Question

Give an example of a real number, an imaginary number, and a complex number. Explain how you know what type each one is.
Real number:
Imaginary Number:
Complex number:

Answer 1

Sure! Here’s an example of each type of number, along with explanations for how we classify them:

**Real number:**
Example: \( 3.14 \)
Explanation: A real number is any number that can be found on the number line, including both rational numbers (like integers and fractions) and irrational numbers. The number \( 3.14 \) is a rational number because it can be expressed as a fraction (\( \frac{314}{100} \)). It does not have an imaginary part.

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**Imaginary number:**
Example: \( 5i \)
Explanation: An imaginary number is defined as a number that can be written as a real number multiplied by the imaginary unit \( i \), which is the square root of -1. The number \( 5i \) is purely imaginary because it is composed only of the imaginary unit \( i \) and has no real part (the real part is \( 0 \)).

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**Complex number:**
Example: \( 2 + 3i \)
Explanation: A complex number is defined as a number that has both a real part and an imaginary part and can be expressed in the form \( a + bi \), where \( a \) is the real part and \( b \) is the coefficient of the imaginary part. In this example, \( 2 + 3i \) has a real part \( 2 \) and an imaginary part \( 3i \), making it a complex number.

To summarize:
- **Real number:** \( 3.14 \) (no imaginary part)
- **Imaginary number:** \( 5i \) (no real part)
- **Complex number:** \( 2 + 3i \) (has both real and imaginary parts)