Number Systems - 9th

1.1 Introduction

Mathematics is the language of the universe. Every calculation, measurement, and pattern around us is governed by numbers. From counting objects to performing complex scientific computations, numbers are at the core of human knowledge.
The system through which we represent and work with numbers is called the Number System.
India has played a remarkable role in shaping the modern number system. The concept of zero (0) and the decimal (place value) system were first discovered and refined here. Without these contributions, modern mathematics, science, and technology could not exist.
An ancient Sanskrit verse beautifully describes the glory of mathematics:

“यथा शिखा मयूराणां नागानां मणयो यथा। तद्वद्वेदाङ्गशास्त्राणां गणितं मूर्ध्नि स्थितम्॥”
(As the crest jewel is to the peacock, as the gem is to the cobra, so mathematics is the crown of all sciences.)

1.2 What is a Number System?

A Number System is a way of writing and expressing numbers. Just as languages like Hindi or English allow us to communicate with words, a number system allows us to communicate with quantities.
For example:

• The symbol 3 represents three objects.
• The number 21 is different from 12, because of the place value system.

1.3 Types of Numbers

Numbers are classified into different types for better understanding and usage.

1. Natural Numbers (N)

• Counting numbers starting from 1, 2, 3, …
• Denoted by N.
• Example: 5, 18, 234.

2. Whole Numbers (W)

• All natural numbers along with zero.
• Denoted by W.
• Example: 0, 7, 100.

3. Integers (Z)

• Whole numbers and their negatives.
• Denoted by Z.
• Example: –3, –2, –1, 0, 1, 2, 3.

4. Rational Numbers (Q)

• Numbers that can be expressed as p/q, where p and q are integers and q ≠ 0.
• Examples: 1/2, –3/7, 4.

5. Irrational Numbers

• Numbers that cannot be expressed as fractions of integers.
• Their decimal expansions are non-terminating and non-repeating.
• Examples: √2, √3, π.
• Baudhayana (Ancient Indian mathematician) calculated √2 while constructing fire altars.
  Baudhayana Sulbasutra:
  “dīrghasyākṣaṇayā rajjuḥ pārśvamānī tiryaṅmānī ca yat pṛthagbhūte kurutastadubhayāṅ karoti”
  (This describes the Pythagoras theorem long before Pythagoras.)

6. Real Numbers (R)

• Combination of rational and irrational numbers.
• Denoted by R.

1.4 Representation of Numbers

Numbers can be represented in different ways for better understanding. Let’s explore:

1.4.1 Number Line

• A number line is a straight line where numbers are placed at equal distances.
• All natural numbers (1, 2, 3, …) lie on the right side.
• Whole numbers include 0.
• Integers extend on both sides (…, -3, -2, -1, 0, 1, 2, 3, …).
• Fractions and rational numbers (like ½, -½) lie between integers.
• Irrational numbers like √2 can also be represented using geometric constructions.
  
  Baudhayana Sulbasutra (approx. 800 BCE):
  He gave a method to construct √2 geometrically while building fire altars.
  This shows that irrational numbers were understood in ancient India through practical geometry.

1.4.2 Decimal Representation

• Every number can be written in decimal form (base 10).
• Example: ½ = 0.5, ⅓ = 0.333…
• Non-terminating and non-repeating decimals represent irrational numbers.
  Example: π - 3.14159265...

Aryabhata (5th century CE) used place value notation without a symbol for zero initially, later developed by Brahmagupta (7th century CE).

स्थानात् स्थानं दशगुणं स्यात्
(From one place to the next, the value increases tenfold – explaining the decimal system.)

1.4.3 Geometrical Representation

Numbers were often explained using geometry in Vedic and later mathematical traditions.

• Natural numbers → counted objects (stones, sticks).
• Fractions → division of a unit length or area.
• Irrationals → diagonal of a square (√2), side of a triangle (√3).

For example, to represent √2:

• Draw a square of side 1 unit.
• The diagonal = √2 units.
• Place this on a number line using compass construction.

1.4.4 Symbolic Representation

• Modern math uses symbols like π, e, i.
• Ancient Indians used Sanskrit alphabets as number codes.
  Example: In the Katapayadi system, consonants were given numerical values, so verses encoded astronomical numbers.

1.4.5 Real-Life Representation

• Temperature: +10°C and -5°C show integers on a number line.
• Money: ₹100 profit (+100) vs. ₹50 loss (-50).
• Fractions: Sharing pizza into equal parts.

1.5 Laws of Exponents

Exponents (or powers) are a way of writing repeated multiplication in short form.
For example:
2⁵=2×2×2×2×2=32
Here, 2 is the base and 5 is the exponent (or power).

