Question
What is a rational number? What is an irrational number? What happens when you add them together?
Answers
GPT-4o mini
A **rational number** is a number that can be expressed as the quotient or fraction of two integers, where the numerator is an integer and the denominator is a non-zero integer. For example, \( \frac{1}{2}, -3, and 4.75 \) are all rational numbers because they can be expressed as \( \frac{1}{2}, \frac{-3}{1}, \) and \( \frac{19}{4} \) respectively.
An **irrational number**, on the other hand, is a number that cannot be expressed as a simple fraction or quotient of two integers. The decimal representation of an irrational number is non-terminating and non-repeating. Examples of irrational numbers include \( \pi \) (approximately \( 3.14159 \)), the square root of 2 (\( \sqrt{2} \)), and \( e \) (the base of the natural logarithm).
When you add a rational number to an irrational number, the result is always an irrational number. This is because the sum of a rational number and an irrational number cannot be expressed as a fraction of two integers; the presence of the irrational number means that the decimal representation of the sum will also be non-terminating and non-repeating.
For example, if you take the rational number \( 2 \) and add it to the irrational number \( \sqrt{2} \):
\[
2 + \sqrt{2}
\]
This sum is irrational.
An **irrational number**, on the other hand, is a number that cannot be expressed as a simple fraction or quotient of two integers. The decimal representation of an irrational number is non-terminating and non-repeating. Examples of irrational numbers include \( \pi \) (approximately \( 3.14159 \)), the square root of 2 (\( \sqrt{2} \)), and \( e \) (the base of the natural logarithm).
When you add a rational number to an irrational number, the result is always an irrational number. This is because the sum of a rational number and an irrational number cannot be expressed as a fraction of two integers; the presence of the irrational number means that the decimal representation of the sum will also be non-terminating and non-repeating.
For example, if you take the rational number \( 2 \) and add it to the irrational number \( \sqrt{2} \):
\[
2 + \sqrt{2}
\]
This sum is irrational.