Figure It Out

1. 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.

In a stick-based number system, each stick represents one unit. Mathematical operations can be performed directly using collections of sticks without using any numerals.

Addition:
To add two collections, place all the sticks from both collections together in a single group. The resulting collection represents the sum. In this method, addition simply means combining groups of sticks.

Subtraction:
To subtract, remove from one collection the same number of sticks as are present in the other collection. The sticks that remain represent the difference. Thus, subtraction can be viewed as taking away sticks from a collection.

Multiplication:
Multiplication can be understood as repeated addition. Create several equal groups of sticks and combine them into one larger collection. The total collection obtained represents the product.

Division:
Division can be performed by arranging the sticks into equal groups. The number of groups formed or the number of sticks in each group gives the result. Therefore, division is an operation of equal sharing or equal grouping.

This method shows that arithmetic operations can be performed visually and practically even without using modern numerals or number names.

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2. 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!

A simple way to extend the system is to continue using combinations of letters after the single letters are exhausted.

Suppose the letters a to z represent the first twenty-six values. After reaching z, we can begin using two-letter combinations such as aa, ab, ac, ad, and so on. When all two-letter combinations are used, three-letter combinations such as aaa, aab, aac, and others can be introduced.

For example:

a, b, c, ..., z,
aa, ab, ac, ..., az,
ba, bb, bc, ..., zz,
aaa, aab, aac, ...

Since there is no limit to the length of letter strings that can be created, every natural number can be represented. This idea is similar to the way spreadsheet columns are named A, B, C, ..., Z, AA, AB, AC, and so on.

Thus, by continuously increasing the number of letters in each string, the system can be extended indefinitely to represent all numbers.

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3. Try making your own number system.

One possible number system can be created using geometric symbols instead of numerals.

For example:

● = first value
▲ = second value
■ = third value
★ = fourth value
◆ = fifth value

After these basic symbols, combinations of symbols can be used to represent larger values. For instance, two-symbol combinations can represent values larger than those represented by single symbols.

Another possibility is to use colours, shapes, or patterns instead of letters and numerals. Different combinations of these symbols can be assigned different values according to a fixed rule.

The most important feature of any number system is that each symbol or combination of symbols must represent one unique value. As long as this rule is followed, many different number systems can be designed.

This activity helps us understand that number systems are human inventions. Different civilizations have used different symbols and methods to represent numbers, but all number systems serve the same purpose of counting, measuring, and performing calculations.

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Figure It Out

1. Represent the following numbers in the Roman system.

(i) 1222

Roman numerals:
1000 = M
200 = CC
20 = XX
2 = II

Therefore,
1222 = 1000 + 200 + 20 + 2
= M + CC + XX + II

Answer: MCCXXII

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(ii) 2999

Roman numerals:
2000 = MM
900 = CM
90 = XC
9 = IX

Therefore,
2999 = 2000 + 900 + 90 + 9
= MM + CM + XC + IX

Answer: MMCMXCIX

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(iii) 302

Roman numerals:
300 = CCC
2 = II

Therefore,
302 = 300 + 2
= CCC + II

Answer: CCCII

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(iv) 715

Roman numerals:
700 = DCC
10 = X
5 = V

Therefore,
715 = 700 + 10 + 5
= DCC + X + V

Answer: DCCXV

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Figure It Out

1. 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?

Many indigenous communities developed their number systems according to their daily needs, environment, and cultural practices. Different objects often had different importance in their lives, such as fish, coconuts, canoes, or tools. Using separate counting sequences helped them keep track of these objects more easily and reduced confusion.

For example, if one counting sequence was used for coconuts and another for fish, people could immediately understand what was being counted without needing additional explanations. This made communication faster and more efficient.

Such systems also reflect the close relationship between mathematics and culture. Number systems are not only tools for counting but are also influenced by the way people live, work, trade, and organize their societies.

