Solution

The question asks whether zero is a rational number and if it can be expressed in the form \( \dfrac{p}{q} \), where \( p \) and \( q \) are integers and \( q\neq0 \). This tests the understanding of the definition of rational numbers.

Step 1: Recall the definition of a rational number

A rational number is any number that can be expressed in the form \( \dfrac{p}{q} \), where \( p \) and \( q \) are integers, and \( q \) is not equal to zero. The integer \( p \) is the numerator, and the integer \( q \) is the denominator.

Step 2: Consider zero as a fraction

We need to determine if zero can be written as a fraction with an integer numerator and a non-zero integer denominator. We can write zero as:

\( \dfrac{0}{1} \), \( \dfrac{0}{2} \), \( \dfrac{0}{3} \), \( \dfrac{0}{-1} \), \( \dfrac{0}{-2} \), and so on.

Step 3: Verify the conditions for a rational number

In each of these fractions, the numerator \( p \) is 0, which is an integer. The denominator \( q \) can be any non-zero integer (e.g., 1, 2, 3, -1, -2). Since the denominator is never zero, the condition \( q\neq0 \) is satisfied.

Step 4: Simplify the fractions

Any fraction with 0 as the numerator and a non-zero denominator simplifies to 0. For example:

\( \dfrac{0}{1}=0 \), \( \dfrac{0}{2}=0 \), \( \dfrac{0}{-3}=0 \)

Step 5: Conclude

Yes, zero is a rational number because it can be written in the form \( \dfrac{p}{q} \), where \( p \) and \( q \) are integers and \( q\neq0 \). For example, \( 0=\dfrac{0}{1}=\dfrac{0}{2}=\dfrac{0}{3} \). The denominator \( q \) can also be a negative integer.

Solution

We need to find six rational numbers between 3 and 4. The key idea here is to express 3 and 4 as equivalent rational numbers with a common denominator that allows us to easily insert six rational numbers in between.

Step 1: Determine the common denominator

Since we want to find six rational numbers between 3 and 4, we choose a denominator that is greater than 6. A convenient choice is 6 + 1 = 7. This ensures enough “space” between the two numbers when expressed as fractions.

Step 2: Convert 3 and 4 to equivalent fractions with the chosen denominator

We express 3 and 4 with the denominator 7:

\( 3=\dfrac{3\times7}{7}=\dfrac{21}{7} \)

\( 4=\dfrac{4\times7}{7}=\dfrac{28}{7} \)

Step 3: Identify the rational numbers between the two fractions

Now we need to find six rational numbers between \( \dfrac{21}{7} \) and \( \dfrac{28}{7} \). We can simply increment the numerator by 1 for each subsequent rational number:

\( \dfrac{22}{7},\dfrac{23}{7},\dfrac{24}{7},\dfrac{25}{7},\dfrac{26}{7},\dfrac{27}{7} \)

These six rational numbers lie between \( \dfrac{21}{7} \) (which is 3) and \( \dfrac{28}{7} \) (which is 4).

Final Answer: The six rational numbers between 3 and 4 are \( \dfrac{22}{7},\dfrac{23}{7},\dfrac{24}{7},\dfrac{25}{7},\dfrac{26}{7},\dfrac{27}{7} \).

This method works because by choosing a denominator slightly larger than the number of rationals we want to find, we create enough fractional “slots” between our two original numbers to fit the required number of rational numbers. Any denominator larger than 7 would also work, leading to a different (but equally valid) set of rational numbers between 3 and 4.

Solution

We are asked to find five rational numbers between \( \dfrac{3}{5} \) and \( \dfrac{4}{5} \). To do this, we will convert the fractions to equivalent fractions with a larger denominator so that we can easily identify rational numbers between them.

Step 1: Find a common denominator

We need to find a common denominator for \( \dfrac{3}{5} \) and \( \dfrac{4}{5} \). Since we want to find five rational numbers between them, we should aim for a denominator that is large enough to accommodate these numbers. Multiplying the numerator and denominator of both fractions by 10 will give us a large enough gap.

Step 2: Convert to equivalent fractions

Convert \( \dfrac{3}{5} \) to an equivalent fraction:

\( \dfrac{3}{5}=\dfrac{3\times10}{5\times10}=\dfrac{30}{50} \)

Convert \( \dfrac{4}{5} \) to an equivalent fraction:

\( \dfrac{4}{5}=\dfrac{4\times10}{5\times10}=\dfrac{40}{50} \)

Step 3: Identify rational numbers between the equivalent fractions

Now we need to find five rational numbers between \( \dfrac{30}{50} \) and \( \dfrac{40}{50} \). We can simply choose any five numerators between 30 and 40, keeping the denominator as 50.

For example, we can choose 31, 32, 33, 34, and 35.

Step 4: List the rational numbers

The five rational numbers are:

\( \dfrac{31}{50},\dfrac{32}{50},\dfrac{33}{50},\dfrac{34}{50},\dfrac{35}{50} \)

Final Answer: The five rational numbers between \( \dfrac{3}{5} \) and \( \dfrac{4}{5} \) are \( \dfrac{31}{50},\dfrac{32}{50},\dfrac{33}{50},\dfrac{34}{50},\dfrac{35}{50} \).

Solution

This question tests our understanding of the definitions of natural numbers, whole numbers, integers, and rational numbers, and how these sets relate to each other.

(i) Every natural number is a whole number.

Step 1: Define Natural Numbers

Natural numbers are the counting numbers, starting from 1: \( 1,2,3,4,… \)

Step 2: Define Whole Numbers

Whole numbers include all natural numbers and zero: \( 0,1,2,3,4,… \)

Step 3: Compare the two sets

Since every natural number is also found in the set of whole numbers, the statement is true.

Final Answer: True, since the collection of whole numbers contains all the natural numbers.

(ii) Every integer is a whole number.

Step 1: Define Integers

Integers include all whole numbers and their negative counterparts: \( …,-3,-2,-1,0,1,2,3,… \)

Step 2: Define Whole Numbers (again for clarity)

Whole numbers include all natural numbers and zero: \( 0,1,2,3,4,… \)

Step 3: Compare the two sets

Integers include negative numbers, which are not whole numbers. For example, -1 is an integer but not a whole number.

Final Answer: False, for example \( -2 \) is not a whole number.

(iii) Every rational number is a whole number.

Step 1: Define Rational Numbers

Rational numbers can be expressed in the form \( \dfrac{p}{q} \), where \( p \) and \( q \) are integers and \( q\neq0 \).

Step 2: Define Whole Numbers (again for clarity)

Whole numbers include all natural numbers and zero: \( 0,1,2,3,4,… \)

Step 3: Compare the two sets

Rational numbers include fractions and decimals that are not integers. For example, \( \dfrac{1}{2} \) is a rational number but not a whole number.

Final Answer: False, for example \( \dfrac{1}{2} \) is a rational number but not a whole number.