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

Understand the Question

The alphabet has $26$ letters. In the base system, numbers $1$ to $26$ are assigned to single letters from $\text{'a'}$ to $\text{'z'}$ .
To extend this system to all numbers, we group numbers into consecutive blocks of $26$ .
In each successive block, we repeat the same letter an additional time:
- Numbers $1$ to $26 \rightarrow$ single letters ( $\text{'a'}$ to $\text{'z'}$ )
- Numbers $27$ to $52 \rightarrow$ two repeated letters ( $\text{'aa'}$ to $\text{'zz'}$ )
- Numbers $53$ to $78 \rightarrow$ three repeated letters ( $\text{'aaa'}$ to $\text{'zzz'}$ )
- and so on for all positive integers.

Step 1 · Establish the Pattern in Blocks of 26

Block 1 (Numbers $1$ to $26$ ): Single letters

1 \to \text{'a'}, \quad 2 \to \text{'b'}, \quad \dots, \quad 26 \to \text{'z'}

Block 2 (Numbers $27$ to $52$ ): Two identical letters

27 \to \text{'aa'}, \quad 28 \to \text{'bb'}, \quad \dots, \quad 52 \to \text{'zz'}

Block 3 (Numbers $53$ to $78$ ): Three identical letters

53 \to \text{'aaa'}, \quad 54 \to \text{'bbb'}, \quad \dots, \quad 78 \to \text{'zzz'}

Step 2 · Formulate the General Rule for Number $N$

To find the representation for any positive integer $N$ :

Number of repetitions: Divide $(N - 1)$ by $26$ . The quotient plus $1$ gives the number of times the letter is repeated.
Letter to use: The remainder of $(N - 1) \div 26$ plus $1$ gives the alphabetical position of the letter.

Example for $N = 70$ :

\begin{aligned} (70 - 1) \div 26 &= 69 \div 26 \\[0.6em] &= 2 \text{ with a remainder of } 17 \end{aligned}

Repetitions: $2 + 1 = 3$
Letter: $17 + 1 = 18^{\text{th}}$ letter $\to \text{'r'}$
Representation: $\text{'rrr'}$

Answer

Numbers can be represented in blocks of $26$ by repeating the $((N-1) \bmod 26 + 1)^{\text{th}}$ letter $\left(\lfloor (N-1)/26 \rfloor + 1\right)$ times: $1\text{--}26 \to \text{'a'}\dots\text{'z'}$ , $27\text{--}52 \to \text{'aa'}\dots\text{'zz'}$ , $53\text{--}78 \to \text{'aaa'}\dots\text{'zzz'}$ , and so on.

Common Mistakes

Off-by-One Division Error: Dividing $N$ directly by $26$ instead of $(N - 1)$ causes incorrect results for exact multiples of $26$ (such as $26$ or $52$ ).
Confusing with Positional Base-26: Mistaking this repeated-letter scheme ( $\text{aa}, \text{bb}, \dots$ ) for positional base-26 spreadsheet notation ( $\text{AA}, \text{AB}, \dots, \text{BA}$ ). Both extend the alphabet, but use completely different rules.

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 $(+, -, \times, \div)$ for numbers occurring in this system, without using Hindu numerals. Use this to evaluate the following:

(i) $(\text{ukasar-ukasar-ukasar-ukasar-urapon}) + (\text{ukasar-ukasar-ukasar-urapon})$

(ii) $(\text{ukasar-ukasar-ukasar-ukasar-urapon}) - (\text{ukasar-ukasar-ukasar-ukasar})$

(iii) $(\text{ukasar-ukasar-ukasar-ukasar-urapon}) \times (\text{ukasar-ukasar})$

(iv) $(\text{ukasar-ukasar-ukasar-ukasar-ukasar-ukasar-ukasar-ukasar}) \div (\text{ukasar-ukasar})$

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?