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Sagot :
Certainly! Let's go through each question step-by-step and provide detailed explanations.
### 1. Is zero a rational number? Can you write it in the form [tex]\(\frac{p}{q}\)[/tex], where [tex]\(p\)[/tex] and [tex]\(q\)[/tex] are integers and [tex]\(q \neq 0\)[/tex]?
Yes, zero is a rational number. A rational number is defined as any number that can be expressed in the form [tex]\( \frac{p}{q} \)[/tex], where [tex]\( p \)[/tex] and [tex]\( q \)[/tex] are integers and [tex]\( q \neq 0 \)[/tex].
To write zero in the form [tex]\( \frac{p}{q} \)[/tex]:
- We can choose [tex]\( p = 0 \)[/tex] and [tex]\( q = 1 \)[/tex] (or any non-zero integer for [tex]\( q \)[/tex]). Thus, [tex]\( \frac{0}{1} = 0 \)[/tex].
So, zero can indeed be written in the form [tex]\( \frac{p}{q} \)[/tex], where [tex]\( p \)[/tex] and [tex]\( q \)[/tex] are integers and [tex]\( q \neq 0 \)[/tex]. Therefore, the answer is True, and zero is a rational number.
### 2. Find six rational numbers between 3 and 4.
Finding rational numbers between two integers involves identifying fractions that lie between those numbers. One straightforward approach is to divide the interval into equal parts.
Let's find six rational numbers between 3 and 4 by dividing the interval into seven equal parts:
[tex]\[ \text{Rational Numbers} = \left\{ 3 + \frac{i}{7} \mid i \text{ is an integer from 1 to 6} \right\} \][/tex]
- When [tex]\( i = 1 \)[/tex], we get [tex]\( 3 + \frac{1}{7} = 3.142857 \)[/tex]
- When [tex]\( i = 2 \)[/tex], we get [tex]\( 3 + \frac{2}{7} = 3.285714 \)[/tex]
- When [tex]\( i = 3 \)[/tex], we get [tex]\( 3 + \frac{3}{7} = 3.428571 \)[/tex]
- When [tex]\( i = 4 \)[/tex], we get [tex]\( 3 + \frac{4}{7} = 3.571429 \)[/tex]
- When [tex]\( i = 5 \)[/tex], we get [tex]\( 3 + \frac{5}{7} = 3.714286 \)[/tex]
- When [tex]\( i = 6 \)[/tex], we get [tex]\( 3 + \frac{6}{7} = 3.857143 \)[/tex]
Therefore, the six rational numbers between 3 and 4 are:
[tex]\[ 3.142857, 3.285714, 3.428571, 3.571429, 3.714286, 3.857143 \][/tex]
### 3. Find five rational numbers between [tex]\(\frac{3}{5}\)[/tex] and [tex]\(\frac{4}{5}\)[/tex].
To find rational numbers between two given fractions, we can use a similar approach of dividing the interval into equal parts:
[tex]\[ \text{Rational Numbers} = \left\{ \frac{3}{5} + i \times \frac{\frac{4}{5} - \frac{3}{5}}{6} \mid i \text{ is an integer from 1 to 5} \right\} \][/tex]
The difference between [tex]\(\frac{4}{5}\)[/tex] and [tex]\(\frac{3}{5}\)[/tex] is:
[tex]\[ \frac{4}{5} - \frac{3}{5} = \frac{1}{5} \][/tex]
So, we divide [tex]\( \frac{1}{5} \)[/tex] by 6, giving us the step size of:
[tex]\[ \text{Step Size} = \frac{1}{30} \][/tex]
Using this step size, we find the five rational numbers:
- When [tex]\( i = 1 \)[/tex], we get [tex]\( \frac{3}{5} + \frac{1}{30} = 0.633333 \)[/tex]
- When [tex]\( i = 2 \)[/tex], we get [tex]\( \frac{3}{5} + 2 \times \frac{1}{30} = 0.666667 \)[/tex]
- When [tex]\( i = 3 \)[/tex], we get [tex]\( \frac{3}{5} + 3 \times \frac{1}{30} = 0.700000 \)[/tex]
- When [tex]\( i = 4 \)[/tex], we get [tex]\( \frac{3}{5} + 4 \times \frac{1}{30} = 0.733333 \)[/tex]
- When [tex]\( i = 5 \)[/tex], we get [tex]\( \frac{3}{5} + 5 \times \frac{1}{30} = 0.766667 \)[/tex]
Therefore, the five rational numbers between [tex]\(\frac{3}{5}\)[/tex] and [tex]\(\frac{4}{5}\)[/tex] are:
[tex]\[ 0.633333, 0.666667, 0.700000, 0.733333, 0.766667 \][/tex]
