• Numbers represent a common concept hidden in equal groups, illustrated by { , , } and {A, B, C}, each containing /// elements, named "Three".

• Ascending order arrangement enables development of counting systems.

1.1 History of numerals and the system of numeration:

• The development of number concepts and counting systems is linked to human civilization, though undocumented.

• Ancient people compared groups via one-to-one relations or conceived numbers using tally marks, fingers, rope knots, pebbles, etc.

• Word numerals originated in the Vedic age, such as eka (1), dasa (10), satam (100), sahasram ($10^3$), ajutam ($10^4$), and in ancient Assam: chandra (one), netra/bhuj/pakha (two), Ram/bahni/guna (three), veda/yuga (four).

• Physical postures (hands, fingers) were used alongside word numerals; medieval Europeans used fingers for numbers up to ten thousand.

• Old Grouping System: In ancient Egypt (3400 B.C.), numbers were expressed by combining symbols. Examples:

  • 1: /

  • 10: ∩

  • 100: (Coil of rope)

  • 1000:

  • 10,000: (pointing finger)

  • 100,000: (fish like Aari, burbot)

  • 1,000,000: (a giant holding up a big mass)

  • Example: 12322 was represented as ∩∩ .

• Ancient Babylon: Used triangular symbols for ‘one’ and ‘ten’ and a subtraction symbol, using a place value system in 60 reaching 'sixty.'

• Mayans: Used 20 as a base but not consistently; positional digits were written vertically, with values from the third place as 18 × 20, 18 × $20^2$, 18 × $20^3$, etc., points for ones (.) and dashes for fives (–) were used to write numbers upto 20.

• Place Value System (Decimal System): The best system, invented in India, uses ten digits (0-9) to write any number.

1.2 Base of a number:

• Grouping helped overcome the problem of assigning unique names/symbols to every number.

• South Americans used hands (five fingers) to denote 'six'.

• In 'base of 5', numbers are counted in groups of five (signal with a hand).

• In base ten, numbers above ten were named as ten-one, ten-two, ten-three etc.

  • Eleven = One left over ten.

  • Twelve = One left over two.

  • Twenty = Two tens.

  • Hundred = Ten times ten.

• Aristotle considered ten as the base due to ten fingers.

• Various bases were used: Siberians (base 3), Germans (base 5), ancient Americans/Mayans (base 20), Babylonians (base 60).

• Base 60 is still used for time (seconds/minutes) and angles.

• 'Twelve’ has historical significance, possibly due to twelve lunar months in a year, also present in measures like foot, pound, dozen, gross.

• India has used base ten since ancient times; it is now used worldwide. For example, four thousand nine hundred forty eight will be: $4 × 10^3 + 9 × 10^2 + 4 × 10 + 8 × 1$ or 4948.

• The symbols for writing numbers are called digits (e.g., 0-9 for base ten).

• A number written with digits is called a numeral; for example, ‘4948’ represents ‘four thousand nine hundred forty eight’.

Numerical Value

• Digits in a number have different values based on their place (place-value or weight).

• The value is determined by the power of 10 in that place. -Thousand: $4 × 10^3$ (4000 units) -Hundred: $9 × 10^2$ (900 units)

  • Tens: $4 × 10^1$ (40 units)

  • Ones:$8 × 10^0$ (8 units)

• The base of a system determines the value of each digit.

• Example: 1101 in base 2 has the numerical value: $1 × 2^3 + 1× 2^2 + 0 × 2 + 1$ . Also, in base B, the numeral b1b2b3b4 according to positional value has the numerical expansion :$b4 + b3B + b2B^2 + b1B^3$, where b1, b2, b3, b4 are the digits.

General Numeral Expansion

• Numeral N = b1, b2, b3 ……bn in base B has the expansion bn + bn–1.B + bn–2B2 + ….. + b1Bn–1.

• Digits in base B system are 0, 1, 2,… B–1 (total B).

• Base 2 (Binary): digits 0, 1 (2 digits)

• Base 3: digits 0, 1, 2 (3 digits)

• Base 5: digits 0, 1, 2, 3, 4 (5 digits)

• Base 12: digits 0, 1, 2, 3 ….. 11 (12 digits)

• The number of distinct digits in a system is the base.

1.3 The Roman System

• Literal numeration system used in chapters of books, clock dials, etc.

• Seven symbols: I, V, X, L, C, D, and M.

  • I, X, C, M: 1, 10, 100, 1000 respectively.

  • V, L, D: 5, 50, 500 respectively.

Numbers

• Roman: I, II, III, IV, V, VI, VII, VIII, IX, X

• Decimal: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10

Principles of Writing Roman Numerals:

1. Addition and Subtraction: A smaller digit to the right is added; to the left, subtracted (e.g., VI = 6, IV = 4).

2. Between Larger Digits: A smaller digit between two larger digits is subtracted from the right digit (e.g., XIV = 14).

3. Repetition Limitations: V, L, and D are never repeated.

4. Subtraction Limitations: V, L, and D are not subtracted from any digit (e.g., not VC for 95).

5. Bar Notation: A bar over a digit increases its value 1000 times (e.g., V = 5000).

6. Subtraction Pairs: Only certain pairs allow subtraction:

   • (I, V), (I, X), (X, L), (X, C), (C, D), (C, M).

• Roman System is a grouping system without place-value, lacking zero.

• Digits I, X, C, M are generally not recommended to be repeated more than three times.

1.3.2 : Indo-Arabic numeral system :

All over the world this system is basically known as the Hindu-Arabic sys- tem. Already it is mentioned that the birthplace of this system is India. Through travels and trades the Arabians spread it to West Europe (exact time not known). So its name is Indo-Arabic.
The number symbols 0, 1, 2, 3….9 that we are using now-a-days are also called the Indo-Arabic numerals.
There are evidences that the forms of these symbols were engraved in the stone-pillars erected by the king Ashoka in around 250 B.C.
From this it is understood that the digits for writing the numbers today were originated in India itself.
However the zero, the decimal system and the idea of place-value originated only after few centuries from that. But even for that, the scholars have estimated that these came to light in full forms much before the 8th century.
Proofs for these are inscribed in ancient pillars, stone-scripts, caves, stone-walls and etc.
Along with the passes of time and the development of printing technology the old number symbols have taken the shape of today undergoing a lot of changes.
Likewise the Indian word ‘Sunya’ (i.e. empty or vacant) becomes ‘Sifr’ in Arab. Later on, in almost the thirteenth century it got the name ‘Cifer’ after getting a name ‘Cifra’ in Germany.
In fact both the words ‘zero’ and ‘cifer’ meaning Sunya are said to be originated from the Arabic word ‘Zephirum’.
Until tenth century the forms of the Indo-Arabic number-symbols were not as those of today. They have taken the shape of today coming across a lot of changes.
International Indo-Arbian digits The modern form of this is shown in the following table.
In course of time the number of Hindu-Arabian numerals become ten combining the symbol ‘0’ (Sunya) along with these nine.
These ten digits i.e., 1, 2, 3, ….9 are now the international digits which are in vogue mostly all over the world.
Depending on place in different parts of India these digits are used in slight varied forms
(see the table beolw).