What Is The Square Root Of 4? How To Find The Square Root Of 4?

Square roots of negative numbers can be discussed within the framework of complex numbers. More generally, square roots can be considered in any context in which a notion of the «square» of a mathematical object is defined. These include function spaces and square matrices, among other mathematical structures.

According to historian of mathematics D.E. Smith, Aryabhata’s method for finding the square root was first introduced in Europe by Cataneo—in 1546.

The principal square root function   (usually just referred to as the «square root function») is a function that maps the set of nonnegative real numbers onto itself. In geometrical terms, the square root function maps the area of a square to its side length.

The square root of x is rational if and only if x is a rational number that can be represented as a ratio of two perfect squares. (See square root of 2 for proofs that this is an irrational number, and quadratic irrational for a proof for all non-square natural numbers.) The square root function maps rational numbers into algebraic numbers, the latter being a superset of the rational numbers).

For all real numbers x,

<span data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0640b9c1359d9e56c1047817a6abaeb9d12f0eb9" data-alt="
\sqrt{x^2} = \left|x\right| =
\begin{cases}
x, & \mbox{if }x \ge 0 \\
-x, & \mbox{if }x       (see absolute value)

For all nonnegative real numbers x and y,

The square root function is continuous for all nonnegative x, and differentiable for all positive x. If f denotes the square root function, whose derivative is given by:

The square root of a nonnegative number is used in the definition of Euclidean norm (and distance), as well as in generalizations such as Hilbert spaces. It defines an important concept of standard deviation used in probability theory and statistics. It has a major use in the formula for roots of a quadratic equation; quadratic fields and rings of quadratic integers, which are based on square roots, are important in algebra and have uses in geometry. Square roots frequently appear in mathematical formulas elsewhere, as well as in many physical laws.

A positive number has two square roots, one positive, and one negative, which are opposite to each other. When talking of the square root of a positive integer, it is usually the positive square root that is meant.

The square roots of an integer are algebraic integers—more specifically quadratic integers.

The square root of a positive integer is the product of the roots of its prime factors, because the square root of a product is the product of the square roots of the factors. Since   only roots of those primes having an odd power in the factorization are necessary. More precisely, the square root of a prime factorization is

As decimal expansions

As expansions in other numeral systems

As with before, the square roots of the perfect squares (e.g., 0, 1, 4, 9, 16) are integers. In all other cases, the square roots of positive integers are irrational numbers, and therefore have non-repeating digits in any standard positional notation system.

The square roots of small integers are used in both the SHA-1 and SHA-2 hash function designs to provide nothing up my sleeve numbers.

As periodic continued fractions

One of the most intriguing results from the study of irrational numbers as continued fractions was obtained by Joseph Louis Lagrange c.. Lagrange found that the representation of the square root of any non-square positive integer as a continued fraction is periodic. That is, a certain pattern of partial denominators repeats indefinitely in the continued fraction. In a sense these square roots are the very simplest irrational numbers, because they can be represented with a simple repeating pattern of integers.

Square roots of positive numbers are not in general rational numbers, and so cannot be written as a terminating or recurring decimal expression. Therefore in general any attempt to compute a square root expressed in decimal form can only yield an approximation, though a sequence of increasingly accurate approximations can be obtained.

where and ₁₀ are the natural and base-10 logarithms.

as it allows one to adjust the estimate by some amount and measure the square of the adjustment in terms of the original estimate and its square. Furthermore,   when is close to 0, because the tangent line to the graph of   at  , as a function of alone, is  . Thus, small adjustments to can be planned out by setting   to , or  .

1. Start with an arbitrary positive start value x. The closer to the square root of a, the fewer the iterations that will be needed to achieve the desired precision.
2. Replace x by the average (x + a/x) / 2 between x and a/x.
3. Repeat from step 2, using this average as the new value of x.

That is, if an arbitrary guess for   is x₀, and x_{n + 1} = (x_n + a/x_n) / 2, then each x_n is an approximation of   which is better for large n than for small n. If a is positive, the convergence is quadratic, which means that in approaching the limit, the number of correct digits roughly doubles in each next iteration. If , the convergence is only linear.

Using the identity

The time complexity for computing a square root with n digits of precision is equivalent to that of multiplying two n-digit numbers.

Another useful method for calculating the square root is the shifting nth root algorithm, applied for .

