Try making your own number system.

Question

Homi · Accepted Answer

We can create a number system by choosing a base and assigning symbols to its powers.

Step 1 — Choosing a base and symbols We first choose a base for our number system. Let us choose 3 as our base. Next, we assign unique symbols to the powers of this base. Let 'A' represent the value of our base raised to the power of 0. $A = 3^0$ $= 1$ Let 'B' represent the value of our base raised to the power of 1. $B = 3^1$ $= 3$ Let 'C' represent the value of our base raised to the power of 2. $C = 3^2$ $= 9$ We can continue this pattern for higher powers if needed.

$\boxed{\text{Symbols: A=1, B=3, C=9}}$

Diagram 1

Step 2 — Rules for representing numbers In our new number system, we write numbers by combining these symbols. Each symbol can be used a maximum of (base - 1) times. Since our base is 3, each symbol can be used at most 2 times. To represent a number, we find the largest symbol that fits into the number. We use that symbol as many times as possible (up to 2 times). Then, we find the remaining value and repeat the process with the next smaller symbol.

Step 3 — Representing numbers 1 to 12 Let us now represent numbers from 1 to 12 using our new system.

For 1: We need the value 1. The symbol 'A' represents 1. We use 'A' once. 1: A

For 2: We need the value 2. We can use 'A' twice (1 + 1). 2: AA

For 3: We need the value 3. The symbol 'B' represents 3. We use 'B' once. 3: B

For 4: We need the value 4. The largest symbol less than or equal to 4 is 'B' (which is 3). We use 'B' once. Remaining value: $4 - 3 = 1$ . Now we need 1. We use 'A' once. 4: BA

For 5: We need the value 5. We use 'B' once (3). Remaining value: $5 - 3 = 2$ . Now we need 2. We use 'A' twice (1 + 1). 5: BAA

For 6: We need the value 6. We can use 'B' twice (3 + 3). This is allowed because we can use 'B' up to 2 times. 6: BB

For 7: We need the value 7. We use 'B' twice (3 + 3 = 6). Remaining value: $7 - 6 = 1$ . Now we need 1. We use 'A' once. 7: BBA

For 8: We need the value 8. We use 'B' twice (3 + 3 = 6). Remaining value: $8 - 6 = 2$ . Now we need 2. We use 'A' twice (1 + 1). 8: BBAA

For 9: We need the value 9. The symbol 'C' represents 9. We use 'C' once. 9: C

For 10: We need the value 10. We use 'C' once (9). Remaining value: $10 - 9 = 1$ . Now we need 1. We use 'A' once. 10: CA

For 11: We need the value 11. We use 'C' once (9). Remaining value: $11 - 9 = 2$ . Now we need 2. We use 'A' twice (1 + 1). 11: CAA

For 12: We need the value 12. We use 'C' once (9). Remaining value: $12 - 9 = 3$ . Now we need 3. We use 'B' once. 12: CB

Answer

1: A 2: AA 3: B 4: BA 5: BAA 6: BB 7: BBA 8: BBAA 9: C 10: CA 11: CAA 12: CB

Question 3

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