Certainly! Let's categorize the given numbers [tex]\( 500, 0.11, -9, \sqrt{5}, \frac{22}{7}, \frac{1}{10}, \)[/tex] and [tex]\( 0 \)[/tex] into the specified sets of numbers.
### a.) Natural numbers:
Natural numbers are positive integers starting from [tex]\(1\)[/tex] onwards. In this case, the only number that fits this criteria is:
[tex]\[ [500] \][/tex]
### b.) Whole numbers:
Whole numbers include all natural numbers and additionally [tex]\(0\)[/tex]. The numbers that fit this criteria are:
[tex]\[ [500, 0] \][/tex]
### c.) Integers:
Integers include all whole numbers and their negative counterparts. The numbers that fit this criteria are:
[tex]\[ [500, -9, 0] \][/tex]
### d.) Rational numbers:
Rational numbers are numbers that can be expressed as the quotient or fraction [tex]\(\frac{p}{q}\)[/tex] of two integers, where [tex]\( p \)[/tex] and [tex]\( q \neq 0 \)[/tex]. The numbers that fit this criteria are:
[tex]\[ [500, 0.11, 2.23606797749979, 3.142857142857143, 0.1, 0] \][/tex]
### e.) Irrational numbers:
Irrational numbers cannot be expressed as a simple fraction; their decimal representation is non-repeating and non-terminating. The numbers that fit this criteria are:
[tex]\[ [0.11, 2.23606797749979, 3.142857142857143, 0.1] \][/tex]
### f.) Real numbers:
Real numbers include all the numbers that exist on the number line, encompassing both rational and irrational numbers. Therefore, all the given numbers fall into this category:
[tex]\[ [500, 0.11, -9, 2.23606797749979, 3.142857142857143, 0.1, 0] \][/tex]
To summarize, the categorizations are:
- Natural numbers: [tex]\( [500] \)[/tex]
- Whole numbers: [tex]\( [500, 0] \)[/tex]
- Integers: [tex]\( [500, -9, 0] \)[/tex]
- Rational numbers: [tex]\( [500, 0.11, 2.23606797749979, 3.142857142857143, 0.1, 0] \)[/tex]
- Irrational numbers: [tex]\( [0.11, 2.23606797749979, 3.142857142857143, 0.1] \)[/tex]
- Real numbers: [tex]\( [500, 0.11, -9, 2.23606797749979, 3.142857142857143, 0.1, 0] \)[/tex]
