A Story of Numbers | FIO

Question 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

Understand the Question

In a stick-based tally system, each stick represents a single unit ( $1$ ).
Without using number names or Hindu-Arabic numerals ( $0, 1, 2, \dots$ $0, 1, 2, \dots$ ), all arithmetic operations are performed through direct physical manipulation of stick collections:
- Addition: Combining distinct collections into a single pile.
- Subtraction: Removing sticks from a larger pile equal to the smaller pile.
- Multiplication: Repeatedly grouping collections of sticks.
- Division: Distributing sticks equally into groups one by one.

Step 1 · Adding Collections of Sticks

To add two numbers, combine both stick collections into a single group and count the total sticks. Diagram 1

For example, to add $|||$ and $||$ :

\begin{aligned} \text{First number } (A): &\ ||| \\ \text{Second number } (B): &\ || \\ \text{Combine } A \text{ and } B: &\ |||\quad || \\ \text{Count the total sticks: } &\ ||||| \end{aligned}

Step 2 · Subtracting Collections of Sticks

To subtract one number from another, remove from the larger collection a number of sticks equal to the smaller collection, then count the remaining sticks.For example, to subtract $||$ from $|||||$ :

\begin{aligned} \text{Starting number } (A): &\ ||||| \\ \text{Number to subtract } (B): &\ || \\ \text{Remove } B \text{ sticks from } A: &\ ||||| \text{ minus } || \text{ leaves } ||| \\ \text{Count the remaining sticks: } &\ ||| \end{aligned}

Step 3 · Multiplying Collections of Sticks

To multiply two numbers, repeat the first collection of sticks as many times as there are sticks in the second collection, combine them, and count the total.For example, to multiply $|||$ by $||$ :

\begin{aligned} \text{First number } (A): &\ ||| \\ \text{Second number } (B): &\ || \\ \text{Repeat } A, B \text{ times: } &\ ||| \text{ and } ||| \\ \text{Combine all repeated groups: } &\ |||\quad ||| \\ \text{Count the total sticks: } &\ |||||| \end{aligned}

Step 4 · Dividing Collections of Sticks

To divide a collection of sticks, create as many empty piles as there are sticks in the divisor collection. Distribute the dividend sticks one by one into the piles until none remain; the number of sticks in any one pile is the answer.For example, to divide $||||||$ by $||$ :

\begin{aligned} \text{Starting number } (A): &\ |||||| \\ \text{Number to divide by } (B): &\ || \\ \text{Create 2 empty groups: } &\ \text{Group 1, Group 2} \\ \text{Distribute sticks one by one: } & \\ \text{Group 1: } |, &\ \text{Group 2: } | \\ \text{Group 1: } ||, &\ \text{Group 2: } || \\ \text{Group 1: } |||, &\ \text{Group 2: } ||| \\ \text{Count sticks in one group: } &\ ||| \end{aligned}

Answer

Addition: Combine all sticks from both collections into one group and count the total.

Subtraction: Remove sticks from the larger collection equal in count to the smaller collection; count the remaining sticks.
Multiplication: Repeat the first collection as many times as there are sticks in the second collection, combine them, and count the total.
Division: Distribute the total sticks one by one into as many equal groups as indicated by the divisor collection; count the sticks in any one group.

Common Mistakes

Using Hindu-Arabic Numerals: Relying on standard digits ( $0, 1, 2, \dots$ ) or number names instead of purely stick/tally representations.
Unequal Division: Distributing sticks unevenly across piles rather than placing one stick at a time cyclically into each pile.

Question 1

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