Question

1. consider the set of real numbers and the set of complex numbers.

a. is every real number also a complex number? explain.

Explanation:

Define complex numbers and real numbers

$$ \mathbb{C} = \{a + bi \mid a, b \in \mathbb{R}, i^2 = -1\} $$

Express real numbers in complex form

$$ \forall a \in \mathbb{R}, \quad a = a + 0i \in \mathbb{C} $$

Determine subset relationship

$$ \mathbb{R} \subset \mathbb{C} $$

Answer:

Yes. Every real number \(a\) can be written in the form \(a + 0i\), which is a complex number with an imaginary part of \(0\). Therefore, the set of real numbers is a subset of the set of complex numbers.