Question

how to know if number is real number

Answer 1

A real number is any value that represents a quantity along a continuous line. This includes both rational and irrational numbers. Here’s how you can determine if a number is a real number:

1. **Definition**: Real numbers include:
- **Rational Numbers**: These can be expressed as a fraction \( \frac{a}{b} \), where \( a \) and \( b \) are integers, and \( b \neq 0 \). Examples include \( 1, -2, \frac{1}{2}, 0.75 \).
- **Irrational Numbers**: Numbers that cannot be expressed as a simple fraction. They have non-repeating, non-terminating decimal expansions. Examples include \( \pi \) and \( \sqrt{2} \).

2. **Identify the Number Type**:
- If the number can be written as a fraction of two integers, it is rational (and thus a real number).
- If the number is a whole number, negative number, or a decimal, check if it can be expressed as a fraction.
- If the number is a decimal that repeats or terminates, it is also rational.

3. **Check for Imaginary Parts**: If a number has an imaginary component (like \( 3 + 4i \)), it is not a real number. Here, \( i \) is the imaginary unit.

4. **Common Examples**:
- Real Numbers: \( -3, 0, \frac{4}{5}, 0.333..., \sqrt{2} \)
- Not Real Numbers: \( 3 + 2i \)

5. **Using a Number Line**: If you can plot the number on a number line, it is a real number.

In contrast, any number that falls into the category of complex numbers (like imaginary numbers) is not considered a real number. If you follow these guidelines, you should be able to identify whether a number is real or not.