A Story of Numbers | IT

Question 13

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?

Solution

Number systems are built by grouping smaller units into larger, distinct units.

Step 1 — Understanding Grouping by Ten

Let us first understand how our everyday number system works. We use a system based on grouping by ten. This means we count individual items, called units.

Diagram 1

When we have 10 units, we group them together. This new group is called a 'ten'.

$10 \text{ units} = 1 \text{ ten}$

Then, we group 10 'tens' together. This new group is called a 'hundred'.

$10 \text{ tens} = 1 \text{ hundred}$

Each new landmark number is 10 times the previous one. This '10' is our grouping number.

Step 2 — Grouping by Five

Now, let us imagine a system where we group by five instead of ten. We start with single units.

When we have 5 units, we group them together. This new group could be called a 'five'.

$5 \text{ units} = 1 \text{ five}$

Next, we group 5 'fives' together. This new group would be called a 'twenty-five'.

$5 \text{ fives} = 1 \text{ twenty-five}$

We can continue this process to form larger numbers. So, yes, we can definitely get a number system by grouping together 5 collections of the previous landmark number.

$\boxed{\text{Yes, a system grouping by 5 is possible.}}$

Step 3 — Generalizing the Grouping Number

Let us consider if any positive integer can be used for grouping. Let this positive integer be 'b'.

We can start with single units. Then, we group 'b' single units together. This forms one 'group of b'.

$b \text{ units} = 1 \text{ group of } b$

Next, we group 'b' 'groups of b' together. This forms one 'group of b-squared'.

$b \text{ groups of } b = 1 \text{ group of } b^2$

This process creates a number system where 'b' is the base. However, there is one important condition for 'b'. If $b=1$ , then 1 unit would make 1 'group of one'. This does not create a larger, distinct unit. So, a grouping number of 1 does not make a useful number system. The grouping number 'b' must be greater than 1.

Any positive integer greater than 1 can be used as the grouping number.

$\boxed{\text{Any positive integer } b > 1 \text{ can be used.}}$

Answer

(i) Yes, we can get a number system by grouping together 5 collections of size equal to the previous landmark number. (ii) Yes, this 5 can be replaced by any positive integer 'b', as long as 'b' is greater than 1.

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?

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

Q. How do we ensure that all cows have returned safely after grazing?

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

Q. Do we have fewer cows than our neighbour?

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

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) LXXXVII + 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 × L, L × D, V × D, VII × IX.

Q13

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 $\text{?}$ ( $10^2$ )? 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 ∩?

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?