NCERT Solutions for Class 8 Maths Ganita Prakash Chapter 3 — Figure it Out (Pages 60–61)#
The Gumulgal people of Australia counted in 2s. They had two words, urapon for 1 and ukasar for 2, and joined them to make bigger numbers: ukasar-urapon is 3, ukasar-ukasar is 4, and so on.
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? #
Solution
Different objects were probably counted in different ways and for different purposes. Some possible reasons:
- Some things come naturally in pairs or groups (pairs of shoes, bunches of fruit, boats with a fixed crew), so their counting words may count the groups, not single items.
- Separate words make it clear what is being counted just from the number word, the way we say “a dozen eggs” or “a ream of paper”.
- Important goods for trade or ceremonies (fish, coconuts, canoes) may have had their own counting traditions.
These systems grew from everyday life, before a single abstract number sequence for everything was settled on.
2. 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: #
A number in this system is a string of ukasars, with one urapon at the end if the number is odd. So we can work with the ukasars and urapons directly:
- Add: put all the ukasars and urapons together; two urapons make one more ukasar.
- Subtract: remove matching words; if an urapon has to be taken from an ukasar, break the ukasar into two urapons first.
- Multiply: write the first number once for every word-value of the second; for example, × ukasar-ukasar means “write it four times”, that is, two copies, then two copies again.
- Divide: see how many copies of the second number can be taken out of the first.
(i) (ukasar-ukasar-ukasar-ukasar-urapon) + (ukasar-ukasar-ukasar-urapon)
Putting everything together gives 7 ukasars and 2 urapons. The two urapons make one more ukasar:
Sum = ukasar-ukasar-ukasar-ukasar-ukasar-ukasar-ukasar-ukasar (8 ukasars)
Check: $9 + 7 = 16$, and $16 = 8 \times 2$.
(ii) (ukasar-ukasar-ukasar-ukasar-urapon) – (ukasar-ukasar-ukasar)
Removing 3 ukasars from 4 ukasars and an urapon leaves 1 ukasar and the urapon:
Difference = ukasar-urapon
Check: $9 - 6 = 3$.
(iii) (ukasar-ukasar-ukasar-ukasar-urapon) × (ukasar-ukasar)
ukasar-ukasar is 4, so we write the first number 4 times. That gives 16 ukasars and 4 urapons, and the 4 urapons make 2 more ukasars:
Product = 18 ukasars (ukasar written 18 times)
Check: $9 \times 4 = 36$, and $36 = 18 \times 2$.
(iv) (ukasar-ukasar-ukasar-ukasar-ukasar-ukasar-ukasar-ukasar) ÷ (ukasar-ukasar)
We take out groups of ukasar-ukasar from the 8 ukasars. Exactly 4 groups come out, and nothing is left. So the answer is “4”, which is:
Quotient = ukasar-ukasar
Check: $16 \div 4 = 4$.
3. Identify the features of the Hindu number system that make it efficient when compared to the Roman number system. #
Solution
| Hindu number system | Roman numerals |
|---|---|
| Place value: the same digit means different values in different places (the 3 in 375 means 300). | Each symbol has a fixed value wherever it is written. |
| Has a symbol for zero, which marks empty places (as in 302). | No symbol for zero. |
| Only ten symbols (0–9) write every number, however large. | New symbols are needed for bigger landmark numbers. |
| Numbers are short: 2999 has 4 digits. | Numbers get long: 2999 is MMCMXCIX. |
| Easy calculation: column addition, long multiplication and division work digit by digit. | Calculation is very hard; people needed an abacus. |
4. Using the ideas discussed in this section, try refining the number system you might have made earlier. #
Solution
The key idea of this section is landmark numbers: new symbols for bigger groups, so that long strings of the same symbol are not needed.
For example, if your earlier system used only one symbol, say ●, written again and again, refine it by adding landmark symbols:
| Symbol | Value |
|---|---|
| ● | 1 |
| ▲ | 5 |
| ■ | 25 |
| ★ | 125 |
Now 38 is written ■ ▲▲ ●●● instead of 38 dots. Making each landmark the same multiple of the one before (here 5 times) prepares the system for the next idea in the chapter, the base.