The Modulus of a Complex Number

In the setting of The Complex Numbers, with the Real Numbers Identified with a Subset of the Complex Numbers, let real and imaginary parts be as in Real and Imaginary Parts of a Complex Number §parts and square roots as in The k-th Root and the Square Root of a Nonnegative Real Number §square-root, and let $z\in\mathbb{C}$.

¶ The modulus of $z$ is the real number

$$|z|=\sqrt{(\operatorname{Re}z)^{2}+(\operatorname{Im}z)^{2}},$$

the argument of the square root being a sum of squares of real numbers and hence $\ge0$.

¶ For a real number $x$, read in $\mathbb{C}$, we have $x=x+0\,i$ by Rules of Arithmetic in a Commutative Ring: Zero, Signs and Squares, and No Zero Divisors in a Field §zero, so $\operatorname{Re}x=x$ and $\operatorname{Im}x=0$ by The Complex Numbers, with the Real Numbers Identified with a Subset of the Complex Numbers §canonical; hence, as $0^{2}=0$ by Powers in a Commutative Ring, a Field and an Ordered Field: Exponent Laws, Factorisation, Geometric Sums, Monotonicity and Bernoulli's Inequality §product, the modulus of $x$ is $\sqrt{x^{2}+0^{2}}=\sqrt{x^{2}}$, which is the absolute value of $x$ by Properties of k-th Roots: Inverse to Powers, Absolute Values, Products, Monotonicity, Subadditivity, Differences and Limits §abs. The notation $|x|$ is therefore unambiguous.