### 4. State whether the following statements are true or false. Give reasons for your answers.
(i) Every natural number is a whole number.
This statement is True. Natural numbers are the set of positive integers starting from 1 (i.e., [tex]\( \{1, 2, 3, \ldots\} \)[/tex]). Whole numbers are the set of non-negative integers (i.e., [tex]\( \{0, 1, 2, 3, \ldots\} \)[/tex]). Since every natural number is included in the set of whole numbers by definition, this statement is true.
(ii) Every integer is a whole number.
This statement is False. Integers include positive numbers, negative numbers, and zero (i.e., [tex]\( \{\ldots, -2, -1, 0, 1, 2, \ldots\} \)[/tex]), while whole numbers are only the non-negative integers (i.e., [tex]\( \{0, 1, 2, 3, \ldots\} \)[/tex]). Because negative integers are not included in the set of whole numbers, this statement is false.
(iii) Every rational number is a whole number.
This statement is False. Rational numbers are numbers that can be expressed as the quotient or fraction of two integers (i.e., [tex]\( \frac{p}{q} \)[/tex] where [tex]\( p \)[/tex] and [tex]\( q \)[/tex] are integers and [tex]\( q \neq 0 \)[/tex]). These include fractions such as [tex]\( \frac{1}{2} \)[/tex] or [tex]\( \frac{3}{4} \)[/tex], which are not whole numbers. Therefore, not every rational number is a whole number, making this statement false.
So, the answers to the statements are:
(i) True
(ii) False
(iii) False
### 1. Is zero a rational number? Can you write it in the form [tex]\(\frac{p}{q}\)[/tex], where [tex]\(p\)[/tex] and [tex]\(q\)[/tex] are integers and [tex]\(q \neq 0\)[/tex]?
Yes, zero is a rational number. A rational number is defined as any number that can be expressed in the form [tex]\( \frac{p}{q} \)[/tex], where [tex]\( p \)[/tex] and [tex]\( q \)[/tex] are integers and [tex]\( q \neq 0 \)[/tex].
To write zero in the form [tex]\( \frac{p}{q} \)[/tex]:
- We can choose [tex]\( p = 0 \)[/tex] and [tex]\( q = 1 \)[/tex] (or any non-zero integer for [tex]\( q \)[/tex]). Thus, [tex]\( \frac{0}{1} = 0 \)[/tex].
So, zero can indeed be written in the form [tex]\( \frac{p}{q} \)[/tex], where [tex]\( p \)[/tex] and [tex]\( q \)[/tex] are integers and [tex]\( q \neq 0 \)[/tex]. Therefore, the answer is True, and zero is a rational number.
### 2. Find six rational numbers between 3 and 4.
Finding rational numbers between two integers involves identifying fractions that lie between those numbers. One straightforward approach is to divide the interval into equal parts.