The square of any positive or negative number is positive, and the square of 0 is 0. Therefore, no negative number can have a real square root. However, it is possible to work with a more inclusive set of numbers, called the complex numbers, that does contain solutions to the square root of a negative number. This is done by introducing a new number, denoted by i (sometimes by j, especially in the context of electricity where «i» traditionally represents electric current) and called the imaginary unit, which is defined such that i² = −1. Using this notation, we can think of i as the square root of −1, but we also have (−i)² = i² = −1 and so −i is also a square root of −1. By convention, the principal square root of −1 is i, or more generally, if x is any nonnegative number, then the principal square root of −x is

The right side (as well as its negative) is indeed a square root of −x, since

For every non-zero complex number z there exist precisely two numbers w such that w² = z: the principal square root of z (defined below), and its negative.

Principal square root of a complex number

To find a definition for the square root that allows us to consistently choose a single value, called the principal value, we start by observing that any complex number   can be viewed as a point in the plane,   expressed using Cartesian coordinates. The same point may be reinterpreted using polar coordinates as the pair   where   is the distance of the point from the origin, and   is the angle that the line from the origin to the point makes with the positive real ( ) axis. In complex analysis, the location of this point is conventionally written   If

then the principal square root of   is defined to be the following:

The principal square root function is thus defined using the nonpositive real axis as a branch cut.
If   is a non-negative real number (which happens if and only if  ) then the principal square root of   is   in other words, the principal square root of a non-negative real number is just the usual non-negative square root.
It is important that <span data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b27eb4930922e649015e37b8d782b6b1ad7d55b9" data-alt="{\displaystyle -\pi   because if, for example,   (so  ) then the principal square root is but using   would instead produce the other square root  

The principal square root function is holomorphic everywhere except on the set of non-positive real numbers (on strictly negative reals it is not even continuous). The above Taylor series for   remains valid for complex numbers   with <span data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4e657241d23e0514c31745c2d302fffa61a77ed" data-alt="{\displaystyle |x| 

The above can also be expressed in terms of trigonometric functions:

where is the sign of (except that, here, sgn(0) = 1). In particular, the imaginary parts of the original number and the principal value of its square root have the same sign. The real part of the principal value of the square root is always nonnegative.

For example, the principal square roots of are given by:

where <span data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3aeabfa69185eedc1363619bb534927d46eb8aff" data-alt="{\displaystyle -\pi   and <span data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/679df4a1684cbd40b9ff7ac9bbed56bffe6a2903" data-alt="{\displaystyle -\pi  .

•  
  Counterexample for the principal square root: z = −1 and w = −1
  This equality is valid only when <span data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/616dd651f429490e09f3398a52a279be4c00a279" data-alt="{\displaystyle -\pi  
•  
  Counterexample for the principal square root: and z = −1
  This equality is valid only when <span data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd8739a5027cfcf839f6d2f8105a1e94a115622" data-alt="{\displaystyle -\pi  
•  
  Counterexample for the principal square root: z = −1)
  This equality is valid only when  

A similar problem appears with other complex functions with branch cuts, e.g., the complex logarithm and the relations logz + logw = log(zw) or log(z^*) = log(z)^* which are not true in general.

if the branch includes +i or

if the branch includes −i, while the right-hand side becomes

where the last equality,   is a consequence of the choice of branch in the redefinition of √.

A cube root of   is a number   such that  ; it is denoted  

If is an integer greater than two, a th root of   is a number   such that  ; it is denoted  

Given any polynomial , a root of is a number such that p(y) = 0. For example, the th roots of are the roots of the polynomial (in )  

Abel–Ruffini theorem states that, in general, the roots of a polynomial of degree five or higher cannot be expressed in terms of th roots.

Each element of an integral domain has no more than 2 square roots. The difference of two squares identity u² − v² = (u − v)(u + v) is proved using the commutativity of multiplication. If and are square roots of the same element, then u² − v² = 0. Because there are no zero divisors this implies or u + v = 0, where the latter means that two roots are additive inverses of each other. In other words if an element a square root of an element exists, then the only square roots of are and . The only square root of 0 in an integral domain is 0 itself.

In a field of characteristic 2, an element either has one square root or does not have any at all, because each element is its own additive inverse, so that −u = u. If the field is finite of characteristic 2 then every element has a unique square root. In a field of any other characteristic, any non-zero element either has two square roots, as explained above, or does not have any.

