Question
What is irrational, rational
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GPT-4o mini
In mathematics, the terms "rational" and "irrational" refer to two different types of numbers.
### Rational Numbers:
A rational number is any number that can be expressed as the quotient or fraction \( \frac{a}{b} \), where \( a \) and \( b \) are integers and \( b \neq 0 \). This means that rational numbers can be whole numbers, fractions, or terminating or repeating decimals. Examples of rational numbers include:
- \( \frac{1}{2} \)
- 3 (which can be expressed as \( \frac{3}{1} \))
- -4 (which can be expressed as \( \frac{-4}{1} \))
- 0.75 (which can be expressed as \( \frac{3}{4} \))
- 0.333... (which is \( \frac{1}{3} \))
### Irrational Numbers:
An irrational number, on the other hand, cannot be expressed as a simple fraction. This means that its decimal representation goes on forever without repeating. Examples of irrational numbers include:
- \( \pi \) (approximately 3.14159...)
- \( e \) (the base of natural logarithms, approximately 2.71828...)
- \( \sqrt{2} \) (approximately 1.41421...)
- \( \sqrt{3} \)
Irrational numbers are an important part of the number system, and they help to fill in the gaps between rational numbers on the number line. In summary:
- **Rational numbers**: Can be written as a fraction of two integers.
- **Irrational numbers**: Cannot be written as a fraction; their decimal representation is non-terminating and non-repeating.
### Rational Numbers:
A rational number is any number that can be expressed as the quotient or fraction \( \frac{a}{b} \), where \( a \) and \( b \) are integers and \( b \neq 0 \). This means that rational numbers can be whole numbers, fractions, or terminating or repeating decimals. Examples of rational numbers include:
- \( \frac{1}{2} \)
- 3 (which can be expressed as \( \frac{3}{1} \))
- -4 (which can be expressed as \( \frac{-4}{1} \))
- 0.75 (which can be expressed as \( \frac{3}{4} \))
- 0.333... (which is \( \frac{1}{3} \))
### Irrational Numbers:
An irrational number, on the other hand, cannot be expressed as a simple fraction. This means that its decimal representation goes on forever without repeating. Examples of irrational numbers include:
- \( \pi \) (approximately 3.14159...)
- \( e \) (the base of natural logarithms, approximately 2.71828...)
- \( \sqrt{2} \) (approximately 1.41421...)
- \( \sqrt{3} \)
Irrational numbers are an important part of the number system, and they help to fill in the gaps between rational numbers on the number line. In summary:
- **Rational numbers**: Can be written as a fraction of two integers.
- **Irrational numbers**: Cannot be written as a fraction; their decimal representation is non-terminating and non-repeating.