mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, Pythagorean addition is a

binary operation In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two. More specifically, a binary operation ...

on the

real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...

s that computes the length of the

hypotenuse In geometry, a hypotenuse is the side of a right triangle opposite to the right angle. It is the longest side of any such triangle; the two other shorter sides of such a triangle are called '' catheti'' or ''legs''. Every rectangle can be divided ...

of a

right triangle A right triangle or right-angled triangle, sometimes called an orthogonal triangle or rectangular triangle, is a triangle in which two sides are perpendicular, forming a right angle ( turn or 90 degrees). The side opposite to the right angle i ...

, given its two sides. Like the more familiar addition and multiplication operations of

arithmetic Arithmetic is an elementary branch of mathematics that deals with numerical operations like addition, subtraction, multiplication, and division. In a wider sense, it also includes exponentiation, extraction of roots, and taking logarithms. ...

, it is both

associative In mathematics, the associative property is a property of some binary operations that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement for express ...

and

commutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Perhaps most familiar as a pr ...

. This operation can be used in the conversion of

Cartesian coordinates In geometry, a Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called ''coordinates'', which are the signed distances to the point from two fixed perpendicular o ...

polar coordinates In mathematics, the polar coordinate system specifies a given point (mathematics), point in a plane (mathematics), plane by using a distance and an angle as its two coordinate system, coordinates. These are *the point's distance from a reference ...

, and in the calculation of

Euclidean distance In mathematics, the Euclidean distance between two points in Euclidean space is the length of the line segment between them. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, and therefore is o ...

. It also provides a simple notation and terminology for the

diameter In geometry, a diameter of a circle is any straight line segment that passes through the centre of the circle and whose endpoints lie on the circle. It can also be defined as the longest Chord (geometry), chord of the circle. Both definitions a ...

of a

cuboid In geometry, a cuboid is a hexahedron with quadrilateral faces, meaning it is a polyhedron with six Face (geometry), faces; it has eight Vertex (geometry), vertices and twelve Edge (geometry), edges. A ''rectangular cuboid'' (sometimes also calle ...

, the energy-momentum relation in

physics Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge whi ...

, and the overall noise from independent sources of noise. In its applications to

signal processing Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as audio signal processing, sound, image processing, images, Scalar potential, potential fields, Seismic tomograph ...

and propagation of

measurement uncertainty In metrology, measurement uncertainty is the expression of the statistical dispersion of the values attributed to a quantity measured on an interval or ratio scale. All measurements are subject to uncertainty and a measurement result is complet ...

, the same operation is also called addition in quadrature. A scaled version of this operation gives the

quadratic mean In mathematics, the root mean square (abbrev. RMS, or rms) of a set of values is the square root of the set's mean square. Given a set x_i, its RMS is denoted as either x_\mathrm or \mathrm_x. The RMS is also known as the quadratic mean (denoted M ...

or ''root mean square''. It is implemented in many programming libraries as the hypot function, in a way designed to avoid errors arising due to limited-precision calculations performed on computers.

Donald Knuth Donald Ervin Knuth ( ; born January 10, 1938) is an American computer scientist and mathematician. He is a professor emeritus at Stanford University. He is the 1974 recipient of the ACM Turing Award, informally considered the Nobel Prize of comp ...

has written that "Most of the square root operations in computer programs could probably be avoided if ythagorean additionwere more widely available, because people seem to want square roots primarily when they are computing distances."

Definition

According to the

Pythagorean theorem In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite t ...

, for a

with side lengths

a

and

b

, the length of the

can be calculated as

\sqrt.

This formula defines the Pythagorean addition operation, denoted here as

\oplus

: for any two

a

and

b

, the result of this operation is defined to be

a\oplus b = \sqrt.

For instance, the special right triangle based on the

Pythagorean triple A Pythagorean triple consists of three positive integers , , and , such that . Such a triple is commonly written , a well-known example is . If is a Pythagorean triple, then so is for any positive integer . A triangle whose side lengths are a Py ...

(3,4,5)

gives

3\oplus 4=5

.This example is from . uses two more integer Pythagorean triples, (119,120,169) and (19,180,181), as examples. However, the

integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...

result of this example is unusual: for other integer arguments, Pythagorean addition can produce a

quadratic irrational number In mathematics, a quadratic irrational number (also known as a quadratic irrational or quadratic surd) is an irrational number that is the solution to some quadratic equation with rational coefficients which is irreducible over the rational number ...

as its result.

