Chapter 3: The World of Numbers
Exploring the evolution from fractions to the unbroken continuum of the Real Number Line.
📖 Core Vocabulary
Rational Numbers (Q)
Expressible as p/q (q ≠ 0). Their decimals either terminate or repeat in a loop.
Irrational Numbers (I)
Numbers that cannot be written as simple fractions. They are non-terminating and non-repeating (e.g., π, √2).
Real Numbers (R)
The complete continuum. Uniting rational and irrational sets perfectly maps to every single point on the number line leaving no gaps.
📐 Constructing √2
Using the Baudhâyana-Pythagoras theorem on a number line:
Hypotenuse becomes exactly √2
🔄 Cyclic Magic (1/7)
Derived from 1/7, this sequence of digits simply shifts its position in a loop when multiplied.
1 × ... = 142857
2 × ... = 285714
3 × ... = 428571
4 × ... = 571428
5 × ... = 714285
6 × ... = 857142
⚖️ Brahmagupta's Laws & Density
The Arithmetic of Fortunes (+) & Debts (−):
- Fortune × Fortune = Fortune (+ × + = +)
- Debt × Debt = Fortune (− × − = +)
- Fortune × Debt = Debt (+ × − = −)
The Density Average Formula:
To find a rational number strictly between any two given numbers a and b, simply calculate their average:
x = (a + b) / 2
⚙️ Decimal to Fraction Algorithm
Goal: Convert 0.45 (0.4545...) to p/q format.
Let x equal the decimal:
x = 0.4545...
Multiply by 100 (since 2 digits repeat):
100x = 45.4545...
Subtract Step 1 from Step 2:
100x − x = 45.45... − 0.45...
99x = 45
Simplify to p/q:
x = 45 / 99
x = 5 / 11
⚠️ Common Pitfalls & Exam Traps
The "Infinite" Misconception
Not all infinite decimals are irrational. Repeating decimals like 0.333... are Rational (1/3). Only patternless, non-repeating infinite decimals are Irrational.
The 0.999... vs 1 Debate
0.999... is exactly equal to 1. Proven algebraically: If x = 0.999..., then 10x = 9.999...; subtracting gives 9x = 9, meaning x = 1.
The Non-Zero Denominator
Why can't 'q' be Zero? Division by zero is mathematically undefined. Therefore, in any rational fraction p/q, the integer q must never be 0 (p/0 = Error).