Number System

Given: The number zero (0).

Concept: A number is rational if it can be expressed as $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$.

Working:
Yes, zero is a rational number. We can write:
$$0 = \frac{0}{1} = \frac{0}{2} = \frac{0}{3} = \frac{0}{-1} \text{ etc.}$$
In each case, $p = 0$ (an integer) and $q \neq 0$ (an integer).

Conclusion: Zero is indeed a rational number and can be written in the form $\frac{p}{q}$ in infinitely many ways.

Given: Two rational numbers 3 and 4.

Concept: To find $n$ rational numbers between two numbers $a$ and $b$, multiply numerator and denominator to create a gap. Here we need 6 rational numbers, so we write:
$$3 = \frac{3 \times 7}{7} = \frac{21}{7} \quad \text{and} \quad 4 = \frac{4 \times 7}{7} = \frac{28}{7}$$

Working:
The rational numbers between $\frac{21}{7}$ and $\frac{28}{7}$ are:
$$\frac{22}{7},\ \frac{23}{7},\ \frac{24}{7},\ \frac{25}{7},\ \frac{26}{7},\ \frac{27}{7}$$

Answer: Six rational numbers between 3 and 4 are:
$$\frac{22}{7},\ \frac{23}{7},\ \frac{24}{7},\ \frac{25}{7},\ \frac{26}{7},\ \frac{27}{7}$$
(Note: There are infinitely many such rational numbers; this is one possible set.)

Given: $\frac{3}{5}$ and $\frac{4}{5}$.

Concept: To find 5 rational numbers between them, convert both fractions to equivalent fractions with a larger denominator (multiply by 6):
$$\frac{3}{5} = \frac{3 \times 6}{5 \times 6} = \frac{18}{30} \quad \text{and} \quad \frac{4}{5} = \frac{4 \times 6}{5 \times 6} = \frac{24}{30}$$

Working:
The rational numbers between $\frac{18}{30}$ and $\frac{24}{30}$ are:
$$\frac{19}{30},\ \frac{20}{30},\ \frac{21}{30},\ \frac{22}{30},\ \frac{23}{30}$$

Answer: Five rational numbers between $\frac{3}{5}$ and $\frac{4}{5}$ are:
$$\frac{19}{30},\ \frac{20}{30},\ \frac{21}{30},\ \frac{22}{30},\ \frac{23}{30}$$

(i) Every natural number is a whole number.

Answer: TRUE

Reason: The set of natural numbers is $\mathbb{N} = \{1, 2, 3, 4, \ldots\}$ and the set of whole numbers is $\mathbb{W} = \{0, 1, 2, 3, \ldots\}$. Every natural number is present in the set of whole numbers. Hence, every natural number is a whole number.

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(ii) Every integer is a whole number.

Answer: FALSE

Reason: The set of integers is $\mathbb{Z} = \{\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots\}$. Negative integers such as $-1, -2, -3, \ldots$ are integers but they are NOT whole numbers. Hence, every integer is not a whole number.

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(iii) Every rational number is a whole number.

Answer: FALSE

Reason: Rational numbers include fractions such as $\frac{1}{2}, \frac{3}{4}, \frac{-2}{3}$, etc. These are not whole numbers. For example, $\frac{1}{2}$ is a rational number but not a whole number. Hence, every rational number is not a whole number.

(i) Every irrational number is a real number.

Answer: TRUE

Reason: The set of real numbers consists of all rational numbers and all irrational numbers together. Therefore, every irrational number is a part of the collection of real numbers, making this statement true.

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**(ii) Every point on the number line is of the form $\sqrt{m}$, where $m$ is a natural number.

Answer: FALSE

Reason:** Points on the number line include negative numbers (e.g., $-1, -2$), zero, and positive numbers. Negative numbers cannot be expressed as $\sqrt{m}$ where $m$ is a natural number (since square roots of natural numbers are non-negative). Also, numbers like $2, 3$ are on the number line but $\sqrt{m}$ for natural number $m$ gives $\sqrt{1}=1, \sqrt{2}, \sqrt{3}, \sqrt{4}=2, \ldots$ — not every point is covered. Hence the statement is false.

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(iii) Every real number is an irrational number.