Therefore, they used different sequences of number names because these systems were practical, meaningful, and suited to their everyday life.

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2. Consider the extension of the Gumugal 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:

In the Gumugal system:

urapon = 1
ukasar = 2

Larger numbers are formed by combining groups of two and one. Converting the given expressions into values helps us understand the operations.

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

First expression:
2 + 2 + 2 + 2 + 1 = 9

Second expression:
2 + 2 + 2 + 1 = 7

Sum:
9 + 7 = 16

16 in Gumugal form:
ukasar-ukasar-ukasar-ukasar-ukasar-ukasar-ukasar-ukasar

Answer:
ukasar-ukasar-ukasar-ukasar-ukasar-ukasar-ukasar-ukasar

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(ii) (ukasar-ukasar-ukasar-ukasar-urapon) – (ukasar-ukasar-ukasar)

First expression = 9
Second expression = 6

Difference:
9 − 6 = 3

3 in Gumugal form:
ukasar-urapon

Answer:
ukasar-urapon

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

First expression = 9
Second expression = 4

Product:
9 × 4 = 36

36 in Gumugal form consists of eighteen groups of ukasar.

Answer:
eighteen consecutive groups of ukasar

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

Dividend:
2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 = 16

Divisor:
2 + 2 = 4

Quotient:
16 ÷ 4 = 4

4 in Gumugal form:
ukasar-ukasar

Answer:
ukasar-ukasar

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

The Hindu number system is considered one of the most efficient number systems ever developed. It has several important features that make calculations easier than in the Roman number system.

Use of Place Value:
In the Hindu number system, the value of a digit depends on its position. For example, the digit 5 has different values in 5, 50, and 500. Roman numerals do not have a place value system.

Use of Zero:
The Hindu number system includes the symbol 0, which acts as both a number and a placeholder. Roman numerals do not have a symbol for zero.

Fewer Symbols:
Only ten symbols (0–9) are needed to represent any number, regardless of how large it is. Roman numerals require many repeated symbols.

Easy Arithmetic Operations:
Addition, subtraction, multiplication, and division can be performed systematically using standard algorithms. Such calculations are difficult and time-consuming in Roman numerals.

Representation of Very Large Numbers:
Very large numbers can be written compactly in the Hindu system, whereas Roman numerals become lengthy and complicated.

Because of these advantages, the Hindu number system became widely accepted throughout the world.

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

To refine a self-created number system, it is useful to include some of the strengths of the Hindu number system.

First, a fixed set of symbols should be chosen so that each symbol has a unique meaning. Second, a place value system should be introduced so that the position of a symbol affects its value. Third, a symbol representing zero should be included to indicate the absence of a value in a particular place.

The rules for writing numbers should be simple and consistent so that very large numbers can be represented easily. Clear methods for addition, subtraction, multiplication, and division should also be defined.

For example, if shapes are used instead of digits, their positions can determine whether they represent ones, tens, hundreds, or thousands. This would make the system more organized and efficient.

A refined number system should be simple to learn, easy to use for calculations, and capable of representing numbers of any size.

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Figure It Out

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

2. What numbers do these numerals stand for?

There numerals stand for

(i) 276 (ii) 4322

Figure It Out

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

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

Yes, zero (0) is the number, because there is no symbol for it

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

In a base-7 system, each landmark number is obtained by multiplying the previous landmark number by 7.

The landmark numbers are:

7⁰ = 1
7¹ = 7
7² = 49
7³ = 343
7⁴ = 2401
7⁵ = 16807

Therefore, the first few landmark numbers of a base-7 system are:

1, 7, 49, 343, 2401, 16807, ...

In general, for a base-n system, the landmark numbers are powers of n:

n⁰, n¹, n², n³, n⁴, n⁵, ...

That is:

1, n, n², n³, n⁴, n⁵, ...

Each landmark number is obtained by multiplying the previous landmark number by n. These landmark numbers form the foundation of every base-n number system.

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