Properties

The operation

\oplus

and

. Therefore, if three or more numbers are to be combined with this operation, the order of combination makes no difference to the result:

x_1 \oplus x_2 \oplus \cdots \oplus x_n=\sqrt.

Additionally, on the non-negative real numbers, zero is an

identity element In mathematics, an identity element or neutral element of a binary operation is an element that leaves unchanged every element when the operation is applied. For example, 0 is an identity element of the addition of real numbers. This concept is use ...

for Pythagorean addition. On numbers that can be negative, the Pythagorean sum with zero gives the

absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if x is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), ...

x\oplus 0=, x, .

The three properties of associativity, commutativity, and having an identity element (on the non-negative numbers) are the defining properties of a

commutative monoid In abstract algebra, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being . Monoids are semigroups with identity ...

Applications

Distance and diameter

The

between two points in the

Euclidean plane In mathematics, a Euclidean plane is a Euclidean space of Two-dimensional space, dimension two, denoted \textbf^2 or \mathbb^2. It is a geometric space in which two real numbers are required to determine the position (geometry), position of eac ...

, given by their

(x_1,y_1)

and

(x_2,y_2)

, is

(x_1-x_2)\oplus (y_1-y_2).

In the same way, the distance between three-dimensional points

(x_1,y_1,z_1)

and

(x_2,y_2,z_2)

can be found by repeated Pythagorean addition as

(x_1-x_2)\oplus (y_1-y_2)\oplus (z_1-z_2).

Pythagorean addition can also find the length of an interior

diagonal In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word ''diagonal'' derives from the ancient Greek � ...

of a

rectangle In Euclidean geometry, Euclidean plane geometry, a rectangle is a Rectilinear polygon, rectilinear convex polygon or a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that a ...

rectangular cuboid A rectangular cuboid is a special case of a cuboid with rectangular faces in which all of its dihedral angles are right angles. This shape is also called rectangular parallelepiped or orthogonal parallelepiped. Many writers just call these ...

. For a rectangle with sides

a

and

b

, the diagonal length is

a\oplus b

. For a cuboid with side lengths

a

b

, and

c

, the length of a

body diagonal In geometry, a space diagonal (also interior diagonal or body diagonal) of a polyhedron is a line connecting two vertices that are not on the same face. Space diagonals contrast with '' face diagonals'', which connect vertices on the same face (b ...

a\oplus b\oplus c

Coordinate conversion

Pythagorean addition (and its implementation as the hypot function) is often used together with the

atan2 




In computing and mathematics, the function (mathematics), function atan2 is the 2-Argument of a function, argument arctangent. By definition, \theta = \operatorname(y, x) is the angle measure (in radians, with -\pi  0, \\ mu \arctan\left(\fr ...

function (a two-parameter form of the

arctangent In mathematics, the inverse trigonometric functions (occasionally also called ''antitrigonometric'', ''cyclometric'', or ''arcus'' functions) are the inverse functions of the trigonometric functions, under suitably restricted domains. Specific ...

) to convert from

(x,y)

(r,\theta)

\begin
r&=x\oplus y=\mathsf(x,y)\\
\theta&=\mathsf(y,x).\\
\end

Quadratic mean and spread of deviation

The

root mean square In mathematics, the root mean square (abbrev. RMS, or rms) of a set of values is the square root of the set's mean square. Given a set x_i, its RMS is denoted as either x_\mathrm or \mathrm_x. The RMS is also known as the quadratic mean (denote ...

or quadratic mean of a

finite set In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle count and finish counting. For example, is a finite set with five elements. Th ...

n

numbers is

\tfrac

times their Pythagorean sum. This is a

generalized mean In mathematics, generalised means (or power mean or Hölder mean from Otto Hölder) are a family of functions for aggregating sets of numbers. These include as special cases the Pythagorean means (arithmetic mean, arithmetic, geometric mean, ge ...

of the numbers. The

standard deviation In statistics, the standard deviation is a measure of the amount of variation of the values of a variable about its Expected value, mean. A low standard Deviation (statistics), deviation indicates that the values tend to be close to the mean ( ...