Given an odd prime number , let q = p^e for some positive integer . A non-zero element of the field with elements is a quadratic residue if it has a square root in . Otherwise, it is a quadratic non-residue. There are (q − 1)/2 quadratic residues and (q − 1)/2 quadratic non-residues; zero is not counted in either class. The quadratic residues form a group under multiplication. The properties of quadratic residues are widely used in number theory.

Unlike in an integral domain, a square root in an arbitrary (unital) ring need not be unique up to sign. For example, in the ring   of integers modulo 8 (which is commutative, but has zero divisors), the element 1 has four distinct square roots: ±1 and ±3.

Another example is provided by the ring of quaternions   which has no zero divisors, but is not commutative. Here, the element −1 has infinitely many square roots, including , , and . In fact, the set of square roots of −1 is exactly

A square root of 0 is either 0 or a zero divisor. Thus in rings where zero divisors do not exist, it is uniquely 0. However, rings with zero divisors may have multiple square roots of 0. For example, in   any multiple of is a square root of 0.

The square root of a positive number is usually defined as the side length of a square with the area equal to the given number. But the square shape is not necessary for it: if one of two similar planar Euclidean objects has the area a times greater than another, then the ratio of their linear sizes is  .

A square root can be constructed with a compass and straightedge. In his Elements, Euclid (fl. 300 BC) gave the construction of the geometric mean of two quantities in two different places: Proposition II.14 and Proposition VI.13. Since the geometric mean of a and b is  , one can construct   simply by taking .

The construction is also given by Descartes in his La Géométrie, see figure 2 on page 2. However, Descartes made no claim to originality and his audience would have been quite familiar with Euclid.

Euclid’s second proof in Book VI depends on the theory of similar triangles. Let AHB be a line segment of length with AH = a and HB = b. Construct the circle with AB as diameter and let C be one of the two intersections of the perpendicular chord at H with the circle and denote the length CH as h. Then, using Thales’ theorem and, as in the proof of Pythagoras’ theorem by similar triangles, triangle AHC is similar to triangle CHB (as indeed both are to triangle ACB, though we don’t need that, but it is the essence of the proof of Pythagoras’ theorem) so that AH:CH is as HC:HB, i.e. a/h = h/b, from which we conclude by cross-multiplication that h² = ab, and finally that  . When marking the midpoint O of the line segment AB and drawing the radius OC of length (a + b)/2, then clearly OC > CH, i.e.   (with equality if and only if ), which is the arithmetic–geometric mean inequality for two variables and, as noted above, is the basis of the Ancient Greek understanding of «Heron’s method».

Another method of geometric construction uses right triangles and induction:   can be constructed, and once   has been constructed, the right triangle with legs 1 and   has a hypotenuse of  . Constructing successive square roots in this manner yields the Spiral of Theodorus depicted above.

• Dauben, Joseph W. (2007). «Chinese Mathematics I». In Katz, Victor J. (ed.). The Mathematics of Egypt, Mesopotamia, China, India, and Islam. Princeton: Princeton University Press. ISBN 978-0-691-11485-9.
• Gel’fand, Izrael M.; Shen, Alexander (1993). Algebra (3rd ed.). Birkhäuser. p. 120. ISBN 0-8176-3677-3.
• Joseph, George (2000). The Crest of the Peacock. Princeton: Princeton University Press. ISBN 0-691-00659-8.
• Smith, David (1958). History of Mathematics. Vol. 2. New York: Dover Publications. ISBN 978-0-486-20430-7.
• Selin, Helaine (2008), Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures, Springer, Bibcode:2008ehst.book…..S, ISBN 978-1-4020-4559-2.

There are four all-Harshad numbers: 1, 2, 4, and 6. 12, which is divisible by four thrice over, is a Harshad number in all bases except octal.

A four-sided plane figure is a quadrilateral or quadrangle, sometimes also called a tetragon. It can be further classified as a rectangle or oblong, kite, rhombus, and square.

Four-dimensional space is the highest-dimensional space featuring more than three regular convex figures:

• Two-dimensional: infinitely many regular polygons.
• Three-dimensional: five regular polyhedra; the five Platonic solids which are the tetrahedron, cube, octahedron, dodecahedron, and icosahedron.
• Four-dimensional: six regular polychora; the 5-cell, 8-cell or tesseract, 16-cell, 24-cell, 120-cell, and 600-cell. The 24-cell, made of regular octahedra, has no analogue in any other dimension; it is self-dual, with its 24-cell honeycomb dual to the 16-cell honeycomb.
• Five-dimensional and every higher dimension: three regular convex  —polytopes, all within the infinite family of regular  —simplexes,  —hypercubes, and  —orthoplexes.

