A Story of Numbers

Quickly count the number of objects in each of the following boxes:

Up to what group size could you immediately see the number of objects without counting?

Suppose you are using the number system that uses sticks to represent numbers, as in Method 1. Without using either the number names or the numerals of the Hindu number system, give a method for adding, subtracting, multiplying and dividing two numbers or two collections of sticks.

One way of extending the number system in Method 2 is by using strings with more than one letter—for example, we could use 'aa' for 27. How can you extend this system to represent all the numbers? There are many ways of doing it!

Try making your own number system.

Represent the following numbers in the Roman system.

(i) 1222 (ii) 2999 (iii) 302 (iv) 715

A group of indigenous people in a Pacific island use different sequences of number names to count different objects. Why do you think they do this?

Consider the extension of the Gumulgal number system beyond 6 in the same way of counting by 2s. Come up with ways of performing the different arithmetic operations (+, −, ×, ÷) for numbers occurring in this system, without using Hindu numerals. Use this to evaluate the following:

(i) (ukasar-ukasar-ukasar-ukasar-urapon) + (ukasar-ukasar-ukasar-urapon)

(ii) (ukasar-ukasar-ukasar-ukasar-urapon) − (ukasar-ukasar-ukasar-ukasar)

(iii) (ukasar-ukasar-ukasar-ukasar-urapon) × (ukasar-ukasar)

(iv) (ukasar-ukasar-ukasar-ukasar-ukasar-ukasar-ukasar-ukasar) ÷ (ukasar-ukasar)

Identify the features of the Hindu number system that make it efficient when compared to the Roman number system.

Using the ideas discussed in this section, try refining the number system you might have made earlier.

Represent the following numbers in the Egyptian system: 10458, 1023, 2660, 784, 1111, 70707.

What numbers do these numerals stand for?

Write the following numbers in the above base-5 system using the symbols in Table 2: 15, 50, 137, 293, 651.

Is there a number that cannot be represented in our base-5 system above? Why or why not?

Compute the landmark numbers of a base-7 system. In general, what are the landmark numbers of a base-n system?

Add the following Egyptian numerals:

Add the following numerals that are in the base-5 system that we created:

Remember that in this system, 5 times a landmark number gives the next one!

Can there be a number whose representation in Egyptian numerals has one of the symbols occurring 10 or more times? Why not?

Create your own number system of base 4, and represent numbers from 1 to 16.

Give a simple rule to multiply a given number by 5 in the base-5 system that we created.

Represent the following numbers in the Mesopotamian system —

(i) 63 (ii) 132 (iii) 200 (iv) 60 (v) 3605

Why do you think the Chinese alternated between the Zong and Heng symbols? If only the Zong symbols were to be used, how would 41 be represented? Could this numeral be interpreted in any other way if there is no significant space between two successive positions?

Form a base-2 place value system using 'ukasar' and 'urapon' as the digits. Compare this system with that of the Gumulgal's.

Where in your daily lives, and in which professions, do the Hindu numerals, and 0, play an important role? How might our lives have been different if our number system and 0 hadn't been invented or conceived of?

The ancient Indians likely used base 10 for the Hindu number system because humans have 10 fingers, and so we can use our fingers to count. But what if we had only 8 fingers? How would we be writing numbers then? What would the Hindu numerals look like if we were using base 8 instead? Base 5? Try writing the base-10 Hindu numeral 25 as base-8 and base-5 Hindu numerals, respectively. Can you write it in base-2?

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 20? 50? 100?

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:

1. urapon
2. ukasar
3. ukasar-urapon
4. ukasar-ukasar
5. ukasar-ukasar-urapon
6. 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:

1. urapon
2. ukasar
3. ukasar-urapon
4. ukasar-ukasar
5. ukasar-ukasar-urapon
6. 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) LXXXVII + LXXVIII

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: V × L, L × D, V × D, VII × IX.

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 $\cap$ (that is 10)? Find the following products—

Each landmark number is a power of 10 and so multiplying it with 10 increases the power by 1, which is the next landmark number.

What is any landmark number multiplied by $\text{?}$ ($10^2$)? 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: 2907 + 43. 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: 2907 + 43. 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 10?

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?