FIO-1

Chapter: A STORY OF NUMBERS
Class: 8 (Class 8)
Category: figure_it_out

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Question

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.

Question diagram(s):

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We can perform arithmetic operations by physically manipulating collections of sticks, where each stick represents one unit.

Step 1 — Adding Collections of Sticks

To add two numbers, we combine their stick representations. Let us say we want to add a number represented by $A$ sticks and another number represented by $B$ sticks. We simply put all the $A$ sticks and all the $B$ sticks into one single group. Then, we count the total number of sticks in this new group. For example, let us add 3 sticks and 2 sticks.

$\text{First number (A): } |||$

$\text{Second number (B): } ||$

$\text{Combine A and B: } ||| \text{ } ||$

$\text{Count the total sticks: } |||||$

$\boxed{5 \text{ sticks}}$

Step 2 — Subtracting Collections of Sticks

To subtract one number from another, we remove sticks from the larger collection. Let us say we want to subtract a number represented by $B$ sticks from a number represented by $A$ sticks, where $A$ is greater than $B$. We start with the group of $A$ sticks. Then, we take away $B$ sticks from this group. We count the number of sticks remaining. For example, let us subtract 2 sticks from 5 sticks.

$\text{Starting number (A): } |||||$

$\text{Number to subtract (B): } ||$

$\text{Remove B sticks from A: } ||||| \text{ minus } || \text{ leaves } |||$

$\text{Count the remaining sticks: } |||$

$\boxed{3 \text{ sticks}}$

Step 3 — Multiplying Collections of Sticks

To multiply two numbers, we perform repeated addition. Let us say we want to multiply a number represented by $A$ sticks by a number represented by $B$ sticks. We take the group of $A$ sticks and repeat it $B$ times. Then, we combine all these repeated groups into one large group and count the total sticks. For example, let us multiply 3 sticks by 2.

$\text{First number (A): } |||$

$\text{Second number (B): } ||$

$\text{Repeat A, B times: } ||| \text{ and } |||$

$\text{Combine all repeated groups: } ||| \text{ } |||$

$\text{Count the total sticks: } ||||||$

$\boxed{6 \text{ sticks}}$

Step 4 — Dividing Collections of Sticks

To divide a collection of sticks by another number, we distribute them equally into groups. Let us say we want to divide a number represented by $A$ sticks by a number represented by $B$ sticks. We create $B$ empty groups or piles. Then, we take the $A$ sticks one by one and place them sequentially into each of the $B$ groups, cycling through the groups until all $A$ sticks are used. The number of sticks in any one of these $B$ groups will be our answer. For example, let us divide 6 sticks by 2.

$\text{Starting number (A): } ||||||$

$\text{Number to divide by (B): } ||$

$\text{Create 2 empty groups: Group 1, Group 2}$

$\text{Distribute sticks one by one: }$

$\text{Group 1: } | \text{ , Group 2: } |$

$\text{Group 1: } || \text{ , Group 2: } ||$

$\text{Group 1: } ||| \text{ , Group 2: } |||$

$\text{Count sticks in one group: } |||$

$\boxed{3 \text{ sticks}}$

Answer

(i) To add two numbers, combine all sticks from both collections and count the total. (ii) To subtract one number from another, remove sticks from the larger collection equal to the smaller collection, then count the remaining sticks. (iii) To multiply two numbers, repeat the first collection of sticks as many times as indicated by the second number, then combine and count the total. (iv) To divide two numbers, distribute the total sticks one by one into as many equal groups as the divisor indicates; the number of sticks in each group is the result.