The fourth dimension is also the highest dimension where regular self-intersecting figures exist:

• Two-dimensional: infinitely many regular star polygons.
• Three-dimensional: four regular star polyhedra, the regular Kepler-Poinsot star polyhedra.
• Four-dimensional: ten regular star polychora, the Schläfli–Hess star polychora. They contain cells of Kepler-Poinsot polyhedra alongside regular tetrahedra, icosahedra and dodecahedra.
• Five-dimensional and every higher dimension: zero regular star-polytopes; uniform star polytopes in dimensions   >   are the most symmetric, which mainly originate from stellations of regular  -polytopes.

Altogether, sixteen (or 16 = 4²) regular convex and star polychora are generated from symmetries of four (4) Coxeter Weyl groups and point groups in the fourth dimension: the   simplex,   hypercube,   icositetrachoric, and   hexacosichoric groups; with the   demihypercube group generating two alternative constructions.

There are also sixty-four (or 64 = 4³) four-dimensional Bravais lattices, and sixty-four uniform polychora in the fourth dimension based on the same  ,  ,   and   Coxeter groups, and extending to prismatic groups of uniform polyhedra, including one special non-Wythoffian form, the grand antiprism. There are also two infinite families of duoprisms and antiprismatic prisms in the fourth dimension.

Four-dimensional differential manifolds have some unique properties. There is only one differential structure on   except when   =  , in which case there are uncountably many.

There are four Hopf fibrations of hyperspheres:

While the shape of the character for the digit 4 has an ascender in most modern typefaces, in typefaces with text figures the glyph usually has a descender, as, for example, in  .

• Four Noble Truths – Dukkha, Samudaya, Nirodha, Magga^[19]
• Four sights – observations which affected Prince Siddhartha deeply and made him realize the sufferings of all beings, and compelled him to begin his spiritual journey—an old man, a sick man, a dead man, and an ascetic^[20]
• Four Great Elements – earth, water, fire, and wind^[21]
• Four Heavenly Kings^[22]
• Four Foundations of Mindfulness – contemplation of the body, contemplation of feelings, contemplation of mind, contemplation of mental objects^[19]
• Four Right Exertions^[23]
• Four Bases of Power^[24]
• Four jhānas^[25]
• Four arūpajhānas^[26]
• Four Divine Abidings – loving-kindness, compassion, sympathetic joy, and equanimity^[27]
• Four stages of enlightenment – stream-enterer, once-returner, non-returner, and arahant^[28]
• Four main pilgrimage sites – Lumbini, Bodh Gaya, Sarnath, and Kusinara^[19]

• The Tetragrammaton is the four-letter name of God.^[29]
• Ezekiel has a vision of four living creatures: a man, a lion, an ox, and an eagle.^[30]
• The four Matriarchs (foremothers) of Judaism are Sarah, Rebekah, Leah, and Rachel.^[31]
• The Four Species (lulav, hadass, aravah and etrog) are taken as one of the mitzvot on the Jewish holiday of Sukkot. (Judaism)^[32]
• The Four Cups of Wine to drink on the Jewish holiday of Passover. (Judaism)^[33]
• The Four Questions to be asked on the Jewish holiday of Passover. (Judaism)^[33]
• The Four Sons to be dealt with on the Jewish holiday of Passover. (Judaism)^[33]
• The Four Expressions of Redemption to be said on the Jewish holiday of Passover. (Judaism)^[34]
• The four Gospels: Matthew, Mark, Luke, and John. (Christianity)^[35]
• The Four Horsemen of the Apocalypse ride in the Book of Revelation. (Christianity)^[36]
• The four holy cities of Judaism: Jerusalem, Hebron, Safed, and Tiberius^[37]

• There are four Vedas: Rigveda, Samaveda, Yajurveda and Atharvaveda.^[38]
• In Puruṣārtha, there are four aims of human life: Dharma, Artha, Kāma, Moksha.^[39]
• The four stages of life Brahmacharya (student life), Grihastha (household life), Vanaprastha (retired life) and Sannyasa (renunciation).^[40]
• The four primary castes or strata of society: Brahmana (priest/teacher), Kshatriya (warrior/politician), Vaishya (landowner/entrepreneur) and Shudra (servant/manual laborer).^[41]
• The swastika symbol is traditionally used in Hindu religions as a sign of good luck and signifies good from all four directions.^[42]
• The god Brahma has four faces.^[43]
• There are four yugas: Satya, Dvapara, Treta and Kali^[44]

