Class 9 Mathematics Chapter 1 Number System

Key Concept: Rational + Irrational

b) Irrational

[Solution Description] According to the properties of irrational numbers, the sum of a rational number and an irrational number is always irrational. Therefore, when $7$ and $\pi$ are added, the result is irrational.

Key Concept: Basic Operations

b) $\frac{1}{4}$

[Solution Description] To solve for the difference of two fractions with the same denominator, subtract their numerators and keep the common denominator. Here, it involves subtracting 3 from 5 in the numerator:

$$\frac{5}{8} – \frac{3}{8} = \frac{5-3}{8} = \frac{2}{8} = \frac{1}{4}$$

Key Concept: Complex Identification, Real World Application

c) $\sqrt{5} + \sqrt{3}$

[Solution Description] To determine which expression results in an irrational number, we evaluate each option:

– For $\sqrt{12} – 2$: Simplifying gives $2\sqrt{3} – 2$, where $\sqrt{3}$ is irrational. Thus, this yields an irrational result.

– For $\frac{\pi}{\pi + 1}$: The division of a constant with $\pi$ does not inherently result in an irrational number, as $\pi + 1$ is also transcendental and combines to give a rational approximation.

– For $\sqrt{5} + \sqrt{3}$: Both terms are individually irrational numbers. Their sum remains irrational.

– For $\frac{\sqrt{8}}{2}$: Simplifies to $\frac{2\sqrt{2}}{2} = \sqrt{2}$, which is irrational.

The correct choice here is $\sqrt{5} + \sqrt{3}$.

Key Concept: Equivalent Fractions

b) $\frac{4}{6}$

[Solution Description] An equivalent fraction can be obtained by multiplying the numerator and denominator by the same number. Multiplying both the numerator and denominator of $\frac{2}{3}$ by 2 gives $\frac{4}{6}$. Therefore, $\frac{4}{6}$ is equivalent to $\frac{2}{3}$.

Key Concept: Natural vs Whole

b) No

[Solution Description] A natural number is any positive integer starting from 1, such as 1, 2, 3, and so on. Zero is not included in the set of natural numbers. Therefore, zero is not considered a natural number.

Key Concept: Identify Repeating Block

a) 1

[Solution Description] To find the repeating block for $\frac{1}{9}$, divide 1 by 9.

$1 \div 9 = 0.11111…$

Here, the single digit ‘1’ repeats indefinitely. Therefore, the repeating block is ‘1’.

Key Concept: Rational Operations

b) Commutative Property

[Solution Description] This equation demonstrates the commutative property of addition, which states that changing the order of addends does not change the sum.

Key Concept: Historical Myths

c) Hippacus of Croton

[Solution Description] Hippacus of Croton is a Pythagorean who is often linked with myths concerning the discovery of irrational numbers, such as the irrationality of $\sqrt{2}$.

Key Concept: Non-Terminating Recurring

b) $\frac{5}{12}$

[Solution Description] A rational number $\frac{p}{q}$ has a non-terminating recurring decimal expansion if the denominator $q$ in its simplest form contains any prime factors other than 2 or 5. Let’s test each:

– For $4/25: q = 25 = 5^2$, consists only of 5’s.

– For $5/12: q = 12 = 2^2 \times 3$, includes a factor of 3.

– For $17/64: q = 64 = 2^6$, consists only of 2’s.

– For $1/10: q = 10 = 2 \times 5$, consists only of 2’s and 5’s.

The number $5/12$ meets the conditions for having a non-terminating recurring decimal expansion.

Key Concept: Negative Exponent

d) $\frac{1}{27}$

[Solution Description] To evaluate $3^{-3}$, use the negative exponent rule: $a^{-n} = \frac{1}{a^n}$. Therefore, $3^{-3} = \frac{1}{3^3} = \frac{1}{27}$.

Key Concept: Commutative Law

d) Both A and C

[Solution Description] The commutative property states that the order of addition or multiplication does not affect the result. For example, $a + b = b + a$ and $a \times b = b \times a$.

