Answer :
Certainly! Let's explore how we can choose one rational and one irrational number between the two given numbers: [tex]\( 0.3003000300003\ldots \)[/tex] and [tex]\( 0.303003000300003\ldots \)[/tex].
### Step-by-Step Solution:
1. Identifying the Number Characteristics:
- Rational Numbers: A rational number can be expressed as a fraction of two integers (i.e., [tex]\( \frac{a}{b} \)[/tex] where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are integers and [tex]\( b \neq 0 \)[/tex]). Usually, these numbers either terminate (like 0.25) or repeat (like 0.333...).
- Irrational Numbers: An irrational number cannot be expressed as a simple fraction. Its decimal representation neither terminates nor repeats. Examples include [tex]\(\pi\)[/tex] and [tex]\(\sqrt{2}\)[/tex].
2. Choosing a Rational Number:
- One simple way to choose a rational number between the two given numbers is to select a decimal that either terminates or has a repeating pattern and lies strictly between the two bounds.
- Consider [tex]\( 0.301 \)[/tex]. This number is clearly between [tex]\( 0.3003000300003\ldots \)[/tex] and [tex]\( 0.303003000300003\ldots \)[/tex]. Also, it terminates, making it rational.
3. Choosing an Irrational Number:
- To find an irrational number, we need a decimal representation that does not terminate or repeat and lies strictly between the two bounds.
- Let's consider a number like [tex]\( 0.301301301301\ldots \)[/tex]. We are creating a non-repeating, non-terminating decimal representation by continuously choosing digits such that it never settles into a repeating pattern.
- By choosing [tex]\( 0.301301301301...\)[/tex], it will keep expanding in a manner that creates no repeating block of digits, thus making it irrational.
### Final Answer:
- Rational Number: [tex]\( 0.301 \)[/tex]
- Irrational Number: [tex]\( 0.301301301301301\ldots \)[/tex] (non-terminating and non-repeating)
These selections ensure that both numbers lie between [tex]\( 0.3003000300003\ldots \)[/tex] and [tex]\( 0.303003000300003\ldots \)[/tex] and satisfy their respective definitions of rationality and irrationality.
### Step-by-Step Solution:
1. Identifying the Number Characteristics:
- Rational Numbers: A rational number can be expressed as a fraction of two integers (i.e., [tex]\( \frac{a}{b} \)[/tex] where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are integers and [tex]\( b \neq 0 \)[/tex]). Usually, these numbers either terminate (like 0.25) or repeat (like 0.333...).
- Irrational Numbers: An irrational number cannot be expressed as a simple fraction. Its decimal representation neither terminates nor repeats. Examples include [tex]\(\pi\)[/tex] and [tex]\(\sqrt{2}\)[/tex].
2. Choosing a Rational Number:
- One simple way to choose a rational number between the two given numbers is to select a decimal that either terminates or has a repeating pattern and lies strictly between the two bounds.
- Consider [tex]\( 0.301 \)[/tex]. This number is clearly between [tex]\( 0.3003000300003\ldots \)[/tex] and [tex]\( 0.303003000300003\ldots \)[/tex]. Also, it terminates, making it rational.
3. Choosing an Irrational Number:
- To find an irrational number, we need a decimal representation that does not terminate or repeat and lies strictly between the two bounds.
- Let's consider a number like [tex]\( 0.301301301301\ldots \)[/tex]. We are creating a non-repeating, non-terminating decimal representation by continuously choosing digits such that it never settles into a repeating pattern.
- By choosing [tex]\( 0.301301301301...\)[/tex], it will keep expanding in a manner that creates no repeating block of digits, thus making it irrational.
### Final Answer:
- Rational Number: [tex]\( 0.301 \)[/tex]
- Irrational Number: [tex]\( 0.301301301301301\ldots \)[/tex] (non-terminating and non-repeating)
These selections ensure that both numbers lie between [tex]\( 0.3003000300003\ldots \)[/tex] and [tex]\( 0.303003000300003\ldots \)[/tex] and satisfy their respective definitions of rationality and irrationality.