of a collection of observations is the quadratic mean of their individual deviations from the mean. When two or more independent random variables are added, the standard deviation of their sum is the Pythagorean sum of their standard deviations. Thus, the Pythagorean sum itself can be interpreted as giving the amount of overall noise when combining independent sources of noise. If the

engineering tolerance Engineering tolerance is the permissible limit or limits of variation in: # a physical dimension; # a measured value or physical property of a material, manufactured object, system, or service; # other measured values (such as temperature, hum ...

s of different parts of an assembly are treated as independent noise, they can be combined using a Pythagorean sum. In

experimental science An experiment is a procedure carried out to support or refute a hypothesis, or determine the efficacy or likelihood of something previously untried. Experiments provide insight into cause-and-effect by demonstrating what outcome occurs when ...

s such as

, addition in quadrature is often used to combine different sources of

. However, this method of

propagation of uncertainty In statistics, propagation of uncertainty (or propagation of error) is the effect of variables' uncertainties (or errors, more specifically random errors) on the uncertainty of a function based on them. When the variables are the values of ex ...

applies only when there is no correlation between sources of uncertainty, and it has been criticized for conflating experimental noise with

systematic error Observational error (or measurement error) is the difference between a measurement, measured value of a physical quantity, quantity and its unknown true value.Dodge, Y. (2003) ''The Oxford Dictionary of Statistical Terms'', OUP. Such errors are ...

Other

The energy-momentum relation in

, describing the energy of a moving particle, can be expressed as the Pythagorean sum

E = mc^2 \oplus pc,

where

m

is the

rest mass The invariant mass, rest mass, intrinsic mass, proper mass, or in the case of bound systems simply mass, is the portion of the total mass of an object or system of objects that is independent of the overall motion of the system. More precisely, ...

of a particle,

p

is its

momentum In Newtonian mechanics, momentum (: momenta or momentums; more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. ...

c

is the

speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant exactly equal to ). It is exact because, by international agreement, a metre is defined as the length of the path travelled by light in vacuum during a time i ...

, and

E

is the particle's resulting relativistic energy. When combining signals, it can be a useful design technique to arrange for the combined signals to be

orthogonal In mathematics, orthogonality (mathematics), orthogonality is the generalization of the geometric notion of ''perpendicularity''. Although many authors use the two terms ''perpendicular'' and ''orthogonal'' interchangeably, the term ''perpendic ...

in polarization or

phase Phase or phases may refer to: Science *State of matter, or phase, one of the distinct forms in which matter can exist *Phase (matter), a region of space throughout which all physical properties are essentially uniform *Phase space, a mathematica ...

, so that they add in quadrature. In early

radio engineering Broadcast engineering or radio engineering is the field of electrical engineering, and now to some extent computer engineering and information technology, which deals with radio and television broadcasting. Audio engineering and RF engineering ...

, this idea was used to design

directional antenna A directional antenna or beam antenna is an antenna that radiates or receives greater radio wave power in specific directions. Directional antennas can radiate radio waves in beams, when greater concentration of radiation in a certain directio ...

s, allowing signals to be received while nullifying the interference from signals coming from other directions. When the same technique is applied in software to obtain a directional signal from a radio or

ultrasound Ultrasound is sound with frequency, frequencies greater than 20 Hertz, kilohertz. This frequency is the approximate upper audible hearing range, limit of human hearing in healthy young adults. The physical principles of acoustic waves apply ...

phased array In antenna (radio), antenna theory, a phased array usually means an electronically scanned array, a computer-controlled Antenna array, array of antennas which creates a radio beam, beam of radio waves that can be electronically steered to point ...

, Pythagorean addition may be used to combine the signals. Other recent applications of this idea include improved efficiency in the frequency conversion of

laser A laser is a device that emits light through a process of optical amplification based on the stimulated emission of electromagnetic radiation. The word ''laser'' originated as an acronym for light amplification by stimulated emission of radi ...

s. In the

psychophysics Psychophysics is the field of psychology which quantitatively investigates the relationship between physical stimulus (physiology), stimuli and the sensation (psychology), sensations and perceptions they produce. Psychophysics has been described ...

haptic perception Haptic perception ( "palpable", ''haptikόs'' "suitable for touch") means literally the ability "to grasp something", and is also known as stereognosis. Perception in this case is achieved through the active exploration of surfaces and objects by a ...

