NCERT Class 9 Maths Solutions: Number System

Question:

The question asks about the relationship between irrational numbers and real numbers. You need to understand the definitions of both irrational numbers and real numbers and how they are categorized within the number system. Specifically, recall that real numbers encompass both rational and irrational numbers.


The statement “Every irrational number is a real number” is true. The set of real numbers (denoted by R) is composed of two disjoint sets: the set of rational numbers (denoted by Q) and the set of irrational numbers (denoted by I). This means that any number that belongs to the set of irrational numbers also belongs to the set of real numbers. Irrational numbers are numbers that cannot be expressed as a simple fraction p/q, where p and q are integers and q is not zero. Examples of irrational numbers include pi (π) and the square root of 2 (√2). These numbers, along with all rational numbers (like 1/2, -3, 0, 5), collectively form the set of real numbers.

The final answer is $\boxed{A}$.

Question:

Understanding the definitions of rational numbers and whole numbers is crucial. Rational numbers can be expressed as a fraction p/q where p and q are integers and q is not zero. Whole numbers are non-negative integers (0, 1, 2, 3, …).


The statement “Every rational number is a whole number” is false. A rational number is any number that can be expressed as a fraction p/q, where p and q are integers and q is not equal to zero. Whole numbers are non-negative integers, meaning they are 0, 1, 2, 3, and so on. While all whole numbers can be expressed as rational numbers (e.g., 5 can be written as 5/1), not all rational numbers are whole numbers. For example, 1/2 is a rational number because it can be written as a fraction of two integers, but it is not a whole number as it is not an integer. Other examples include -3/4 or 7/3.

Therefore, the statement is false.

The correct answer is B.

Question:

Understanding the definitions of natural numbers and whole numbers and their relationship. Natural numbers are counting numbers starting from 1. Whole numbers include natural numbers and zero.


The statement “Every natural number is a whole number” is True.
Reasoning:
Natural numbers are defined as the set {1, 2, 3, 4, …}.
Whole numbers are defined as the set {0, 1, 2, 3, 4, …}.
By comparing these two sets, we can see that all the numbers present in the set of natural numbers (1, 2, 3, etc.) are also present in the set of whole numbers. The set of whole numbers simply includes zero in addition to all the natural numbers. Therefore, every natural number is indeed a whole number.

The final answer is $\boxed{A}$.

Question:

Understanding the definitions of real numbers, rational numbers, and irrational numbers. A real number is any number that can be found on the number line. Rational numbers can be expressed as a fraction p/q, where p and q are integers and q is not zero. Irrational numbers cannot be expressed as a simple fraction.


The statement “Every real number is an irrational number” is false. Real numbers include both rational and irrational numbers. Rational numbers are a subset of real numbers, and irrational numbers are also a subset of real numbers. For example, the number 2 is a real number, but it is also a rational number because it can be written as 2/1. Therefore, not all real numbers are irrational.

Answer: B

Question:

Understanding the definitions of integers and whole numbers.
Integers are the set of whole numbers and their negative counterparts.
Whole numbers are the set of non-negative integers (0, 1, 2, 3, …).


The statement “Every integer is a whole number” is false.
Integers include positive numbers (like 1, 2, 3), negative numbers (like -1, -2, -3), and zero (0).
Whole numbers include only non-negative numbers: 0, 1, 2, 3, and so on.
Since integers include negative numbers, which are not part of the whole numbers, not every integer is a whole number. For example, -5 is an integer but not a whole number.

Answer:
B

Question:

The question asks for numbers with decimal expansions that are non-terminating and non-recurring. These are the defining characteristics of irrational numbers. Therefore, the core concept is understanding what an irrational number is and how to identify or construct such numbers.


Irrational numbers are numbers that cannot be expressed as a simple fraction p/q, where p and q are integers and q is not zero. Their decimal expansions are infinite and do not repeat in a predictable pattern.

To write three numbers whose decimal expansions are non-terminating and non-recurring, we need to provide examples of irrational numbers. Here are three such examples:

1. The mathematical constant pi (π):
π = 3.1415926535…
The decimal expansion of π goes on forever without repeating any sequence of digits.

2. The square root of any non-perfect square integer:
For example, the square root of 2 (√2):
√2 ≈ 1.4142135623…
The decimal expansion of √2 is non-terminating and non-recurring. Other examples include √3, √5, √7, etc.

