CBSE Class 9 Maths Chapter 1: Number Systems NCERT Solutions

Yes, zero is a rational number. According to the definition, a rational number is any number that can be expressed in the form <math>\frac{p}{q}</math>, where 'p' and 'q' are integers and 'q' is not equal to zero.

Zero can be written as <math>\frac{0}{1}</math>. Here, p = 0 (an integer) and q = 1 (a non-zero integer). Thus, zero satisfies the definition of a rational number.

Other valid representations include <math>\frac{0}{2}</math>, <math>\frac{0}{-5}</math>, etc., all of which fit the criteria for a rational number.

To find rational numbers between two given numbers, we can use the method of finding a common denominator. We need to find six rational numbers between 3 and 4.

First, we can express 3 and 4 with a denominator that allows us to insert six numbers. A common technique is to add 1 to the number of rational numbers required and use that as the denominator. So, we will use a denominator of 6 + 1 = 7.

Convert 3 and 4 to fractions with a denominator of 7:

$3\; =\; \backslash frac\{3\; \backslash times\; 7\}\{7\}\; =\; \backslash frac\{21\}\{7\}$

$4\; =\; \backslash frac\{4\; \backslash times\; 7\}\{7\}\; =\; \backslash frac\{28\}\{7\}$

Now, we need to find six rational numbers between <math>\frac{21}{7}</math> and <math>\frac{28}{7}</math>. We can do this by taking integers between 21 and 28 as numerators, keeping the denominator as 7.

The integers between 21 and 28 are 22, 23, 24, 25, 26, and 27.

Therefore, the six rational numbers between 3 and 4 are:

$\backslash frac\{22\}\{7\},\; \backslash frac\{23\}\{7\},\; \backslash frac\{24\}\{7\},\; \backslash frac\{25\}\{7\},\; \backslash frac\{26\}\{7\},\; \backslash frac\{27\}\{7\}$

To find five rational numbers between <math>\frac{3}{5}</math> and <math>\frac{4}{5}</math>, we can use the method of finding a common denominator or the average method. Since the denominators are already the same, we can increase the denominator to find more numbers.

We need to find five rational numbers, so we will add 1 to 5, which is 6, and use this as our new denominator. We will convert <math>\frac{3}{5}</math> and <math>\frac{4}{5}</math> to equivalent fractions with a denominator of 6.

Multiply the numerator and denominator by 6:

$\backslash frac\{3\}\{5\}\; =\; \backslash frac\{3\; \backslash times\; 6\}\{5\; \backslash times\; 6\}\; =\; \backslash frac\{18\}\{30\}$

$\backslash frac\{4\}\{5\}\; =\; \backslash frac\{4\; \backslash times\; 6\}\{5\; \backslash times\; 6\}\; =\; \backslash frac\{24\}\{30\}$

Now we need to find five rational numbers between <math>\frac{18}{30}</math> and <math>\frac{24}{30}</math>. We can choose numerators between 18 and 24.

The integers between 18 and 24 are 19, 20, 21, 22, and 23.

So, the five rational numbers are:

$\backslash frac\{19\}\{30\},\; \backslash frac\{20\}\{30\},\; \backslash frac\{21\}\{30\},\; \backslash frac\{22\}\{30\},\; \backslash frac\{23\}\{30\}$

These can be simplified:

$\backslash frac\{19\}\{30\},\; \backslash frac\{2\}\{3\},\; \backslash frac\{7\}\{10\},\; \backslash frac\{11\}\{15\},\; \backslash frac\{23\}\{30\}$

(Note: There are infinitely many rational numbers between any two given rational numbers, so other valid sets of answers are possible.)

1. True. The set of natural numbers is {1, 2, 3, ...}. The set of whole numbers is {0, 1, 2, 3, ...}. Since all natural numbers are included in the set of whole numbers (along with zero), every natural number is a whole number.
2. False. The set of integers includes positive numbers, negative numbers, and zero {..., -3, -2, -1, 0, 1, 2, 3, ...}. The set of whole numbers only includes non-negative numbers {0, 1, 2, 3, ...}. Therefore, negative integers (like -1, -2, -3) are integers but not whole numbers.
3. False. A rational number can be expressed in the form <math>\frac{p}{q}</math> where p and q are integers and q ≠ 0. Whole numbers are {0, 1, 2, 3, ...}. Many rational numbers, such as <math>\frac{2}{3}</math>, <math>\frac{-5}{7}</math>, or <math>\frac{11}{4}</math>, cannot be expressed as whole numbers.