, Pythagorean addition has been proposed as a model for the perceived intensity of

vibration Vibration () is a mechanical phenomenon whereby oscillations occur about an equilibrium point. Vibration may be deterministic if the oscillations can be characterised precisely (e.g. the periodic motion of a pendulum), or random if the os ...

when two kinds of vibration are combined. In

image processing An image or picture is a visual representation. An image can be two-dimensional, such as a drawing, painting, or photograph, or three-dimensional, such as a carving or sculpture. Images may be displayed through other media, including a pr ...

, the Sobel operator for

edge detection Edge or EDGE may refer to: Technology Computing * Edge computing, a network load-balancing system * Edge device, an entry point to a computer network * Adobe Edge, a graphical development application * Microsoft Edge, a web browser developed b ...

consists of a

convolution In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions f and g that produces a third function f*g, as the integral of the product of the two ...

step to determine the

gradient In vector calculus, the gradient of a scalar-valued differentiable function f of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p gives the direction and the rate of fastest increase. The g ...

of an image followed by a Pythagorean sum at each pixel to determine the magnitude of the gradient.

Implementation

In a 1983 paper,

Cleve Moler Cleve Barry Moler (born August 17, 1939) is an American mathematician and computer programmer specializing in numerical analysis. In the mid to late 1970s, he was one of the authors of LINPACK and EISPACK, Fortran libraries for numerical compu ...

and Donald Morrison described an

iterative method In computational mathematics, an iterative method is a Algorithm, mathematical procedure that uses an initial value to generate a sequence of improving approximate solutions for a class of problems, in which the ''i''-th approximation (called an " ...

for computing Pythagorean sums, without taking square roots. This was soon recognized to be an instance of

Halley's method In numerical analysis, Halley's method is a root-finding algorithm used for functions of one real variable with a continuous second derivative. Edmond Halley was an English mathematician and astronomer who introduced the method now called by his ...

, and extended to analogous operations on

matrices Matrix (: matrices or matrixes) or MATRIX may refer to: Science and mathematics * Matrix (mathematics), a rectangular array of numbers, symbols or expressions * Matrix (logic), part of a formula in prenex normal form * Matrix (biology), the ...

. Although many modern implementations of this operation instead compute Pythagorean sums by reducing the problem to the

square root In mathematics, a square root of a number is a number such that y^2 = x; in other words, a number whose ''square'' (the result of multiplying the number by itself, or y \cdot y) is . For example, 4 and −4 are square roots of 16 because 4 ...

function, they do so in a way that has been designed to avoid errors arising from the limited-precision calculations performed on computers. If calculated using the natural formula,

r = \sqrt,

the squares of very large or small values of

x

and

y

may exceed the range of machine precision when calculated on a computer. This may to an inaccurate result caused by

arithmetic underflow The term arithmetic underflow (also floating-point underflow, or just underflow) is a condition in a computer program where the result of a calculation is a number of more precise absolute value than the computer can actually represent in memory ...

and overflow, although when overflow and underflow do not occur the output is within two ulp of the exact result. Common implementations of the hypot function rearrange this calculation in a way that avoids the problem of overflow and underflow and are even more precise. If either input to hypot is infinite, the result is infinite. Because this is true for all possible values of the other input, the

IEEE 754 The IEEE Standard for Floating-Point Arithmetic (IEEE 754) is a technical standard for floating-point arithmetic originally established in 1985 by the Institute of Electrical and Electronics Engineers (IEEE). The standard #Design rationale, add ...

floating-point standard requires that this remains true even when the other input is not a number (NaN).

Calculation order

The difficulty with the naive implementation is that

x^2+y^2

may overflow or underflow, unless the intermediate result is computed with

extended precision Extended precision refers to floating-point number formats that provide greater precision than the basic floating-point formats. Extended-precision formats support a basic format by minimizing roundoff and overflow errors in intermediate value ...

. A common implementation technique is to exchange the values, if necessary, so that

, x, \ge, y,

, and then to use the equivalent form

r = , x,  \sqrt.

