Background
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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

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> numbers that become negative under squaring operation.
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[![enter image description here](https://i.stack.imgur.com/UlMfW.png)](https://i.stack.imgur.com/UlMfW.png)

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