square root — from formula anatomyThe square root of a number x is a value that, when multiplied by itself, gives x. For example, the square root of 9 is 3, because 3 × 3 = 9. It is one of the most fundamental operations in mathematics, appearing in formulas across physics, from the Pythagorean theorem (c = √(a² + b²)) to the root-mean-square speed of gas molecules and the standard deviation in statistics.
The concept dates back to ancient Babylonians (c. 1800 BCE) who had a method for approximating square roots. The symbol √ was introduced by Christoph Rudolff in 1525.
The square root is the inverse operation of squaring. For positive numbers, there are two roots (positive and negative), but the principal square root is the positive one. It can be computed by various algorithms, such as the Babylonian method (iterative averaging) or using logarithms and exponentials in calculators.
Used in virtually every quantitative science: calculating distances, standard deviations, wave amplitudes, and in algorithms like least-squares fitting.
The square root of 2 is irrational, meaning it cannot be expressed as a fraction of two integers — a discovery that reportedly shocked the Pythagoreans.
| 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 |
| T | variable | tunneling probability — transmission probability |
| z1 | variable | left turning point of the barrier — position metre |
| z2 | variable | right turning point of the barrier — position metre |
| m | variable | mass of the tunneling particle — mass kilogram |
| V | variable | potential energy as a function of position — potential energy joule |
| E | variable | energy of the particle — energy joule |
| ħ | constant | fundamental quantum of action divided by 2π — reduced planck constant = 1.054571817e-34 joule second |
| ∫ | operator | sums the integrand over the interval — integral |
| √ | operator | takes the non-negative root — square root |
| exp | operator | raises e to the given power — exponential |
| H | variable | Hubble parameter — hubble parameter per second |
| z | variable | redshift — redshift |
| H0 | variable | Hubble constant — hubble constant per second |
| Ωm | variable | matter density parameter — density parameter |
| Ωr | variable | radiation density parameter — density parameter |
| Ωk | variable | curvature density parameter — density parameter |
| ΩΛ | variable | cosmological constant density parameter — density parameter |
| Ωx | variable | dark energy density parameter (general) — density parameter |
| f(z) | variable | dark energy equation-of-state function — equation of state function |
| sqrt | operator | square root — square root |
| \text{ASD}(f) | variable | Amplitude spectral density at frequency f — amplitude spectral density per root hertz |
| S(f) | variable | Power spectral density at frequency f — power spectral density per hertz |
| f | variable | Frequency — frequency hertz |
| \sqrt{} | operator | Mathematical operation taking the square root — square root |
| P | variable | orbital period — period second |
| a | variable | semi-major axis of the orbit — length metre |
| M | variable | mass of the central body — mass kilogram |
| G | constant | constant of universal gravitation — gravitational constant = 6.67430e-11 cubic metre per kilogram per second squared |
| sqrt | operator | takes the square root — square root |
| d | variable | Euclidean distance — distance metre |
| x₁ | variable | x-coordinate of point 1 — coordinate metre |
| x₀ | variable | x-coordinate of point 0 — coordinate metre |
| y₁ | variable | y-coordinate of point 1 — coordinate metre |
| y₀ | variable | y-coordinate of point 0 — coordinate metre |
| √ | operator | takes the square root of the sum — square root |
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