Answer: FALSE

Reason: Real numbers include both rational and irrational numbers. For example, $2, \frac{3}{4}, 0$ are real numbers but they are rational, not irrational. Hence, not every real number is irrational.

Answer: No, the square roots of all positive integers are not irrational.

Example:
$$\sqrt{4} = 2, \quad \sqrt{9} = 3, \quad \sqrt{16} = 4$$
Here, $\sqrt{4} = 2$ is a rational number (it can be written as $\frac{2}{1}$).

Conclusion: The square roots of perfect squares like $1, 4, 9, 16, 25, \ldots$ are rational numbers. Only the square roots of non-perfect-square positive integers are irrational.

Concept: We use the Pythagorean theorem to construct $\sqrt{5}$.

Steps:

Step 1: Draw a number line and mark the origin $O$ (representing 0) and point $A$ representing 2, so $OA = 2$ units.

Step 2: At point $A$, draw $AB$ perpendicular to the number line such that $AB = 1$ unit.

Step 3: Join $OB$. By the Pythagorean theorem:
$$OB = \sqrt{OA^2 + AB^2} = \sqrt{2^2 + 1^2} = \sqrt{4 + 1} = \sqrt{5}$$

Step 4: With $O$ as centre and $OB$ as radius, draw an arc that cuts the number line at point $P$.

Conclusion: The point $P$ on the number line represents $\sqrt{5}$, since $OP = OB = \sqrt{5}$.

This is a classroom activity. Below is the mathematical justification:

Step 1: Start at point $O$. Draw $OP_1 = 1$ unit along the number line.
$$OP_1 = \sqrt{1} = 1$$

Step 2: Draw $P_1P_2 \perp OP_1$, with $P_1P_2 = 1$ unit.
$$OP_2 = \sqrt{OP_1^2 + P_1P_2^2} = \sqrt{1^2 + 1^2} = \sqrt{2}$$

Step 3: Draw $P_2P_3 \perp OP_2$, with $P_2P_3 = 1$ unit.
$$OP_3 = \sqrt{OP_2^2 + P_2P_3^2} = \sqrt{(\sqrt{2})^2 + 1^2} = \sqrt{2+1} = \sqrt{3}$$

Step 4: Draw $P_3P_4 \perp OP_3$, with $P_3P_4 = 1$ unit.
$$OP_4 = \sqrt{(\sqrt{3})^2 + 1^2} = \sqrt{3+1} = \sqrt{4} = 2$$

General Pattern: At each step $n$:
$$OP_n = \sqrt{n}$$

Conclusion: By continuing this process, we obtain a spiral (called the square root spiral or Theodorus spiral) where the distance from $O$ to $P_n$ equals $\sqrt{n}$, representing $\sqrt{2}, \sqrt{3}, \sqrt{4}, \ldots$ on the plane.

**(i) $\frac{36}{100}$**
$$\frac{36}{100} = 0.36$$
Type: Terminating decimal

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**(ii) $\frac{1}{11}$**
Performing long division: $1 \div 11$:
$$\frac{1}{11} = 0.090909\ldots = 0.\overline{09}$$
Type: Non-terminating recurring (repeating block: 09)

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**(iii) $4\frac{1}{8}$**
$$4\frac{1}{8} = \frac{33}{8}$$
Performing long division: $33 \div 8$:
$$\frac{33}{8} = 4.125$$
Type: Terminating decimal

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**(iv) $\frac{3}{13}$**
Performing long division: $3 \div 13$:
$$\frac{3}{13} = 0.230769230769\ldots = 0.\overline{230769}$$
Type: Non-terminating recurring (repeating block: 230769)

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**(v) $\frac{2}{11}$**
Performing long division: $2 \div 11$:
$$\frac{2}{11} = 0.181818\ldots = 0.\overline{18}$$
Type: Non-terminating recurring (repeating block: 18)

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**(vi) $\frac{329}{400}$**
Performing long division: $329 \div 400$:
$$\frac{329}{400} = 0.8225$$
Type: Terminating decimal

Given: $\frac{1}{7} = 0.\overline{142857}$

Concept: Since $\frac{2}{7} = 2 \times \frac{1}{7}$, $\frac{3}{7} = 3 \times \frac{1}{7}$, etc., we can multiply the repeating block. Also, the remainders while dividing $1$ by $7$ cycle through $1, 3, 2, 6, 4, 5$ — each remainder corresponds to a cyclic permutation of the block $142857$.