Let's find six rational numbers between 3 and 4 by dividing the interval into seven equal parts:
[tex]\[ \text{Rational Numbers} = \left\{ 3 + \frac{i}{7} \mid i \text{ is an integer from 1 to 6} \right\} \][/tex]
- When [tex]\( i = 1 \)[/tex], we get [tex]\( 3 + \frac{1}{7} = 3.142857 \)[/tex]
- When [tex]\( i = 2 \)[/tex], we get [tex]\( 3 + \frac{2}{7} = 3.285714 \)[/tex]
- When [tex]\( i = 3 \)[/tex], we get [tex]\( 3 + \frac{3}{7} = 3.428571 \)[/tex]
- When [tex]\( i = 4 \)[/tex], we get [tex]\( 3 + \frac{4}{7} = 3.571429 \)[/tex]
- When [tex]\( i = 5 \)[/tex], we get [tex]\( 3 + \frac{5}{7} = 3.714286 \)[/tex]
- When [tex]\( i = 6 \)[/tex], we get [tex]\( 3 + \frac{6}{7} = 3.857143 \)[/tex]
Therefore, the six rational numbers between 3 and 4 are:
[tex]\[ 3.142857, 3.285714, 3.428571, 3.571429, 3.714286, 3.857143 \][/tex]
### 3. Find five rational numbers between [tex]\(\frac{3}{5}\)[/tex] and [tex]\(\frac{4}{5}\)[/tex].
To find rational numbers between two given fractions, we can use a similar approach of dividing the interval into equal parts:
[tex]\[ \text{Rational Numbers} = \left\{ \frac{3}{5} + i \times \frac{\frac{4}{5} - \frac{3}{5}}{6} \mid i \text{ is an integer from 1 to 5} \right\} \][/tex]
The difference between [tex]\(\frac{4}{5}\)[/tex] and [tex]\(\frac{3}{5}\)[/tex] is:
[tex]\[ \frac{4}{5} - \frac{3}{5} = \frac{1}{5} \][/tex]
So, we divide [tex]\( \frac{1}{5} \)[/tex] by 6, giving us the step size of:
[tex]\[ \text{Step Size} = \frac{1}{30} \][/tex]
Using this step size, we find the five rational numbers:
- When [tex]\( i = 1 \)[/tex], we get [tex]\( \frac{3}{5} + \frac{1}{30} = 0.633333 \)[/tex]
- When [tex]\( i = 2 \)[/tex], we get [tex]\( \frac{3}{5} + 2 \times \frac{1}{30} = 0.666667 \)[/tex]
- When [tex]\( i = 3 \)[/tex], we get [tex]\( \frac{3}{5} + 3 \times \frac{1}{30} = 0.700000 \)[/tex]
- When [tex]\( i = 4 \)[/tex], we get [tex]\( \frac{3}{5} + 4 \times \frac{1}{30} = 0.733333 \)[/tex]
- When [tex]\( i = 5 \)[/tex], we get [tex]\( \frac{3}{5} + 5 \times \frac{1}{30} = 0.766667 \)[/tex]
Therefore, the five rational numbers between [tex]\(\frac{3}{5}\)[/tex] and [tex]\(\frac{4}{5}\)[/tex] are:
[tex]\[ 0.633333, 0.666667, 0.700000, 0.733333, 0.766667 \][/tex]
### 4. State whether the following statements are true or false. Give reasons for your answers.
(i) Every natural number is a whole number.
This statement is True. Natural numbers are the set of positive integers starting from 1 (i.e., [tex]\( \{1, 2, 3, \ldots\} \)[/tex]). Whole numbers are the set of non-negative integers (i.e., [tex]\( \{0, 1, 2, 3, \ldots\} \)[/tex]). Since every natural number is included in the set of whole numbers by definition, this statement is true.
(ii) Every integer is a whole number.
This statement is False. Integers include positive numbers, negative numbers, and zero (i.e., [tex]\( \{\ldots, -2, -1, 0, 1, 2, \ldots\} \)[/tex]), while whole numbers are only the non-negative integers (i.e., [tex]\( \{0, 1, 2, 3, \ldots\} \)[/tex]). Because negative integers are not included in the set of whole numbers, this statement is false.
(iii) Every rational number is a whole number.
This statement is False. Rational numbers are numbers that can be expressed as the quotient or fraction of two integers (i.e., [tex]\( \frac{p}{q} \)[/tex] where [tex]\( p \)[/tex] and [tex]\( q \)[/tex] are integers and [tex]\( q \neq 0 \)[/tex]). These include fractions such as [tex]\( \frac{1}{2} \)[/tex] or [tex]\( \frac{3}{4} \)[/tex], which are not whole numbers. Therefore, not every rational number is a whole number, making this statement false.
So, the answers to the statements are:
(i) True
(ii) False
(iii) False
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