How to find the Euclidean norm of a complex number?

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I would like to know how to find the Euclidean norm of a complex number, like $10+i$ or $2-i$?

I would appreciate a clear and easy explanation with the necessary details.

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2 Answers

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For $x + iy \in \mathbb C$ the Euclidean norm is defined as $\| x + iy \| := \sqrt{x^2 + y^2}$.

Now you need to fill in the numbers and compute.

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I am adding an answer especially to clear up OP's confusion about multiplying conjugates and its relation with the Euclidean norm.

Let $a+b i$ be a complex number. Note that its euclidean norm, which I'll denote by $\| \cdot \|_e$, is given by $$\|a+bi\|_e=\sqrt {a^2+b^2}$$

Now what is the conjugate of $a+bi$? We know that it is given by $\overline{a+bi}$ which equals $a-bi$.

So, $$\begin{align}(a+bi)(a-bi)&=a^2-abi+abi-b^2i^2\\ &=a^2-\not{abi}+\not{abi}+b^2 ~~~~\mbox{as $i^2=-1$}\\ &=\|a+bi\|_e^2\end{align}$$

So, to get its euclidean norm, it helps to multiply by its conjugate and take its positive square root.

That is, for a complex number $z$, we have that $$\|z\|_e=\sqrt{z\bar z}$$ where $\bar z$ denotes the conjugate of $z$.

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