A worked example: the 1D vector -7.5
Enter a single-component vector, -7.5, and add zero to it (leaving it unchanged): the result is -7.5, with a magnitude of 7.5. That magnitude formula is √((-7.5)²) = √56.25 = 7.5 — and 7.5 is also exactly what the absolute value calculator returns for |-7.5|. With only one component, "square it, then take the square root" and "drop the sign" produce the identical answer.
The formula: √(x²) is |x|
A vector's magnitude formula is √(sum of squared components). Drop down to one dimension and the sum has only one term, so magnitude becomes √(x²) — and squaring a number then taking the square root always erases its sign, which is precisely what absolute value does. Test it on -20: a 1D vector of -20 has magnitude √((-20)²) = √400 = 20, matching |-20| = 20 exactly.
A second confirmation: (x, 0) also reduces to |x|
The same reduction shows up in two dimensions whenever the second component is zero, since a zero term contributes nothing under the square root. The vector (-9, 0) has magnitude √((-9)² + 0²) = √81 = 9, matching |-9| = 9. The vector (15, 0) has magnitude √(15² + 0²) = √225 = 15, matching |15| = 15 — padding a number with a zero component doesn't change its "length" at all.
Why this connection matters for distance
Subtracting two 1D vectors and taking the magnitude of the result is exactly how far apart two numbers are, regardless of which one is larger. Subtracting -5 from 8 gives a component of 13, with magnitude 13; subtracting 8 from -5 gives a component of -13, but the magnitude is still 13 — matching |13| and |-13|, both of which equal 13. That's the same reason absolute value shows up in distance and error calculations: it's a one-dimensional stand-in for the same "how far, ignoring direction" question that vector magnitude answers in any number of dimensions.