Here, the expressions are: $6 + 4 = 4 + 6$,

$8 – 3 \neq 3 – 8, \quad \text{(commutative property doesn’t apply to subtraction)}$

$5 \times 3 = 3 \times 5$

Options A and C satisfy the commutative property, but since each option is separate, we look for statements like A) or C).

Key Concept: Definition Recall

c) $\frac{4}{5}$

[Solution Description] A rational number can be expressed as $\frac{p}{q}$ where $p$ and $q$ are integers, and $q \neq 0$. Among the given options, $\frac{4}{5}$ satisfies this definition.

Key Concept: Whole vs Natural

c) 0

[Solution Description] Whole numbers include zero and all positive integers, while natural numbers start from 1. Thus, 0 is a whole number that is not a natural number.

Key Concept: Recognize Termination

a) Yes

[Solution Description]

The decimal number 0.375 can be expressed as a fraction $\frac{375}{1000}$. Simplifying this fraction gives us:

$\frac{375}{1000} = \frac{3}{8}$

The denominator 8 is composed of the prime factors $2^3$, which indicates that 0.375 is a terminating decimal.

Key Concept: Repeating Block

b) 31

[Solution Description] Dividing 7 by 22 results in $7 \div 22 = 0.3181818181…$, or $0.\overline{31}81$. The repeating block is “31”.

Key Concept: Decimal to Fraction

b) $\frac{7}{9}$

[Solution Description] Let $x = 0.777…$. Multiply both sides by 10: $10x = 7.777…$. Subtracting these equations, we get $9x = 7$, so $x = \frac{7}{9}$.

Key Concept: Basic Closure

d) All of the above

[Solution Description] Rational numbers are closed under addition, subtraction, and multiplication but not division by zero. Therefore, all these operations except division by zero result in a rational number.

Key Concept: Pythagorean Theorem Application

d) Pythagorean Theorem

[Solution Description]

The Pythagorean Theorem states that in a right triangle, the square of the hypotenuse ($c$) is equal to the sum of the squares of the other two sides ($a$ and $b$), or $c^2 = a^2 + b^2$. Using this theorem, one can construct a right triangle with legs of length 1 unit each, where the hypotenuse will measure $\sqrt{2}$. This demonstrates the application of the Pythagorean Theorem in locating square roots on the number line.

Key Concept: Decimal Form

a) Terminating

[Solution Description] The number $\frac{7}{8}$ can be converted to its decimal form by performing the division 7 divided by 8. Doing this gives us 0.875. Since there is no repeating block and the division stops after a few steps, it is a terminating decimal.

Key Concept: Identify Equivalent

a) $\frac{6}{10}$

[Solution Description]

To find an equivalent fraction for $\frac{3}{5}$, we can multiply both the numerator and denominator by the same number. Let’s multiply by 2:

$\frac{3 \times 2}{5 \times 2} = \frac{6}{10}$

Thus, $\frac{6}{10}$ is equivalent to $\frac{3}{5}$.

Key Concept: Symbol Recognition

c) Z

[Solution Description] The set of integers is represented by the symbol $Z$, derived from the German word “zahlen,” meaning “to count.”

Key Concept: Real Number Line

a) $\sqrt{8}$

[Solution Description] To determine which number falls between 2 and 3, consider:

– $\sqrt{8} \approx 2.828427125$

– $\sqrt{3} \approx 1.732050807$

– $\sqrt{10} \approx 3.162277660$

– The decimal 2.3 is rational, not irrational.

Thus, $\sqrt{8}$ is the correct choice as it lies between 2 and 3.

Key Concept: Historical Context

b) Lambert

[Solution Description]

– Johann Lambert was the first to prove that $\pi$ is an irrational number in 1768.

– Other mathematicians listed did significant work in mathematics but did not prove the irrationality of $\pi$.

Key Concept: Decimal Expansion

c) 0.10100100010000…

[Solution Description] A decimal expansion that neither terminates nor repeats is indicative of an irrational number. The expansion $0.10100100010000…$ continues without repetition or pattern, making it irrational.

Key Concept: Real Number Line

c) Real numbers

[Solution Description] Rational and irrational numbers together comprise the real numbers. Real numbers are represented on the real number line.

