A Story of Numbers | FIO

Question 8

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

Solution

Understand the Question

A number system is built by progressively expanding sets of numbers to solve new mathematical problems:

Counting objects $\rightarrow$ Natural Numbers ( $\mathbb{N}$ )
Representing zero/nothing $\rightarrow$ Whole Numbers ( $\mathbb{W}$ )
Representing negative values/debts $\rightarrow$ Integers ( $\mathbb{Z}$ )
Representing parts or fractions $\rightarrow$ Rational Numbers ( $\mathbb{Q}$ )
Non-repeating, non-terminating values $\rightarrow$ Irrational Numbers ( $\mathbb{I}$ )
All points on a continuous number line $\rightarrow$ Real Numbers ( $\mathbb{R}$ )

Step 1 · Natural Numbers and Whole Numbers

Natural Numbers ( $\mathbb{N}$ ): Used for counting. $\text{Natural Numbers (N)} = \{1, 2, 3, \dots \}$
Whole Numbers ( $\mathbb{W}$ ): Natural numbers including zero. $\text{Whole Numbers (W)} = \{0, 1, 2, 3, \dots \}$

Step 2 · Integers

Includes whole numbers and their negative counterparts: $\text{Integers (Z)} = \{\dots, -3, -2, -1, 0, 1, 2, 3, \dots \}$

Step 3 · Rational Numbers

Numbers that can be expressed as a ratio of two integers $\dfrac{p}{q}$ , where $q \neq 0$ : $\text{Rational Numbers (Q)} = \left\{\dfrac{p}{q} \mid p, q \in \mathbb{Z}, q \neq 0 \right\}$

Every integer $p$ is also rational since it can be written as $\dfrac{p}{1}$ .

Step 4 · Irrational Numbers

Numbers that cannot be written in the form $\dfrac{p}{q}$ (their decimal expansions are non-terminating and non-repeating), such as $\sqrt{2}$ and $\pi$ : $\text{Irrational Numbers (I)} = \{\text{Numbers that cannot be written as a fraction}\}$

Step 5 · Real Numbers

The complete collection of all rational and irrational numbers, corresponding to every point on the number line: $\text{Real Numbers (R)} = \{\text{All Rational and Irrational Numbers}\}$

Answer

The refined number system hierarchically consists of:

Natural Numbers ( $\mathbb{N}$ ): $\{1, 2, 3, \dots\}$
Whole Numbers ( $\mathbb{W}$ ): $\{0, 1, 2, \dots\}$
Integers ( $\mathbb{Z}$ ): $\{\dots, -1, 0, 1, \dots\}$
Rational Numbers ( $\mathbb{Q}$ ): $\left\{\dfrac{p}{q} \mid p, q \in \mathbb{Z}, q \neq 0\right\}$
Irrational Numbers ( $\mathbb{I}$ ): Numbers not expressible as $\dfrac{p}{q}$
Real Numbers ( $\mathbb{R}$ ): $\mathbb{Q} \cup \mathbb{I}$

Common Mistakes

Forgetting $q \neq 0$ : When defining rational numbers $\dfrac{p}{q}$ , the denominator cannot be $0$ because division by zero is undefined.
Excluding Integers from Rational Numbers: Every integer is rational since any integer $n$ can be expressed as $\dfrac{n}{1}$ .
Confusing Rational and Irrational Decimals: Repeating or terminating decimals (like $0.\bar{3} = \dfrac{1}{3}$ ) are rational; only non-terminating and non-repeating decimals (like $\sqrt{2}$ or $\pi$ ) are irrational.

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.

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!

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?