The World of Numbers · Lesson 1 of 7
The Dawn of Mathematics
“Before calculators did the thinking, people first had to invent the numbers.”
• Understand why counting emerged from practical human needs. • Explain one-to-one correspondence as an early counting idea. • Understand the importance of tally marks and early counting artefacts. • Connect trade, measurement and astronomy with the growth of number systems. • Recognise how powers of 10 prepared the way for the modern place-value system.
The Dawn of Mathematics: The Human Need to Count
Mathematics did not begin with symbols on a classroom board. It began with practical problems: keeping track of animals, measuring food, recording time, exchanging goods and observing the sky. Before written numerals existed, humans still needed ways to answer questions such as: How many animals left the settlement? How many came back? How many days have passed? How much grain was traded? These questions created the need for counting.
Counting Before Number Symbols
Imagine a herder who does not yet have written numerals. One simple method is one-to-one correspondence. For every cow that leaves in the morning, one pebble is placed in a pot. In the evening, one pebble is removed for every cow that returns. If no pebble remains, every cow has returned. If some pebbles remain, the number of remaining pebbles matches the number of missing cows.
A method of matching each object in one collection with exactly one object in another collection. It allows quantities to be compared even without written number symbols.
This kind of matching captures the basic meaning of the natural numbers. Natural numbers arise from counting separate objects one by one: 1, 2, 3, 4, and so on.
The counting numbers 1, 2, 3, 4, ... are called natural numbers and are commonly denoted by N.
A History Written in Bone
Long before paper, people recorded quantities by carving marks into durable materials such as bone. These tally marks are important because they show that humans were not merely recognising small quantities visually; they were deliberately recording repeated counts.
The Lebombo Bone
The Lebombo Bone, found in the Lebombo Mountains region of southern Africa, is dated to roughly 35,000 years ago. It contains 29 deliberately carved notches of fairly uniform size. The pattern has been interpreted as evidence that early humans may have been tracking a recurring cycle such as lunar phases or a menstrual cycle.
A tally is more than a scratch. Once a mark is intentionally made for each counted event or object, the mark becomes a record of quantity.
The Ishango Bone
The Ishango Bone, found near the headwaters of the Nile in the Democratic Republic of Congo and dated to around 20,000 BCE, contains several groups of notches. One grouping contains 11, 13, 17 and 19 marks, which are prime numbers between 10 and 20. Another grouping appears to show doubling. Whatever the exact original purpose, the artefact clearly shows that numerical groupings and structured counting are extremely ancient.
A prime number is a natural number greater than 1 that has exactly two positive factors: 1 and itself.
Problem
What do 11, 13, 17 and 19 have in common?
- 1.Each number is greater than 1.
- 2.11 has factors 1 and 11 only.
- 3.13 has factors 1 and 13 only.
- 4.17 has factors 1 and 17 only.
- 5.19 has factors 1 and 19 only.
- 6.Therefore all four are prime numbers.
The Indian Context: Trade and Astronomy
As settlements grew into large civilisations, counting alone was no longer enough. Trade required reliable weights, measures and records. Astronomy required the tracking of long periods of time and large numerical cycles. These needs encouraged number systems to become more powerful and more systematic.
In urban centres of the Indus Valley Civilisation such as Lothal and Harappa, standardised weights and measures were important for trade. Merchants dealing in goods such as pottery, cotton and precious materials needed a dependable way to compare quantities and keep accounts.
Problem
A trader receives 15 copper ingots for every 2 bags of spices. How many ingots correspond to 12 bags?
- 1.12 bags contain 6 groups of 2 bags.
- 2.Each group gives 15 ingots.
- 3.Total ingots = 6 × 15 = 90.
- 4.So 12 bags correspond to 90 copper ingots.
Large Numbers and Powers of 10
Indian mathematical traditions developed names for very large powers of 10. The chapter notes names extending to 10¹² in Vedic literature and much larger powers in later texts. This is significant because expressing large quantities as powers of 10 leads naturally toward a place-value system.
In a place-value system, the value of a digit depends not only on the digit itself but also on its position. For example, the 5 in 50 represents five tens, while the 5 in 500 represents five hundreds. Powers of 10 make this structure precise.
| Number | Meaning using powers of 10 |
|---|---|
| 7 | 7 × 10⁰ |
| 70 | 7 × 10¹ |
| 700 | 7 × 10² |
| 7000 | 7 × 10³ |
Why Natural Numbers Were Not Enough
Natural numbers are excellent for counting objects, but they do not answer every mathematical question. If we subtract a larger natural number from a smaller one, the result is not a natural number. If we divide one object into equal parts, whole counting numbers are no longer enough. These limitations gradually led mathematics toward zero, negative numbers, fractions and eventually much larger number systems.
Problem
Are natural numbers closed under addition and subtraction?
- 1.For addition, 4 + 7 = 11, and 11 is still a natural number.
- 2.In fact, adding any two natural numbers always gives a natural number.
- 3.For subtraction, 8 − 3 = 5 is natural, but 3 − 8 = −5 is not a natural number.
- 4.Therefore natural numbers are closed under addition but not under subtraction.
Practice Problems
- A trader exchanges 15 ingots for every 2 bags of spices. How many ingots will he receive for 16 bags?
- The numbers 11, 13, 17 and 19 occur in one tally grouping. Explain what they have in common and list the next three prime numbers after 19.
- Give two examples showing that natural numbers are closed under addition.
- Give two examples showing that natural numbers are not closed under subtraction.
- A person counts finger joints using 3 joints on each of four fingers while using the thumb as the counter. How many positions can be counted on one hand? Explain the connection with counting in groups of 12.
Key Takeaways
• Counting began as a practical human need long before written numerals. • One-to-one correspondence allows quantities to be compared without number symbols. • Tally marks on ancient artefacts show early deliberate recording of quantity. • Trade, measurement and astronomy pushed number systems toward larger and more systematic forms. • Powers of 10 prepared the way for place value and the later development of zero.
Next, we study the mathematical revolution of Śhūnya and see how 'nothing' became an actual number with arithmetic rules.
Previous
Start of chapter
Next · Lesson 2
The Revolution of Śhūnya: When Nothing Became Something