NCERT Solutions for Class 8 Maths Ganita Prakash Chapter 3 — Figure it Out (Page 80)#
In the Chinese rod numerals, the digits 1 to 9 have two forms: upright zongs (|, ||, |||, …) and flat hengs (—, =, ≡, …). Zongs are used for units, hundreds, tens of thousands, …, and hengs for tens, thousands, hundreds of thousands, …
1. 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? #
Solution
Why they alternated: the rods for neighbouring places look different, so you can see where one place ends and the next begins. Without this, the digits of a number could run into each other.
41 with only zongs: 4 is |||| and 1 is |, so 41 would be written
|||| |
Other readings: if the gap is not clear, these 5 upright rods could be split up in many ways:
- ||| || → 32
- || ||| → 23
- | || || → 122
- | |||| → 14
- ||||| → 5
So the same numeral could mean 41, 32, 23, 122, 14, 5 and more. Alternating zongs and hengs removes this confusion: 41 is written ≣ | (a heng 4 next to a zong 1).
2. Form a base-2 place value system using ‘ukasar’ and ‘urapon’ as the digits. Compare this system with that of the Gumulgal’s. #
Solution
In base 2 we need only two digits. Let urapon (ur) = 0 and ukasar (uk) = 1. The places stand for $2^0 = 1$, $2^1 = 2$, $2^2 = 4$, $2^3 = 8$, … from the right.
| Number | As powers of 2 | Base-2 system | Gumulgal system |
|---|---|---|---|
| 1 | $1$ | uk | urapon |
| 2 | $2$ | uk ur | ukasar |
| 3 | $2 + 1$ | uk uk | ukasar-urapon |
| 4 | $4$ | uk ur ur | ukasar-ukasar |
| 5 | $4 + 1$ | uk ur uk | ukasar-ukasar-urapon |
| 6 | $4 + 2$ | uk uk ur | ukasar-ukasar-ukasar |
| 7 | $4 + 2 + 1$ | uk uk uk | ukasar-ukasar-ukasar-urapon |
| 8 | $8$ | uk ur ur ur | ukasar-ukasar-ukasar-ukasar |
Comparison:
- The Gumulgal system only adds 2s and 1s, so names keep getting longer, about half as many words as the number itself.
- The base-2 system uses place value: the position of each word gives its value, so it needs very few words even for large numbers (8 needs 4 words, 1000 needs only 10).
- Both systems use just two words, but in the base-2 system “urapon” means 0, not 1.
3. 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? #
Solution
In daily life: prices and bills, phone numbers, dates and time, money and bank balances, marks, sports scores, house numbers, PIN codes.
In professions: shopkeepers and accountants (accounts, GST), bankers (interest, loans), engineers and architects (measurements, drawings), doctors and pharmacists (doses, reports), scientists (data, very large and very small numbers), and computer programmers. Computers themselves work in base 2, using only the digits 0 and 1.
Without our number system and 0:
- Writing large numbers would be long and confusing, as with Roman numerals.
- Calculations like multiplication and division would need an abacus or special training.
- Place value, decimals and negative numbers would be hard to express.
- Much of modern science, technology, banking and computing could not have developed.
4. 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? #
Solution
With 8 fingers we would probably count in base 8. We would use only the eight digits 0 to 7, and the places would stand for $1, 8, 64, 512, \ldots$ So “10” would mean eight, and “100” would mean sixty-four. In base 5 we would use the digits 0 to 4, with places $1, 5, 25, 125, \ldots$
25 in base 8. We take as many 8s as possible:
$25 = 3 \times 8 + 1$
So 25 is written 31 in base 8.
25 in base 5. 25 is itself a landmark number:
$25 = 1 \times 25 + 0 \times 5 + 0 \times 1$
So 25 is written 100 in base 5.
25 in base 2. The places are $16, 8, 4, 2, 1$:
$25 = 16 + 8 + 1$
$25 = 1 \times 16 + 1 \times 8 + 0 \times 4 + 0 \times 2 + 1 \times 1$
So 25 is written 11001 in base 2.