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Question 21

Form a base-2 place value system using 'ukasar' and 'urapon' as the digits. Compare this system with that of the Gumulgal's.

Solution

Understand the Question

A base-2 place value system represents numbers using powers of $2$ ( $2^0=1, 2^1=2, 2^2=4, 2^3=8, \dots$ ), where each new power introduces a new place/symbol (landmark number).
The Gumulgal system is an additive system based on only two base words: 'ur' ( $1$ ) and 'uk' ( $2$ ), where larger numbers are formed by repeatedly adding 'uk' and 'ur'.
We construct both representations for numbers from $1$ to $8$ and compare their structural properties.

Step 1 · Form the Base-2 Place Value System

Let 'ukasar' represent the digit $0$ and 'urapon' represent the digit $1$ .

In a base-2 place value system, each position represents a power of $2$ :

\begin{aligned} 2^0 &= 1 = \mathbf{A} \\ 2^1 &= 2 = \mathbf{B} \\ 2^2 &= 4 = \mathbf{C} \\ 2^3 &= 8 = \mathbf{D} \end{aligned}

Forming numbers from $1$ to $8$ by combining the symbols for the active powers of $2$ :

$1: 1 = 2^0 \rightarrow \mathbf{A}$
$2: 2 = 2^1 \rightarrow \mathbf{B}$
$3: 3 = 2^1 + 2^0 \rightarrow \mathbf{BA}$
$4: 4 = 2^2 \rightarrow \mathbf{C}$
$5: 5 = 2^2 + 2^0 \rightarrow \mathbf{CA}$
$6: 6 = 2^2 + 2^1 \rightarrow \mathbf{CB}$
$7: 7 = 2^2 + 2^1 + 2^0 \rightarrow \mathbf{CBA}$
$8: 8 = 2^3 \rightarrow \mathbf{D}$

Step 2 · Form the Gumulgal Additive System

The Gumulgal system uses 'ur' for $1$ and 'uk' for $2$ , forming numbers by repeated addition:

$1: \text{Symbol for } 1 \rightarrow \mathbf{ur}$
$2: \text{Symbol for } 2 \rightarrow \mathbf{uk}$
$3: 2 + 1 \rightarrow \mathbf{uk - ur}$
$4: 2 + 2 \rightarrow \mathbf{uk - uk}$
$5: 2 + 2 + 1 \rightarrow \mathbf{uk - uk - ur}$
$6: 2 + 2 + 2 \rightarrow \mathbf{uk - uk - uk}$
$7: 2 + 2 + 2 + 1 \rightarrow \mathbf{uk - uk - uk - ur}$
$8: 2 + 2 + 2 + 2 \rightarrow \mathbf{uk - uk - uk - uk}$

Step 3 · Compare the Two Systems

\begin{array}{|c|c|} \hline \text{Base 2-system} & \text{Gumulgal system} \\ \hline 1 : \text{A} & \text{ur} \\ \hline 2 : \text{B} & \text{uk} \\ \hline 3 : \text{BA} & \text{uk - ur} \\ \hline 4 : \text{C} & \text{uk - uk} \\ \hline 5 : \text{CA} & \text{uk - uk - ur} \\ \hline 6 : \text{CB} & \text{uk - uk - uk} \\ \hline 7 : \text{CBA} & \text{uk - uk - uk - ur} \\ \hline 8 : \text{D} & \text{uk - uk - uk - uk} \\ \hline \end{array}

Type of System: The base-2 system is a place value system, whereas the Gumulgal system is an additive system.
Landmark Numbers: The base-2 system has infinitely many landmark numbers ( $\text{A}, \text{B}, \text{C}, \text{D}, \dots$ corresponding to powers of $2$ ). The Gumulgal system has only two landmark numbers ('ur' for $1$ and 'uk' for $2$ ).

Answer

\begin{array}{|c|c|} \hline \text{Base 2-system} & \text{Gumulgal system} \ \hline 1 : \text{A} & \text{ur} \ \hline 2 : \text{B} & \text{uk} \ \hline 3 : \text{BA} & \text{uk - ur} \ \hline 4 : \text{C} & \text{uk - uk} \ \hline 5 : \text{CA} & \text{uk - uk - ur} \ \hline 6 : \text{CB} & \text{uk - uk - uk} \ \hline 7 : \text{CBA} & \text{uk - uk - uk - ur} \ \hline 8 : \text{D} & \text{uk - uk - uk - uk} \ \hline \end{array}

Both systems are based on the number $2$, but the **base-2 place value system** uses multiple landmark numbers for powers of $2$, whereas the **Gumulgal system** is purely additive with only two landmark numbers ('ur' and 'uk').

Common Mistakes

Place Value vs. Additive: Confusing a place value system (where each position represents a higher power like $1, 2, 4, 8$ ) with an additive system (where values are simply repeated and summed like $2+2+2$ ).
Landmark Numbers Count: Assuming Gumulgal has many base numbers; it only has two basic words ( $1$ and $2$ ), making large numbers very long to write.

Question 21

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