Are $\mathbb{C}-\mathbb{R}$ imaginary numbers?

Question

Background

I am teaching senior high school students about the structure of numbers. Start from defining $\mathbb{Q}$ and $\mathbb{R}$ as the rational and real numbers respectively, we can define $\mathbb{R}-\mathbb{Q}$ as the irrational numbers.

I am trying to use the same logic to define imaginary numbers by making use of the relationship between $\mathbb{R}$ and $\mathbb{C}$. Another definition for imaginary numbers is

numbers that become negative under squaring operation.

Let $\mathbb{C}$ and $\mathbb{R}$ be the complex and real number sets respectively. Are $\mathbb{C}-\mathbb{R}$ imaginary numbers?

Answer 1

Imaginary numbers are real multiples of $\mathrm{i}$. So the complex number $1+\mathrm{i} \in \Bbb{C} \smallsetminus \Bbb{R}$ is neither real nor imaginary.

Answer 2

Depends what you mean by "imaginary." Perhaps you mean an element of $\Bbb{C}$ of the form $ai$ for $a\in \Bbb{R}$ in which case this is false. Indeed, in the complex plane you have removed only the "$x$-axis" so that $$\Bbb{C}\setminus \Bbb{R}=\{a+bi:b \ne 0\:\text{and}\:a,b\in \Bbb{R}\}.$$