absolute value — from formula anatomyThe absolute value of a real number is its distance from zero on the number line, always non-negative. For a number x, it is written |x| and equals x if x ≥ 0, and -x if x < 0. It appears throughout physics whenever only the magnitude of a quantity matters, regardless of direction, such as in speed (as opposed to velocity) or in the magnitude of a force.
The notation |x| was introduced by Karl Weierstrass in 1841, though the concept of distance from zero has been implicit in mathematics for centuries.
Absolute value strips the sign from a number, turning negatives into positives. In physics, it is used to compute magnitudes of vectors, like the length of a displacement vector or the magnitude of acceleration, by taking the square root of the sum of squared components—a generalization that reduces to the simple absolute value in one dimension.
It is used in error analysis to express measurement uncertainties, in signal processing to compute signal amplitudes, and in quantum mechanics to find probabilities from wavefunction amplitudes.
The absolute value function is not differentiable at zero, a point that causes subtle issues in optimization problems in physics and engineering.
| D | variable | distance between quantum states — quantum distance |
| ψ1 | variable | first quantum state — wave function |
| ψ2 | variable | second quantum state — wave function |
| ⟨ψ1|ψ2⟩ | operator | computes the complex amplitude of one state in another — inner product |
| |·| | operator | gives the magnitude of the complex amplitude — absolute value |
| √ | operator | takes the non-negative root — square root |
| U | variable | First unitary matrix — unitary matrix |
| V | variable | Second unitary matrix — unitary matrix |
| N | variable | Dimension of the Hilbert space — dimension |
| Tr | operator | Sum of diagonal elements of a matrix — trace operator |
| † | operator | Conjugate transpose of a matrix — hermitian conjugate |
| | | | operator | Magnitude of a complex number — absolute value |
| W | variable | Wigner function of the quantum state — Wigner function |
| X | variable | position quadrature variable — position metre |
| P | variable | momentum quadrature variable — momentum kilogram metre per second |
| ∫ dX dP | operator | integration over the full phase space — double integral |
| | | | operator | absolute value of the Wigner function — absolute value |
| P | variable | Probability distribution over x — probability distribution |
| Q | variable | Probability distribution over x — probability distribution |
| x | variable | Outcome index |
| ∑ | operator | Summation over the discrete sample space — summation |
| |·| | operator | Takes the non-negative magnitude — absolute value |
| a_n | variable | n-th term of the sequence — sequence term |
| a_{n+1} | variable | (n+1)-th term of the sequence — sequence term |
| n | variable | index of the term — index |
| limsup | operator | limit superior (largest accumulation point) — limit superior |
| | | | operator | absolute value (magnitude) — absolute value |
| MAE | variable | Mean absolute error — mean absolute error same as dependent variable |
| n | variable | Number of observations — sample size |
| y_i | variable | True value for observation i — true value same as dependent variable |
| \hat{y}_i | variable | Predicted value for observation i — predicted value same as dependent variable |
| \sum_{i=1}^n | operator | Sum over all observations from 1 to n — summation |
| |\cdot| | operator | Takes the absolute value of the difference — absolute value |
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