1.5.1 Basic Laws of Exponents

1. Multiplication of powers with same base
   a^m×aⁿ=a^m+n
   Example: 2³ x 2⁴ = 2⁷ = 128


2. Division of powers with same base
   \( \frac{a^{m}}{a^{n}} = a^{m-n}, \; a \ne 0 \)
   Example: \( \frac{5^{6}}{5^{2}} = 5^{4} = 625 \)


3. Power of a power
   \( (a^{m})^{n} = a^{m \times n} \)
   Example: \( (3^{2})^{3} = 3^{2 \times 3} \)


4. Power of a product
   \( (ab)^{m} = a^{m} \times b^{m} \)
   Example: \( (2 \times 3)^{4} = 2^{4} \times 3^{4} \)


5. Power of a quotient
   \( \left(\frac{a}{b}\right)^{m} = \frac{a^{m}}{b^{m}}, \; b \ne 0 \)
   Example: \( \left(\frac{2}{3}\right)^{2} = \frac{2^{2}}{3^{2}} = \frac{4}{9}, \; b \ne 0 \)


6. Zero exponent rule
   \( (a)^{0} = 1 , \; a \ne 0 \)
   Example: \( (7)^{0} = 1 \)


7. Negative exponent rule
   \( a^{-m} = \frac{1}{a^m}, \; a \neq 0 \)

In Vedic and later Indian mathematics, large powers were used in astronomy and calendar calculations. To simplify them, sutras (formulas) were memorized.

Vedic Math Sutra:
“ऊर्ध्वतिर्यग्भ्याम्”
(Vertically and crosswise – a method used for multiplication, which is the basis for understanding exponents.)

Aryabhata (5th century CE) also used place values with powers of 10 to describe huge astronomical numbers.

Shloka from Aryabhatiya:
“दशगुणितं स्थानात् स्थानं गच्छति”
(Each place is ten times the previous one – directly showing exponents of 10.)

1.5.2 Real-Life Applications of Exponents

• Population growth → doubles or triples over years.
• Compound interest → uses formula \( A = P \left( 1 + \frac{r}{100} \right)^n \)
• Technology → computer memory (2ⁿ bytes).
• Astronomy → distances written as powers of 10 (e.g., 1 light year = 9.46 x 10¹² km

1.6 Important Properties of Numbers

1.6.1 Closure Property

A set of numbers is said to be closed under an operation if performing that operation on any two numbers in the set always results in another number of the same set.

• Addition: Integers are closed under addition.
  Example: −3+5=2 (an integer).
• Subtraction: Integers are closed under subtraction.
  Example: 7−10=−3.
• Division: Integers are not closed under division.
  Example: 7÷2=3.5 (not an integer).

In Vedic rituals, fractions and ratios were often used, but integers were seen as the “complete” numbers, symbolizing wholeness.

1.6.2 Commutative Property

Changing the order of numbers does not change the result.

• Addition: a+b=b+a
  Example: 4+7=7+4=11
• Multiplication: a×b=b×a
  Example: 3×8=8×3=24

Not true for subtraction or division. Example: 8 − 3 ≠ 3 − 8.

यथा पूर्वम् अक्षरम् – order may change, but essence remains same.

1.6.3 Associative Property

Grouping of numbers does not affect the result.

• Addition: (a+b)+c=a+(b+c)
  Example: (2+3)+4=9=2+(3+4)
• Multiplication: (a×b)×c=a×(b×c)
  Example: (2×3)×4=24=2×(3×4)
Not true for subtraction or division.

1.6.4 Identity Property

There are special numbers that act as an identity under certain operations.

• Additive Identity: a+0=a.
  Example: 15+0=15.
• Multiplicative Identity: a×1=a.
  Example: 9×1=9.

1.6.5 Distributive Property

Multiplication distributes over addition and subtraction.

• a×(b+c)=(a×b)+(a×c)
• a×(b−c)=(a×b)−(a×c)

“परावर्त्य योजयेत्” – rearrange and combine, reflecting distributive thinking.

1.7 Real-Life Applications of Numbers

Numbers are not just symbols on paper – they guide our daily life, science, business, and even spiritual traditions. Let’s explore:

1.7.1 Fractions and Rational Numbers

• Cooking & Recipes: If a recipe needs 1½ cups of rice, we are using fractions.
• Sharing: Dividing a chocolate bar among 4 friends = ¼ each.
• Construction: Carpenters use fractions for measurements (½ inch, ¾ inch).

1.7.2 Negative Numbers

• Temperature: In cold regions, -5°C represents temperatures below zero.
• Banking: If your account balance is ₹500 and you withdraw ₹700, your balance is -₹200.
• Height: A submarine 200 m below sea level = -200 m.

1.7.3 Irrational Numbers

• Architecture: The diagonal of a square uses √2, often in building layouts.
• Circles: π (3.14159…) is used in calculating wheels, domes, and roundabouts.
• Science: √3 and √5 appear in trigonometry and wave calculations.

1.7.4 Exponents in Daily Life

• Population Growth: If a town has 10,000 people and grows 2% yearly, the formula uses exponents.
• Compound Interest: Banks calculate savings with exponents.
• Technology: Computer memory (8 GB = 2³ GB).

1.7.5 Numbers in Nature

• Symmetry in flowers → petals follow numerical patterns.
• Beehives → hexagons (geometry + rational numbers).
• Golden Ratio (1.618…): Appears in art, architecture, and Indian temple design.

1.7.6 Everyday Situations

• Sports: Scoreboards use integers, fractions (run rates), and averages.
• Medicine: Dosages are given in fractions (½ tablet).
• Travel: Distance = rational numbers, speed = ratio, time = calculation.
• Mobile Data Plans: MBs and GBs use exponents of 1024.