The computation of

y/x

cannot overflow unless both

x

and

y

are zero. If

y/x

underflows, the final result is equal to

, x,

, which is correct within the precision of the calculation. The square root is computed of a value between 1 and 2. Finally, the multiplication by

, x,

cannot underflow, and overflows only when the result is too large to represent. One drawback of this rearrangement is the additional division by

x

, which increases both the time and inaccuracy of the computation. More complex implementations avoid these costs by dividing the inputs into more cases: * When

x

is much larger than

y

x\oplus y\approx, x,

, to within machine precision. * When

x^2

overflows, multiply both

x

and

y

by a small scaling factor (e.g. 2⁻⁶⁴ for IEEE single precision), use the naive algorithm which will now not overflow, and multiply the result by the (large) inverse (e.g. 2⁶⁴). * When

y^2

underflows, scale as above but reverse the scaling factors to scale up the intermediate values. * Otherwise, the naive algorithm is safe to use. Additional techniques allow the result to be computed more accurately than the naive algorithm, e.g. to less than one ulp. Researchers have also developed analogous algorithms for computing Pythagorean sums of more than two values.

Fast approximation

The alpha max plus beta min algorithm is a high-speed approximation of Pythagorean addition using only comparison, multiplication, and addition, producing a value whose error is less than 4% of the correct result. It is computed as

a\oplus b\approx \alpha\cdot\max(a,b)+\beta\cdot\min(a,b)

for a careful choice of parameters

\alpha

and

\beta

Programming language support

Pythagorean addition function is present as the hypot function in many

programming language A programming language is a system of notation for writing computer programs. Programming languages are described in terms of their Syntax (programming languages), syntax (form) and semantics (computer science), semantics (meaning), usually def ...

s and their libraries. These include: CSS, D, Fortran, Go,

JavaScript JavaScript (), often abbreviated as JS, is a programming language and core technology of the World Wide Web, alongside HTML and CSS. Ninety-nine percent of websites use JavaScript on the client side for webpage behavior. Web browsers have ...

(since ES2015), Julia,

MATLAB MATLAB (an abbreviation of "MATrix LABoratory") is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks. MATLAB allows matrix manipulations, plotting of functions and data, implementat ...

PHP PHP is a general-purpose scripting language geared towards web development. It was originally created by Danish-Canadian programmer Rasmus Lerdorf in 1993 and released in 1995. The PHP reference implementation is now produced by the PHP Group. ...

, and Python.

C++11 C++11 is a version of a joint technical standard, ISO/IEC 14882, by the International Organization for Standardization (ISO) and International Electrotechnical Commission (IEC), for the C++ programming language. C++11 replaced the prior vers ...

includes a two-argument version of hypot, and a three-argument version for

x\oplus y\oplus z

has been included since

C++17 C17, C-17 or C.17 may refer to: Transportation * , a 1917 British C-class submarine Air * Boeing C-17 Globemaster III, a military transport aircraft * Lockheed Y1C-17 Vega, a six-passenger monoplane * Cierva C.17, a 1928 English experimental ...

. The

Java Java is one of the Greater Sunda Islands in Indonesia. It is bordered by the Indian Ocean to the south and the Java Sea (a part of Pacific Ocean) to the north. With a population of 156.9 million people (including Madura) in mid 2024, proje ...

implementation of hypot can be used by its interoperable JVM-based languages including

Apache Groovy Apache Groovy is a Java-syntax-compatible object-oriented programming language for the Java platform. It is both a static and dynamic language with features similar to those of Python, Ruby, and Smalltalk. It can be used as both a programming l ...

Clojure Clojure (, like ''closure'') is a dynamic programming language, dynamic and functional programming, functional dialect (computing), dialect of the programming language Lisp (programming language), Lisp on the Java (software platform), Java platfo ...

, Kotlin, and Scala. Similarly, the version of hypot included with

Ruby Ruby is a pinkish-red-to-blood-red-colored gemstone, a variety of the mineral corundum ( aluminium oxide). Ruby is one of the most popular traditional jewelry gems and is very durable. Other varieties of gem-quality corundum are called sapph ...

extends to Ruby-based

domain-specific language A domain-specific language (DSL) is a computer language specialized to a particular application domain. This is in contrast to a general-purpose language (GPL), which is broadly applicable across domains. There are a wide variety of DSLs, ranging ...

s such as Progress Chef. In

Rust Rust is an iron oxide, a usually reddish-brown oxide formed by the reaction of iron and oxygen in the catalytic presence of water or air moisture. Rust consists of hydrous iron(III) oxides (Fe2O3·nH2O) and iron(III) oxide-hydroxide (FeO(OH) ...

