The World of Numbers | Exercise 3.1

Question 4

Ancient Indians used the joints of their fingers to count, a practice still seen today. Each finger has 3 joints, and the thumb is used to count them. How many can you count on one hand? How does this relate to the ancient base-12 counting systems?

Solution

Understand the Question

In ancient finger-joint counting, the thumb serves as a pointer to count the $3$ distinct segments (joints) on each of the other $4$ fingers of the same hand. This method allows a person to count up to $12$ on a single hand, providing a natural basis for base-12 (duodecimal) number systems.

(i) How many can you count on one hand?

Step 1 · Count the Total Joints on One Hand

The thumb is used as a pointer to count the joints on the remaining $4$ fingers. Each of these $4$ fingers has $3$ joints. Diagram 1

\begin{aligned} \text{Total joints} &= \text{Number of fingers} \times \text{Joints per finger} \\[0.6em] &= 4 \times 3 \\[0.6em] &= 12 \end{aligned}

Answer

(i) $12$

(ii) How does this relate to the ancient base-12 counting systems?

Step 1 · Relate to Base-12 Counting Systems

Because one hand can count up to $12$ distinct units using the finger joints, counting a complete cycle on one hand naturally represents one dozen ( $12$ ). Using the other hand to keep track of complete dozens allows counting up to $12 \times 5 = 60$ or $12 \times 12 = 144$ . This practical counting method provided the foundation for ancient base-12 (duodecimal) numerical systems.

Answer

(ii) Counting $12$ finger joints on one hand provided a natural, intuitive foundation for developing base-12 (duodecimal) counting systems.

Common Mistakes

Counting Thumb Joints: Counting the joints of the thumb instead of using the thumb exclusively as a pointer. Only the remaining $4$ fingers are counted, yielding $4 \times 3 = 12$ units.
Assuming Base-10 Exclusivity: Mistaking base-10 (counting $10$ whole fingers) for the only hand-counting method, overlooking that the $12$ phalanges of four fingers enable base-12 counting on a single hand.

Question 4

More questions in Exercise 3.1