3. A number constructed with a non-repeating pattern of digits:
Consider the number 0.101001000100001…
In this number, the digit ‘1’ is followed by an increasing number of ‘0’s. This pattern ensures that no sequence of digits will ever repeat indefinitely, making the decimal expansion non-terminating and non-recurring. You can create infinitely many such numbers by varying the number of zeros between the ones or using other digits in a non-repeating pattern. For instance, 0.121121112… is another such example.

Question:

A rational number can be expressed as a fraction p/q, where p and q are integers and q is not zero. Rational numbers have decimal representations that either terminate or repeat. Irrational numbers cannot be expressed as a simple fraction, and their decimal representations are non-terminating and non-repeating.


The given number is 7.478478…
Observe the decimal part of the number: 478478…
The digits ‘478’ are repeating in a sequence.
A decimal number that has a repeating block of digits is a rational number. This is because any repeating decimal can be converted into a fraction of two integers.
To express 7.478478… as a fraction, let x = 7.478478…
Multiply by 1000 (since there are 3 repeating digits): 1000x = 7478.478478…
Subtract the first equation from the second:
1000x – x = 7478.478478… – 7.478478…
999x = 7471
x = 7471/999
Since 7471/999 is a ratio of two integers, where the denominator is not zero, the number 7.478478… is a rational number.

Therefore, 7.478478… is a rational number.

Question:

A rational number is a number that can be expressed as a fraction p/q, where p and q are integers and q is not zero. Rational numbers have decimal representations that are either terminating or repeating. An irrational number cannot be expressed as a simple fraction. Irrational numbers have decimal representations that are non-terminating and non-repeating.


The given number is 1.101001000100001…
Let’s examine the decimal part of this number. The pattern of zeros is increasing: after the first ‘1’ there is one ‘0’, after the second ‘1’ there are two ‘0’s, after the third ‘1’ there are three ‘0’s, and so on. This pattern of adding an extra zero before each subsequent ‘1’ will continue indefinitely.
This means the decimal representation will never terminate and it will never repeat. There is no block of digits that repeats consistently.
Therefore, since the decimal representation is non-terminating and non-repeating, the number 1.101001000100001… is an irrational number.

Question:

A rational number is a number that can be expressed as a fraction p/q where p and q are integers and q is not zero. Rational numbers have decimal representations that either terminate or repeat. An irrational number cannot be expressed as a simple fraction and has a decimal representation that is non-terminating and non-repeating.


The given number is 0.3796.
To classify this number as rational or irrational, we need to examine its decimal representation.
The decimal 0.3796 is a terminating decimal because it ends after a finite number of digits (four digits after the decimal point).
Terminating decimals can always be expressed as a fraction.
We can write 0.3796 as 3796/10000.
In this fraction, the numerator (3796) and the denominator (10000) are both integers, and the denominator is not zero.
Therefore, since 0.3796 can be expressed as a fraction of two integers, it is a rational number.

The number 0.3796 is rational.

Question:

Rational Numbers: Numbers that can be expressed as a fraction p/q, where p and q are integers and q is not zero.
Irrational Numbers: Numbers that cannot be expressed as a fraction p/q. Their decimal representation is non-terminating and non-repeating.
Perfect Squares: Numbers that are the square of an integer (e.g., 1, 4, 9, 16). The square root of a perfect square is an integer.


No, the square roots of all positive integers are not irrational.
If a positive integer is a perfect square, then its square root is an integer, and all integers are rational numbers (since they can be written as themselves divided by 1).

Example of a square root of a number that is a rational number:
Consider the positive integer 9.
The square root of 9 is 3.
Since 3 can be written as 3/1, it is a rational number.
Therefore, the square root of 9 is a rational number.

Question:

A rational number can be expressed as a fraction p/q where p and q are integers and q is not zero. An irrational number cannot be expressed in this form. Key properties of rational and irrational numbers include:
1. The product of a non-zero rational number and an irrational number is always irrational.
2. The sum, difference, product, and quotient of two rational numbers are rational.
3. The sum, difference, product, and quotient of two irrational numbers can be either rational or irrational.


The number in question is 2π.
We know that π is an irrational number.
The number 2 is a rational number (it can be expressed as 2/1).
According to the property that the product of a non-zero rational number and an irrational number is always irrational, 2 multiplied by π will result in an irrational number.
Therefore, 2π is an irrational number.