A Story of Numbers | IT

Question 10

What could be the difficulties with using a number system that counts only in groups of a single particular size? How would you represent a number like 1345 in a system that counts only by 5s?

Solution

Understand the Question

A number system that counts only in fixed groups of a single size (without place value) can only represent exact multiples of that size and requires repeating identical symbols for large numbers.
In a positional base- $5$ system, numbers are represented using digits $0, 1, 2, 3,$ and $4$ with place values as powers of $5$ ( $5^0, 5^1, 5^2, 5^3, 5^4, \dots$ ).
To convert a decimal number to base $5$ , we use successive division by $5$ and record the remainders from bottom to top.

(i) What could be the difficulties with using a number system that counts only in groups of a single particular size?

Step 1 · Identify Limitations of Single-Size Grouping

If a system only counts in groups of $5$ (without sub-units or place value):

Inability to represent intermediate numbers: Numbers like $1, 2, 3, 4, 6, 7$ cannot be represented.
Excessive length: Representing a large number like $1345$ requires writing $\dfrac{1345}{5} = 269$ identical symbols.
Complexity: Reading very long strings of identical symbols and performing basic arithmetic operations (addition, subtraction) becomes extremely difficult.

Answer

(i) Key difficulties include inability to represent numbers that are not exact multiples, overly long representations, and difficult arithmetic.

(ii) How would you represent a number like 1345 in a system that counts only by 5s?

Step 1 · Convert 1345 to Base 5

In a base- $5$ system, place values are powers of $5$ ( $5^0 = 1, 5^1 = 5, 5^2 = 25, 5^3 = 125, 5^4 = 625$ ).

Divide $1345$ successively by $5$ and record the remainders:

\begin{aligned} 1345 \div 5 &= 269 \quad \text{remainder } 0 \\ 269 \div 5 &= 53 \quad \text{remainder } 4 \\ 53 \div 5 &= 10 \quad \text{remainder } 3 \\ 10 \div 5 &= 2 \quad \text{remainder } 0 \\ 2 \div 5 &= 0 \quad \text{remainder } 2 \end{aligned}

Reading the remainders from bottom to top:

$1345_{10} = 20340_5$

Answer

(ii) $20340_5$

Common Mistakes

Reading Remainders Top-to-Bottom: Writing the digits as $04302_5$ instead of reading from the last remainder to the first ( $20340_5$ ).
Confusing Fixed Grouping with Base System: Confusing simple tally grouping ( $269$ groups of $5$ ) with a positional base- $5$ number system ( $20340_5$ ).

More questions in IT

Q. Reema's curiosity was sparked, and questions started swirling in her head:

(i) Since when have humans been counting?

(ii) What was their need for counting? What were they counting?

(iii) Since when have people been writing numbers in the modern form?

(iv) How would the Mesopotamians have written $20$ ? $50$ ? $100$ ?

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Context: Imagine that we are living in the Stone Age, say, around ten thousand years ago. Suppose we have a herd of cows.

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Q. If there are fewer, how many more cows would we need so that we have the same number of cows as our neighbour?

Context: Suppose we have a herd of cows. We want to answer the following questions: Q2. Do we have fewer cows than our neighbour? Q3. If there are fewer, how many more cows would we need so that we have the same number of cows as our neighbour?

Q. How will you use such sticks to answer the other two questions (Q2 and Q3)?

How many numbers can you represent in this way using the sounds of the letters of your language?

Do you see a way of extending this method to represent bigger numbers as well? How?

Context: A group of indigenous people in Australia called the Gumulgal had the following words for their numbers:

urapon
ukasar
ukasar-urapon
ukasar-ukasar
ukasar-ukasar-urapon
ukasar-ukasar-ukasar

Q. Can you see how their number names are formed?

Context: A group of indigenous people in Australia called the Gumulgal had the following words for their numbers:

urapon
ukasar
ukasar-urapon
ukasar-ukasar
ukasar-ukasar-urapon
ukasar-ukasar-ukasar

Q. Can you see how the names of the other numbers are formed?

Q10

What could be the difficulties with using a number system that counts only in groups of a single particular size? How would you represent a number like 1345 in a system that counts only by 5s?

Q11

Do it yourself now:

(b) $\text{LXXXVII} + \text{LXXVIII}$

Q12

How will you multiply two numbers given in Roman numerals, without converting them to Hindu numerals? Try to find the product of the following pairs of landmark numbers: $V \times L$ , $L \times D$ , $V \times D$ , $VII \times IX$ .

Q13

Instead of grouping together 10 collections of size equal to the previous landmark number (as in the case of the Egyptian system), can we get a number system by grouping together 5 collections of size equal to the previous landmark number? Can this 5 be replaced by any positive integer?

Q14

What is any landmark number multiplied by $\cap$ (that is $10$ )? Find the following products—

Each landmark number is a power of $10$ and so multiplying it with $10$ increases the power by $1$ , which is the next landmark number.

Q15

What is any landmark number multiplied by $?$ ( $10^2$ )? Find the following products—

Q16

Find the following products—

Thus, the product of any two landmark numbers is another landmark number!

Q17

Context: Thus, the product of any two landmark numbers is another landmark number!

Q. Does this property hold true in the base-5 system that we created? Does this hold for any number system with a base?

Q18

Now find the following products—

Q19

What would be a simple rule to multiply a number with $\cap$ ?

Q20

Context: To get an idea of how the abacus was used for calculations, let us consider a simple addition problem: $2907 + 43$ . The two numbers were taken on either side of the vertical partition.

Q. How would you use this to find the sum?

Q21

Context: To get an idea of how the abacus was used for calculations, let us consider a simple addition problem: $2907 + 43$ . The two numbers were taken on either side of the vertical partition. The counters along each line were brought together.

Q. What is to be done if the total in a line exceeded $10$ ?

Q22

Look at the representation of 60. What will be the representation for 3,600?

Q23

How is this to be read?

Q24

Represent the following numbers using the Mayan system:

(i) 77 (ii) 100 (iii) 361 (iv) 721

Q25

Where does the Hindu/Indian number system figure in the evolution of ideas of number representation? What are its landmark numbers? And does it use a place value system?