Key Concept: Decimal Expansion

b) Terminating

[Solution Description] To determine the type of decimal expansion, divide $1$ by $4$: $1 \div 4 = 0.25$, which terminates.

Key Concept: Identifying Numbers

c) 4

[Solution Description] On a number line, the number that lies exactly between 3 and 5 is 4.

Key Concept: Basic Exponent Rule

b) 1

[Solution Description] The rule for any non-zero number $a$ raised to the power of zero is $a^0 = 1$. Therefore, $7^0 = 1$.

Key Concept: Identify Natural Numbers

c) 7

[Solution Description] A natural number is a positive integer starting from 1 and continuing indefinitely. Among the options, the only positive integer is 7.

Key Concept: Pythagorean Application

b) Construct a right triangle with legs of 3 and 3 units; mark 3$\sqrt{2}$ units from zero.

[Solution Description]

To find the length of the diagonal of a square with side length 3 units, we use the Pythagorean theorem.

The diagonal $d$ of the square is calculated as follows: $d = \sqrt{3^2 + 3^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2}$.

Therefore, the length of the diagonal is $3\sqrt{2}$. To locate this on the number line, one can construct a right triangle with both legs measuring 3 units and use a compass to draw the diagonal $3\sqrt{2}$.

Key Concept: Locate Rational

c) Between 1 and 2

[Solution Description] To locate $\frac{5}{4}$ on the number line, convert it to a decimal by dividing 5 by 4:

$\frac{5}{4} = 1.25$

So, it lies between 1 and 2 on the number line at 1.25.

Key Concept: Rational Identification

a) Yes

[Solution Description] A number is rational if it has a terminating or non-terminating recurring decimal expansion. The given number 0.141414… is a repeating decimal with the block “14” repeating infinitely. This indicates a recurring pattern, which classifies it as a rational number.

Key Concept: Basic Rationalization

a) $\frac{\sqrt{5}}{5}$

[Solution Description] To rationalize the denominator, multiply both the numerator and the denominator by $\sqrt{5}$.

This produces: $\frac{1}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}} = \frac{\sqrt{5}}{5}$

Hence, the rationalized form is $\frac{\sqrt{5}}{5}$.

Key Concept: Rational/Irrational Classification

d) Irrational

[Solution Description]

The decimal representation 3.14159265358979… does not show any repeating pattern and continues indefinitely.

Such a non-terminating and non-recurring decimal expansion indicates an irrational number.

Hence, the correct answer is Irrational.

Key Concept: Basic Multiplication

c) Irrational

[Solution Description]

The numbers $\sqrt{3}$ and $\pi$ are both irrational. When two irrational numbers are multiplied, the product can be either rational or irrational. However, there is no simplification that makes $\sqrt{3} \times \pi$ a rational number. Therefore, it remains an irrational number.

Key Concept: Locate Numbers

c) 3.5

[Solution Description] The numbers that lie between 3 and 4 are those greater than 3 but less than 4. Here, 3.5 is exactly midway.

Key Concept: Rational and Irrational

b) $\sqrt{27}$

[Solution Description] From the properties of rational and irrational numbers, we know:

A rational number can be expressed as a fraction with integer numerator and non-zero denominator, while an irrational number cannot be written as such.

Now let’s evaluate each option:

1. $\sqrt{16}$: Since $\sqrt{16} = 4$, which is a whole number, it is rational.

2. $\sqrt{27}$: There is no perfect square factor within 27, so $\sqrt{27}$ cannot simplify to a rational number.

3. $\frac{3}{4}$: This is already in the form of a fraction, thus it is rational.

4. $0.75$: Can be represented as $\frac{75}{100}$ or $\frac{3}{4}$ upon simplification, making it rational.

Therefore, $\sqrt{27}$ is an irrational number among the given options.

Key Concept: Convert Complex Decimal

a) $\frac{236}{99}$

[Solution Description]

Let $x = 2.383838…$, then multiplying by 100, we get $100x = 238.3838…$.

Subtracting these equations: $100x – x = 238.3838… – 2.3838…$

results in $99x = 236$

Solving for $x$: $x = \frac{236}{99}$

Therefore, $2.383838… = \frac{236}{99}$.