The World of
Numbers
A journey from the human need to count to the complete number system
Designed with Dedication & Love For Mathematics
Akash Srivastva, TGT (Mathematics)
Learning Journey
3.1
Dawn of Mathematics
Natural Numbers
3.2
Revolution of Śhūnya
Zero
3.3
Integers
Dhana & Ṛiṇa
3.4
Fractions &
Rational Numbers
3.5
Irrational Numbers
3.6
Real Numbers
The Human Need to Count
One-to-One Correspondence
For every cow that left the settlement, a pebble was placed in a pot. When the cows returned, pebbles were removed. An empty pot meant the herd was safe.
Natural Numbers
ℕ = {1, 2, 3, 4, …}
The most basic set of numbers that emerged from the act of counting.
The Ishango Bone
~20,000 BCE • Congo
This bone contains three columns of notches. One column groups the numbers 11, 13, 17, 19 — the prime numbers between 10 and 20. Another column shows doubling patterns.
These artefacts prove that the abstract idea of number is tens of thousands of years old.
Trade, Astronomy & Large Numbers
Indus Valley
Standardised weights and measures at Lothal and Harappa for trade with Mesopotamia.
The Vedas
Rigveda used powers of 10. Names given up to 10¹² (parārdha).
Lalitavistara
Buddha describes numbers up to 10⁵³ (tallakṣaṇa).
This deep engagement with large numbers prepared the ground for the place-value system and zero.
When Nothing Became Something
Śhūnya = 0
The Concept of Śhūnyatā
Bakhshali Manuscript
Early centuries CE. First physical use of a bold dot (bindu) to represent zero.
Brahmagupta (628 CE)
Defined zero mathematically: a − a = 0
Zero as a Fully Operational Number
These rules, written in 628 CE, are still used exactly as Brahmagupta stated them.
Dhana and Ṛiṇa
Dhana (Fortune)
Positive numbers
Wealth / Assets
Ṛiṇa (Debt)
Negative numbers
Debts / Liabilities
Together with zero, they form the set of Integers (ℤ).
Visualising Integers
Brahmagupta’s Rules
5 + 4 = 9
(−5) + (−4) = −9
(−3) × 4 = −12
(−3) × (−4) = 12
Think & Reflect
Why does a negative times a negative equal a positive? Think of it as removal of debt. If someone takes away (−) four of your debts of ₹3 each, you become ₹12 richer. Therefore (−3) × (−4) = +12.
Filling the Spaces
Definition of Rational Number
A number that can be expressed in the form p/q, where p and q are integers and q ≠ 0.
ℚ = { p/q | p, q ∈ ℤ, q ≠ 0 }
What we must remember
½ = 2/4 = 3/6 = …
Key Takeaways
(Dhana & Ṛiṇa)
Thank You
The story of numbers is the story of human thought — from pebbles to philosophy, from emptiness to infinity.
“Zero is the most powerful number. It is the beginning of everything and the end of nothing.”
Designed with Dedication & Love For Mathematics
Akash Srivastva, TGT (Mathematics)