A Story of Numbers | FIO

Question 2

One way of extending the number system in Method 2 is by using strings with more than one letter—for example, we could use 'aa' for 27. How can you extend this system to represent all the numbers? There are many ways of doing it!

Solution

This number system uses letters of the alphabet to represent numbers, and we can extend it by repeating letters.

Step 1 — Understanding the Initial System

We are given a system where numbers 1 to 26 are represented by single letters. The number 1 is represented by 'a'. The number 2 is represented by 'b'. This continues, so the number 26 is represented by 'z'. Each letter corresponds to its position in the alphabet.

Step 2 — Observing the Extension Pattern

The problem suggests extending this system, for example, by using 'aa' for 27. Following this idea, the next 26 numbers (from 27 to 52) are represented by repeating the same letter twice. So, 27 is represented by 'aa'. 28 is represented by 'bb'. 29 is represented by 'cc'. This pattern continues until 52, which is represented by 'zz'. In this group, the letter used is determined by its position after subtracting 26 from the number. For example, for 27, we calculate $27 - 26 = 1$ , so we use the 1st letter ('a') repeated twice. For 52, we calculate $52 - 26 = 26$ , so we use the 26th letter ('z') repeated twice.

Step 3 — Generalizing the Extension Rule

We can generalize this pattern to represent any number. We group numbers into blocks of 26. The first block (numbers 1 to 26) uses single letters. The second block (numbers 27 to 52) uses two identical letters. The third block (numbers 53 to 78) would use three identical letters. This pattern continues for all numbers.

To represent any number $N$ :

Find the number of letters to repeat: Subtract 1 from $N$ . Divide this result by 26. The whole number part of this division, plus 1, tells us how many times to repeat a letter. For example, for number 70: $(70 - 1) \div 26$ $= 69 \div 26$ $= 2 \text{ with a remainder}$ So, we take the whole number part (2) and add 1, which gives 3 letters.
Find which letter to use: Subtract 1 from $N$ . Find the remainder when this result is divided by 26. Add 1 to this remainder. This gives us the position of the letter in the alphabet (1st for 'a', 2nd for 'b', etc.). For example, for number 70: $(70 - 1) = 69$ $69 \text{ divided by } 26 \text{ gives a remainder of } 17$ So, we take the remainder (17) and add 1, which gives 18. The 18th letter of the alphabet is 'r'.
Form the representation: Repeat the determined letter the determined number of times. For number 70, we repeat 'r' three times, so its representation is 'rrr'.

This method allows us to represent any positive whole number using strings of identical letters.

Answer

(i) a, b, c, ..., z are 26 numbers. (ii) aa, bb, ..., zz are 26 more numbers.

More questions in FIO

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.

Try making your own number system.

Represent the following numbers in the Roman system.

(i) 1222 (ii) 2999 (iii) 302 (iv) 715

A group of indigenous people in a Pacific island use different sequences of number names to count different objects. Why do you think they do this?

Consider the extension of the Gumulgal number system beyond 6 in the same way of counting by 2s. Come up with ways of performing the different arithmetic operations (+, −, ×, ÷) for numbers occurring in this system, without using Hindu numerals. Use this to evaluate the following:

(i) (ukasar-ukasar-ukasar-ukasar-urapon) + (ukasar-ukasar-ukasar-urapon)

(ii) (ukasar-ukasar-ukasar-ukasar-urapon) − (ukasar-ukasar-ukasar-ukasar)

(iii) (ukasar-ukasar-ukasar-ukasar-urapon) × (ukasar-ukasar)

(iv) (ukasar-ukasar-ukasar-ukasar-ukasar-ukasar-ukasar-ukasar) ÷ (ukasar-ukasar)

Identify the features of the Hindu number system that make it efficient when compared to the Roman number system.

Using the ideas discussed in this section, try refining the number system you might have made earlier.

Represent the following numbers in the Egyptian system: 10458, 1023, 2660, 784, 1111, 70707.

Q10

What numbers do these numerals stand for?

Q11

Write the following numbers in the above base-5 system using the symbols in Table 2: 15, 50, 137, 293, 651.

Q12

Is there a number that cannot be represented in our base-5 system above? Why or why not?

Q13

Compute the landmark numbers of a base-7 system. In general, what are the landmark numbers of a base-n system?

Q14

Add the following Egyptian numerals:

Q15

Add the following numerals that are in the base-5 system that we created:

Remember that in this system, 5 times a landmark number gives the next one!

Q16

Can there be a number whose representation in Egyptian numerals has one of the symbols occurring 10 or more times? Why not?

Q17

Create your own number system of base 4, and represent numbers from 1 to 16.

Q18

Give a simple rule to multiply a given number by 5 in the base-5 system that we created.

Q19

Represent the following numbers in the Mesopotamian system —

(i) 63 (ii) 132 (iii) 200 (iv) 60 (v) 3605

Q20

Why do you think the Chinese alternated between the Zong and Heng symbols? If only the Zong symbols were to be used, how would 41 be represented? Could this numeral be interpreted in any other way if there is no significant space between two successive positions?

Q21

Form a base-2 place value system using 'ukasar' and 'urapon' as the digits. Compare this system with that of the Gumulgal's.

Q22

Where in your daily lives, and in which professions, do the Hindu numerals, and 0, play an important role? How might our lives have been different if our number system and 0 hadn't been invented or conceived of?

Q23

The ancient Indians likely used base 10 for the Hindu number system because humans have 10 fingers, and so we can use our fingers to count. But what if we had only 8 fingers? How would we be writing numbers then? What would the Hindu numerals look like if we were using base 8 instead? Base 5? Try writing the base-10 Hindu numeral 25 as base-8 and base-5 Hindu numerals, respectively. Can you write it in base-2?