NCERT Solutions for Class 8 Maths Ganita Prakash Chapter 3 — Figure it Out (Page 54)#
In Method 1 a number is a collection of sticks, one stick for each object. In Method 2 a number is a letter: a for 1, b for 2, …, z for 26. These questions ask how far each method can go.
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. #
Solution
Take the two collections of sticks, A and B.
- Adding: put the sticks of A and B together in one pile. The pile is the sum.
- Subtracting (B from A): match the sticks one to one, removing one stick from A for each stick in B. The sticks left in A are the difference. If A runs out first, B was larger.
- Multiplying: for every stick in B, lay out a full copy of collection A. All the copies together are the product.
- Dividing (A by B): keep taking groups, each matching B stick for stick, out of A. The number of groups taken (one stick for each group) is the quotient, and the sticks left over that are too few to make another group are the remainder.
All four methods only need one-to-one matching, never number names.
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! #
Solution
One way: after the 26 single letters, use doubled letters, then tripled letters, and so on:
| Numbers | Codes |
|---|---|
| 1 to 26 | a, b, c, …, z |
| 27 to 52 | aa, bb, cc, …, zz |
| 53 to 78 | aaa, bbb, …, zzz |
and so on, adding one more copy of the letter for every block of 26. Every number gets a code, but codes become very long.
A shorter way: after z, use all pairs of letters in dictionary order, aa, ab, ac, …, az, ba, bb, …, zz, then all triples aaa, aab, …. There are $26 \times 26 = 676$ pairs, so two letters reach $26 + 676 = 702$, and three letters reach much further. This is how spreadsheet columns are named (A, …, Z, AA, AB, …).
Both ways give an unending sequence, so every number can be represented.
3. Try making your own number system. #
Solution
Many answers are possible. Here is one example, using counting in groups of 5:
| Symbol | Value |
|---|---|
| ● | 1 |
| ▲ | 5 (five ●) |
| ■ | 25 (five ▲) |
A number is written by taking as many ■ as possible, then ▲, then ●. For example:
$38 = 25 + 5 + 5 + 1 + 1 + 1$, so 38 is written ■ ▲▲ ●●●.
To reach bigger numbers, add a new symbol for five ■ (125), and so on.