NCERT Solutions for Class 8 Maths Ganita Prakash Chapter 3 — Figure it Out (Page 63)#
In the base-5 system made in the chapter, each landmark number is 5 times the one before. The symbols (Table 2 of the book) are:
| Value | $5^0 = 1$ | $5^1 = 5$ | $5^2 = 25$ | $5^3 = 125$ | $5^4 = 625$ | $5^5 = 3125$ |
|---|---|---|---|---|---|---|
| Symbol | (wave) |
A number is grouped starting from the largest landmark number that fits, just as $143 = 125 + 5 + 5 + 5 + 1 + 1 + 1$ is written .
1. Write the following numbers in the above base-5 system using the symbols in Table 2: 15, 50, 137, 293, 651. #
Solution
| Number | Base-5 numeral | Grouping |
|---|---|---|
| 15 | $5 + 5 + 5$ | |
| 50 | $25 + 25$ | |
| 137 | $125 + 5 + 5 + 1 + 1$ | |
| 293 | $125 + 125 + 25 + 5 + 5 + 5 + 1 + 1 + 1$ | |
| 651 | $625 + 25 + 1$ |
Check for 293: $2 \times 125 + 1 \times 25 + 3 \times 5 + 3 = 250 + 25 + 15 + 3 = 293$.
2. Is there a number that cannot be represented in our base-5 system above? Why or why not? #
Solution
Yes, zero (0) cannot be represented, because there is no symbol for it. Writing no symbols at all would be a blank, not a numeral.
Every positive whole number can be represented: we take as many copies of the largest landmark number that fits, then the next, and so on. Each landmark is used at most 4 times, because 5 copies would make the next landmark. If the number is very large, we only need to add symbols for $5^6, 5^7, \ldots$
3. Compute the landmark numbers of a base-7 system. In general, what are the landmark numbers of a base-n system? #
Solution
The first landmark number is 1, and each next one is 7 times the previous one:
| Landmark | $7^0$ | $7^1$ | $7^2$ | $7^3$ | $7^4$ |
|---|---|---|---|---|---|
| Value | 1 | 7 | 49 | 343 | 2401 |
∴ The landmark numbers of a base-7 system are 1, 7, 49, 343, 2401, …
In general, the landmark numbers of a base-$n$ system are the powers of $n$:
$$n^0 = 1, \quad n^1 = n, \quad n^2, \quad n^3, \quad \ldots$$