Question 25
Where does the Hindu/Indian number system figure in the evolution of ideas of number representation? What are its landmark numbers? And does it use a place value system?
- The Hindu/Indian number system is a foundational milestone in mathematics that introduced the base- positional system and the concept of zero ().
- It uses ten basic digits ( through ) where each digit's value is determined by its position (place value).
- This system spread through the Arab world to Europe (known as Hindu-Arabic numerals) and became the universal number system used worldwide today.
(i) Where does the Hindu/Indian number system figure in the evolution of ideas of number representation?
Step 1 · Historical Significance and Evolution
The Hindu/Indian number system originated in ancient India and represents a major breakthrough in the history of mathematics:
- It introduced the concept of zero (Shunya) and the place-value system, allowing any number to be written using only ten symbols.
- It spread to the Arab world and was later introduced to Europe by Arab scholars, gradually replacing inefficient additive systems like Roman numerals.
- Today, it serves as the universal standard for number representation worldwide.
(i) It is a foundational development in number representation that introduced zero and the base- place-value system, eventually becoming the universal number system.
(ii) What are its landmark numbers?
Step 1 · Identify Landmark Digits
The system uses ten fundamental symbols (digits):
The most critical landmark is the digit zero (), which acts both as a placeholder and a number in its own right, making positional notation possible.
(ii) The ten digits , with zero () being the most significant landmark.
(iii) Does it use a place value system?
Step 1 · Verify and Demonstrate the Place Value System
Yes, it is a positional base- (decimal) place-value system where the value of a digit depends on its position in the number.
For example, in the number :
- The rightmost represents
- The middle represents
- The leftmost represents
(iii) Yes, it uses a decimal (base-) place-value system.
- Additive vs Positional Systems: Confusing place-value systems with non-positional/additive systems like Roman numerals where symbols do not shift value based on position.
- Role of Zero: Overlooking zero as just 'nothing' rather than a necessary placeholder that enables multi-digit positional representation (e.g., distinguishing , , and ).
More questions in IT
Q. Reema's curiosity was sparked, and questions started swirling in her head:
(i) Since when have humans been counting?
(ii) What was their need for counting? What were they counting?
(iii) Since when have people been writing numbers in the modern form?
(iv) How would the Mesopotamians have written ? ? ?
Context: Imagine that we are living in the Stone Age, say, around ten thousand years ago. Suppose we have a herd of cows.
Q. How do we ensure that all cows have returned safely after grazing?
Context: Imagine that we are living in the Stone Age, say, around ten thousand years ago. Suppose we have a herd of cows.
Q. Do we have fewer cows than our neighbour?
Context: Imagine that we are living in the Stone Age, say, around ten thousand years ago. Suppose we have a herd of cows.
Q. If there are fewer, how many more cows would we need so that we have the same number of cows as our neighbour?
Context: Suppose we have a herd of cows. We want to answer the following questions: Q2. Do we have fewer cows than our neighbour? Q3. If there are fewer, how many more cows would we need so that we have the same number of cows as our neighbour?
Q. How will you use such sticks to answer the other two questions (Q2 and Q3)?
How many numbers can you represent in this way using the sounds of the letters of your language?
Do you see a way of extending this method to represent bigger numbers as well? How?
Context: A group of indigenous people in Australia called the Gumulgal had the following words for their numbers:
- urapon
- ukasar
- ukasar-urapon
- ukasar-ukasar
- ukasar-ukasar-urapon
- ukasar-ukasar-ukasar
Q. Can you see how their number names are formed?
Context: A group of indigenous people in Australia called the Gumulgal had the following words for their numbers:
- urapon
- ukasar
- ukasar-urapon
- ukasar-ukasar
- ukasar-ukasar-urapon
- ukasar-ukasar-ukasar
Q. Can you see how the names of the other numbers are formed?
What could be the difficulties with using a number system that counts only in groups of a single particular size? How would you represent a number like 1345 in a system that counts only by 5s?
Do it yourself now:
(b)
How will you multiply two numbers given in Roman numerals, without converting them to Hindu numerals? Try to find the product of the following pairs of landmark numbers: , , , .
Instead of grouping together 10 collections of size equal to the previous landmark number (as in the case of the Egyptian system), can we get a number system by grouping together 5 collections of size equal to the previous landmark number? Can this 5 be replaced by any positive integer?
What is any landmark number multiplied by (that is )? Find the following products—
Each landmark number is a power of and so multiplying it with increases the power by , which is the next landmark number.
What is any landmark number multiplied by ()? Find the following products—
Find the following products—
Thus, the product of any two landmark numbers is another landmark number!
Context: Thus, the product of any two landmark numbers is another landmark number!
Q. Does this property hold true in the base-5 system that we created? Does this hold for any number system with a base?
Now find the following products—
What would be a simple rule to multiply a number with ?
Context: To get an idea of how the abacus was used for calculations, let us consider a simple addition problem: . The two numbers were taken on either side of the vertical partition.
Q. How would you use this to find the sum?
Context: To get an idea of how the abacus was used for calculations, let us consider a simple addition problem: . The two numbers were taken on either side of the vertical partition. The counters along each line were brought together.
Q. What is to be done if the total in a line exceeded ?
Look at the representation of 60. What will be the representation for 3,600?
How is this to be read?
Represent the following numbers using the Mayan system:
(i) 77 (ii) 100 (iii) 361 (iv) 721
Where does the Hindu/Indian number system figure in the evolution of ideas of number representation? What are its landmark numbers? And does it use a place value system?