Question
Irrational numbers can be written as
Answers
GPT-4o mini
Irrational numbers can be represented in several ways:
1. **Decimal Representation**: Irrational numbers have non-repeating and non-terminating decimal expansions. For example, the number \( \pi \) (approximately 3.14159...) and the square root of 2 (approximately 1.41421...) are both irrational.
2. **Radical Form**: Many irrational numbers can be expressed in radical form, such as \( \sqrt{2} \), \( \sqrt{3} \), or more complex forms like \( \sqrt[3]{2} \).
3. **Algebraic Expressions**: Some irrational numbers are solutions to polynomial equations with integer coefficients but are not rational. For example, the number \( \sqrt{2} \) is the solution to the equation \( x^2 - 2 = 0 \).
4. **Transcendental Numbers**: Some irrational numbers, like \( \pi \) and \( e \), are not solutions to any polynomial equation with rational coefficients; they are called transcendental numbers.
In summary, irrational numbers can be written as non-repeating decimals, radicals, or algebraic/ transcendental expressions, but they cannot be expressed as simple fractions.
1. **Decimal Representation**: Irrational numbers have non-repeating and non-terminating decimal expansions. For example, the number \( \pi \) (approximately 3.14159...) and the square root of 2 (approximately 1.41421...) are both irrational.
2. **Radical Form**: Many irrational numbers can be expressed in radical form, such as \( \sqrt{2} \), \( \sqrt{3} \), or more complex forms like \( \sqrt[3]{2} \).
3. **Algebraic Expressions**: Some irrational numbers are solutions to polynomial equations with integer coefficients but are not rational. For example, the number \( \sqrt{2} \) is the solution to the equation \( x^2 - 2 = 0 \).
4. **Transcendental Numbers**: Some irrational numbers, like \( \pi \) and \( e \), are not solutions to any polynomial equation with rational coefficients; they are called transcendental numbers.
In summary, irrational numbers can be written as non-repeating decimals, radicals, or algebraic/ transcendental expressions, but they cannot be expressed as simple fractions.