Question 21

We will define a base-2 number system using given digits and compare it with the Gumulgal number system.

Step 1 — Forming the Base-2 System

A base-2 system, also known as binary, uses two digits. Let us assign 'ukasar' to represent the digit 0. Let 'urapon' represent the digit 1. In a base-2 place value system, each position represents a power of 2. The problem defines specific symbols for these powers of 2: $2^0 = 1 = \mathbf{A}$ $2^1 = 2 = \mathbf{B}$ $2^2 = 4 = \mathbf{C}$ $2^3 = 8 = \mathbf{D}$ And so on for higher powers of 2. To represent a number, we find which powers of 2 add up to it. We list the symbols for the powers of 2 that are present (meaning they have a 'urapon' digit).

Let us form the numbers from 1 to 8 in this base-2 system: 1: $1 = 2^0$ So, 1 is represented as A.

2: $2 = 2^1$ So, 2 is represented as B.

3: $3 = 2^1 + 2^0$ So, 3 is represented as BA.

4: $4 = 2^2$ So, 4 is represented as C.

5: $5 = 2^2 + 2^0$ So, 5 is represented as CA.

6: $6 = 2^2 + 2^1$ So, 6 is represented as CB.

7: $7 = 2^2 + 2^1 + 2^0$ So, 7 is represented as CBA.

8: $8 = 2^3$ So, 8 is represented as D.

$\boxed{\text{Base 2-system:}}$ $\boxed{\text{1 : A}}$ $\boxed{\text{2 : B}}$ $\boxed{\text{3 : BA}}$ $\boxed{\text{4 : C}}$ $\boxed{\text{5 : CA}}$ $\boxed{\text{6 : CB}}$ $\boxed{\text{7 : CBA}}$ $\boxed{\text{8 : D}}$

Step 2 — Understanding the Gumulgal System

The Gumulgal system uses two basic symbols: 'ur' for 1 and 'uk' for 2. Numbers are formed by adding these symbols together. Let us form the numbers from 1 to 8 in the Gumulgal system: 1: The symbol for 1 is ur. So, 1 is represented as ur.

2: The symbol for 2 is uk. So, 2 is represented as uk.

3: $3 = 2 + 1$. We combine 'uk' and 'ur'. So, 3 is represented as uk - ur.

4: $4 = 2 + 2$. We combine two 'uk' symbols. So, 4 is represented as uk - uk.

5: $5 = 2 + 2 + 1$. We combine two 'uk' and one 'ur'. So, 5 is represented as uk - uk - ur.

6: $6 = 2 + 2 + 2$. We combine three 'uk' symbols. So, 6 is represented as uk - uk - uk.

7: $7 = 2 + 2 + 2 + 1$. We combine three 'uk' and one 'ur'. So, 7 is represented as uk - uk - uk - ur.

8: $8 = 2 + 2 + 2 + 2$. We combine four 'uk' symbols. So, 8 is represented as uk - uk - uk - uk.

$\boxed{\text{Gumulgal system:}}$ $\boxed{\text{1 : ur}}$ $\boxed{\text{2 : uk}}$ $\boxed{\text{3 : uk - ur}}$ $\boxed{\text{4 : uk - uk}}$ $\boxed{\text{5 : uk - uk - ur}}$ $\boxed{\text{6 : uk - uk - uk}}$ $\boxed{\text{7 : uk - uk - uk - ur}}$ $\boxed{\text{8 : uk - uk - uk - uk}}$

Step 3 — Comparing the Systems

Let us compare the base-2 system with the Gumulgal system.

Comparison Table: Base 2-system | Gumulgal system 1 : A | ur 2 : B | uk 3 : BA | uk - ur 4 : C | uk - uk 5 : CA | uk - uk - ur 6 : CB | uk - uk - uk 7 : CBA | uk - uk - uk - ur 8 : D | uk - uk - uk - uk

Both systems are related to the number 2. The base-2 system is a place value system. The Gumulgal system is an additive system. The base-2 system has many landmark numbers (A, B, C, D, etc., representing powers of 2). The Gumulgal system has only two landmark numbers ('ur' for 1 and 'uk' for 2).

Answer

Comparison Table: Base 2-system | Gumulgal system 1 : A | ur 2 : B | uk 3 : BA | uk - ur 4 : C | uk - uk 5 : CA | uk - uk - ur 6 : CB | uk - uk - uk 7 : CBA | uk - uk - uk - ur 8 : D | uk - uk - uk - uk

Both have the same base 2, but the base 2 system has many landmark numbers, whereas the Gumulgal system has only two landmark numbers.