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

Mathematics Asked by Unreal Engine 5 Coming Soon on January 5, 2022

# 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?

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.

Answered by Eric Towers on January 5, 2022

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

Answered by Alekos Robotis on January 5, 2022

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