Predictions:
$$\frac{2}{7} = 2 \times 0.\overline{142857} = 0.\overline{285714}$$
$$\frac{3}{7} = 3 \times 0.\overline{142857} = 0.\overline{428571}$$
$$\frac{4}{7} = 4 \times 0.\overline{142857} = 0.\overline{571428}$$
$$\frac{5}{7} = 5 \times 0.\overline{142857} = 0.\overline{714285}$$
$$\frac{6}{7} = 6 \times 0.\overline{142857} = 0.\overline{857142}$$

Observation: Each decimal is a cyclic permutation of the digits $142857$. This happens because the remainders when dividing by 7 cycle through all non-zero residues.

**(i) $0.\overline{6}$**

Let $x = 0.6666\ldots$ $\quad \cdots (1)$

Multiply both sides by 10:
$$10x = 6.6666\ldots \quad \cdots (2)$$

Subtract (1) from (2):
$$10x - x = 6.6666\ldots - 0.6666\ldots$$
$$9x = 6$$
$$x = \frac{6}{9} = \frac{2}{3}$$

$$\boxed{0.\overline{6} = \frac{2}{3}}$$

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**(ii) $0.4\overline{7}$**

Let $x = 0.4777\ldots$ $\quad \cdots (1)$

Multiply both sides by 10:
$$10x = 4.777\ldots \quad \cdots (2)$$

Multiply both sides by 100:
$$100x = 47.777\ldots \quad \cdots (3)$$

Subtract (2) from (3):
$$100x - 10x = 47.777\ldots - 4.777\ldots$$
$$90x = 43$$
$$x = \frac{43}{90}$$

$$\boxed{0.4\overline{7} = \frac{43}{90}}$$

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**(iii) $0.\overline{001}$**

Let $x = 0.001001001\ldots$ $\quad \cdots (1)$

Multiply both sides by 1000:
$$1000x = 1.001001001\ldots \quad \cdots (2)$$

Subtract (1) from (2):
$$1000x - x = 1.001001\ldots - 0.001001\ldots$$
$$999x = 1$$
$$x = \frac{1}{999}$$

$$\boxed{0.\overline{001} = \frac{1}{999}}$$

Let $x = 0.9999\ldots$ $\quad \cdots (1)$

Multiply both sides by 10:
$$10x = 9.9999\ldots \quad \cdots (2)$$

Subtract (1) from (2):
$$10x - x = 9.9999\ldots - 0.9999\ldots$$
$$9x = 9$$
$$x = 1$$

$$\boxed{0.99999\ldots = \frac{1}{1} = 1}$$

Discussion: This result may seem surprising, but it makes sense because $0.9999\ldots$ is a non-terminating recurring decimal and the difference between 1 and $0.9999\ldots$ is $0.0000\ldots = 0$. There is no gap between $0.9999\ldots$ and $1$; they represent the same number. This shows that every non-terminating recurring decimal is a rational number.

Concept: When we divide $1$ by $17$, the remainders at each step can only be $1, 2, 3, \ldots, 16$ (i.e., at most $16$ different non-zero remainders). Once a remainder repeats, the decimal block repeats. Therefore, the maximum length of the repeating block is $\mathbf{16}$.

Verification by long division:

Performing $1 \div 17$:

$$1.000000000000000 \div 17$$

Step-by-step remainders: $10, 100\to 15, 150\to 14, 140\to 4, 40\to 6, 60\to 9, 90\to 5, 50\to 16, 160\to 7, 70\to 2, 20\to 3, 30\to 13, 130\to 11, 110\to 8, 80\to 12, 120\to 1$ (remainder 1 repeats)

$$\frac{1}{17} = 0.\overline{0588235294117647}$$

The repeating block is $0588235294117647$, which has 16 digits.

Conclusion: The maximum number of digits in the repeating block of $\frac{1}{17}$ is $16$, which is confirmed by the division.