, hypot is implemented as a

method Method (, methodos, from μετά/meta "in pursuit or quest of" + ὁδός/hodos "a method, system; a way or manner" of doing, saying, etc.), literally means a pursuit of knowledge, investigation, mode of prosecuting such inquiry, or system. In re ...

floating point In computing, floating-point arithmetic (FP) is arithmetic on subsets of real numbers formed by a ''significand'' (a signed sequence of a fixed number of digits in some base) multiplied by an integer power of that base. Numbers of this form ...

objects rather than as a two-argument function.

Metafont Metafont is a page description language, description language used to define raster fonts. It is also the name of the interpreter (computer software), interpreter that executes Metafont code, generating the bitmap fonts that can be embedded into ...

has Pythagorean addition and

subtraction Subtraction (which is signified by the minus sign, –) is one of the four Arithmetic#Arithmetic operations, arithmetic operations along with addition, multiplication and Division (mathematics), division. Subtraction is an operation that repre ...

as built-in operations, under the symbols ++ and +-+ respectively.

History

The

on which this operation is based was studied in ancient

Greek mathematics Ancient Greek mathematics refers to the history of mathematical ideas and texts in Ancient Greece during Classical antiquity, classical and late antiquity, mostly from the 5th century BC to the 6th century AD. Greek mathematicians lived in cities ...

, and may have been known earlier in

Egyptian mathematics Ancient Egyptian mathematics is the mathematics that was developed and used in Ancient Egypt 3000 to c. , from the Old Kingdom of Egypt until roughly the beginning of Hellenistic Egypt. The ancient Egyptians utilized a numeral system for counti ...

and

Babylonian mathematics Babylonian mathematics (also known as Assyro-Babylonian mathematics) is the mathematics developed or practiced by the people of Mesopotamia, as attested by sources mainly surviving from the Old Babylonian period (1830–1531 BC) to the Seleucid ...

; see . However, its use for computing distances in Cartesian coordinates could not come until after

René Descartes René Descartes ( , ; ; 31 March 1596 – 11 February 1650) was a French philosopher, scientist, and mathematician, widely considered a seminal figure in the emergence of modern philosophy and Modern science, science. Mathematics was paramou ...

invented these coordinates in 1637; the formula for distance from these coordinates was published by

Alexis Clairaut Alexis Claude Clairaut (; ; 13 May 1713 – 17 May 1765) was a French mathematician, astronomer, and geophysicist. He was a prominent Newtonian whose work helped to establish the validity of the principles and results that Isaac Newton, Sir Isaa ...

in 1731. The terms "Pythagorean addition" and "Pythagorean sum" for this operation have been used at least since the 1950s, and its use in signal processing as "addition in quadrature" goes back at least to 1919. From the 1920s to the 1940s, before the widespread use of computers, multiple designers of

slide rule A slide rule is a hand-operated mechanical calculator consisting of slidable rulers for conducting mathematical operations such as multiplication, division, exponents, roots, logarithms, and trigonometry. It is one of the simplest analog ...

s included square-root scales in their devices, allowing Pythagorean sums to be calculated mechanically. Researchers have also investigated

analog circuit Analogue electronics () are electronic systems with a continuously variable signal, in contrast to digital electronics where signals usually take only two levels. The term ''analogue'' describes the proportional relationship between a signal ...

s for approximating the value of Pythagorean sums.

References

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ISO/IEC JTC 1/SC 22 ISO/IEC JTC 1/SC 22 Programming languages, their environments and system software interfaces is a standardization subcommittee of the Joint Technical Committee ISO/IEC JTC 1 of the International Organization for Standardization (ISO) and the Intern ...

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Addition Addition (usually signified by the Plus and minus signs#Plus sign, plus symbol, +) is one of the four basic Operation (mathematics), operations of arithmetic, the other three being subtraction, multiplication, and Division (mathematics), divis ...

Numerical analysis