Question
a) How many natural numbers are there from 1 through 10? b) How many whole numbers are there from 0 through 10?
c) How many rational numbers are there from 1 through 10?
d) How many irrational numbers are there from 1 through 10?
e) How many real numbers are there from 1 through 10?
c) How many rational numbers are there from 1 through 10?
d) How many irrational numbers are there from 1 through 10?
e) How many real numbers are there from 1 through 10?
Answers
Answered by
GPT-4o mini
Let's go through each part of your question:
a) How many natural numbers are there from 1 through 10?
- Natural numbers are typically defined as the positive integers starting from 1. Therefore, the natural numbers from 1 through 10 are: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.
- There are **10 natural numbers** in this range.
b) How many whole numbers are there from 0 through 10?
- Whole numbers include 0 and all positive integers. Therefore, the whole numbers from 0 through 10 are: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.
- There are **11 whole numbers** in this range.
c) How many rational numbers are there from 1 through 10?
- Rational numbers include all numbers that can be expressed as the quotient of two integers (including fractions). Since there are infinitely many fractions (for instance, 1.5, 2.5, 3.25, etc.) between any two numbers, there are actually **infinitely many rational numbers** between 1 and 10.
d) How many irrational numbers are there from 1 through 10?
- Irrational numbers are those that cannot be expressed as a fraction of integers, such as ā2, Ļ, e, etc. Just like rational numbers, there are infinitely many irrational numbers between any two numbers in the range from 1 to 10.
- Thus, there are also **infinitely many irrational numbers** between 1 and 10.
e) How many real numbers are there from 1 through 10?
- Real numbers include both rational and irrational numbers. Since there are infinitely many rational numbers and infinitely many irrational numbers in this range, the number of real numbers from 1 to 10 is also infinite.
- Therefore, there are **infinitely many real numbers** in this range.
In summary:
a) 10 natural numbers
b) 11 whole numbers
c) Infinitely many rational numbers
d) Infinitely many irrational numbers
e) Infinitely many real numbers
a) How many natural numbers are there from 1 through 10?
- Natural numbers are typically defined as the positive integers starting from 1. Therefore, the natural numbers from 1 through 10 are: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.
- There are **10 natural numbers** in this range.
b) How many whole numbers are there from 0 through 10?
- Whole numbers include 0 and all positive integers. Therefore, the whole numbers from 0 through 10 are: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.
- There are **11 whole numbers** in this range.
c) How many rational numbers are there from 1 through 10?
- Rational numbers include all numbers that can be expressed as the quotient of two integers (including fractions). Since there are infinitely many fractions (for instance, 1.5, 2.5, 3.25, etc.) between any two numbers, there are actually **infinitely many rational numbers** between 1 and 10.
d) How many irrational numbers are there from 1 through 10?
- Irrational numbers are those that cannot be expressed as a fraction of integers, such as ā2, Ļ, e, etc. Just like rational numbers, there are infinitely many irrational numbers between any two numbers in the range from 1 to 10.
- Thus, there are also **infinitely many irrational numbers** between 1 and 10.
e) How many real numbers are there from 1 through 10?
- Real numbers include both rational and irrational numbers. Since there are infinitely many rational numbers and infinitely many irrational numbers in this range, the number of real numbers from 1 to 10 is also infinite.
- Therefore, there are **infinitely many real numbers** in this range.
In summary:
a) 10 natural numbers
b) 11 whole numbers
c) Infinitely many rational numbers
d) Infinitely many irrational numbers
e) Infinitely many real numbers