Question 20
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?
The Chinese alternated between vertical and horizontal rod symbols to clearly show different place values and avoid confusion.
Step 1 — Understanding the purpose of alternating symbols
Ancient Chinese mathematicians used small rods to write numbers. These rods were placed in columns, similar to our place value system (like units, tens, hundreds).
They used two main types of symbols:
- Zong symbols: Vertical rods, used for digits in the units place, hundreds place, ten thousands place, and so on.
- Heng symbols: Horizontal rods, used for digits in the tens place, thousands place, hundred thousands place, and so on.
Let us look at how they represented single digits:
- For example, 1 to 5 in the units place would be
|,| |,| | |,| | | |,| | | | |(Zong symbols). - For example, 1 to 5 in the tens place would be
-,=,≡,☰,☱(Heng symbols).
This system of alternating between Zong and Heng symbols for successive place values helped to clearly separate the digits. It made it easy to distinguish which digit belonged to which place value, preventing numbers from looking like one long string of rods.
Step 2 — Representing 41 using only Zong symbols
Let us first see how the number 41 would normally be represented using both Zong and Heng symbols.
- The digit 4 is in the tens place, so it would be represented by four horizontal rods (Heng symbols).
- The digit 1 is in the units place, so it would be represented by one vertical rod (Zong symbol).
- So, 41 would typically be written as:
Now, let us imagine representing 41 using only Zong (vertical) symbols.
- The digit 4 in the tens place would be represented by four vertical rods.
- The digit 1 in the units place would be represented by one vertical rod.
- So, if only Zong symbols were used, 41 would be written as:
Step 3 — Interpreting the all-Zong numeral without space
Consider the representation of 41 using only vertical rods:
If there is no significant space left between the rods representing the '4' (four vertical rods) and the '1' (one vertical rod), a reader might not see them as separate digits.
Instead, they might simply count all the vertical rods together as one group.
Counting all the rods | | | | | gives a total of five rods.
So, this representation could easily be misinterpreted as the number 5.
This potential for misinterpretation is precisely why the Chinese used alternating Zong and Heng symbols.
Answer
(i) The most appropriate reason seems to be that they wanted to avoid misinterpretation while reading the number, as all numerals are represented by rods. (ii) Using only Zong symbols, 41 will be written as four vertical rods followed by one vertical rod: (iii) This can easily be misinterpreted as 5 if no significant space is left between successive positions.
More questions in FIO
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?