• Eid al-Adha lasts for four days, from the 10th to the 14th of Dhul Hijja.^[45]
• The four holy cities of Islam: Mecca, Medina, Jerusalem and Damascus.
• The four tombs in the Green Dome: Muhammad, Abu Bakr, Umar ibn Khattab and Isa ibn Maryam (Jesus).
• There are four Rashidun or Rightly Guided Caliphs: Abu Bakr, Umar ibn al-Khattab, Uthman ibn Affan and Ali ibn Abi Talib.^[46]
• The Four Arch Angels in Islam are: Jibraeel (Gabriel), Mikaeel (Michael), Izraeel (Azrael), and Israfil (Raphael)^[47]
• There are four months in which war is not permitted: Muharram, Rajab, Dhu al-Qi’dah and Dhu al-Hijjah.^[48]
• There are four Sunni schools of fiqh: Hanafi, Shafi`i, Maliki and Hanbali.
• There are four major Sunni Imams: Abū Ḥanīfa, Muhammad ibn Idris ash-Shafi`i, Malik ibn Anas and Ahmad ibn Hanbal.
• There are four books in Islam: Taurāt, Zābūr, Injīl, Qur’ān.^[49]
• Waiting for four months is ordained for those who take an oath for abstention from their wives.^[50]
• The waiting period of the woman whose husband dies is four months and ten days.^[51]
• When Abraham said: «My Lord, show me how You give life to the dead,» Allah said: «Why! Do you have no faith?» Abraham replied: «Yes, but in order that my heart be at rest.» He said: «Then take four birds, and tame them to yourself, then put a part of them on every hill, and summon them; they will come to you flying. [Al-Baqara 2:260]^[52]
• The respite of four months was granted to give time to the mushriks in Surah At-Tawba so that they should consider their position carefully and decide whether to make preparation for war or to emigrate from the country or to accept Islam.^[53]
• Those who accuse honorable women (of unchastity) but do not produce four witnesses, flog them with eighty lashes, and do not admit their testimony ever after. They are indeed transgressors. [An-Noor 24:4]^[54]

• Four Symbols of I Ching^[55]

• In a more general sense, numerous mythological and cosmogonical systems consider Four corners of the world as essentially corresponding to the four points of the compass.^[56]
• Four is the sacred number of the Zia, an indigenous tribe located in the U.S. state of New Mexico.^[57]
• The Chinese, the Koreans, and the Japanese are superstitious about the number four because it is a homonym for «death» in their languages.^[58]
• In Slavic mythology, the god Svetovid has four heads.^[59]

• Four Freedoms: four fundamental freedoms that Franklin D. Roosevelt declared ought to be enjoyed by everyone in the world: Freedom of Speech, Freedom of Religion, Freedom from Want, Freedom from Fear.^[60]
• Gang of Four: Popular name for four Chinese Communist Party leaders who rose to prominence during China’s Cultural Revolution, but were ousted in 1976 following the death of Chairman Mao Zedong. Among the four was Mao’s widow, Jiang Qing. Since then, many other political factions headed by four people have been called «Gangs of Four».^[61]

• Four bits (half a byte) are sometimes called a nibble.^[62]

• A tetramer is an oligomer formed out of four sub-units.^[63]

• Four terrestrial (or rocky) planets in the Solar System: Mercury, Venus, Earth, and Mars.^[64]
• Four giant gas/ice planets in the Solar System: Jupiter, Saturn, Uranus, and Neptune.^[65]
• Four of Jupiter’s moons (the Galilean moons) are readily visible from Earth with a hobby telescope.^[66]
• Messier object M4, a magnitude 7.5 globular cluster in the constellation Scorpius.^[67]
• The Roman numeral IV stands for subgiant in the Yerkes spectral classification scheme.^[68]

• Four is the number of nucleobase types in DNA and RNA – adenine, guanine, cytosine, thymine (uracil in RNA).^[69]
• Many chordates have four feet, legs or leglike appendages (tetrapods).
• The mammalian heart consists of four chambers.^[70]
• Many mammals (Carnivora, Ungulata) use four fingers for movement.
• All insects with wings except flies and some others have four wings.^[71]
• Insects of the superorder Endopterygota, also known as Holometabola, such as butterflies, ants, bees, beetles, fleas, flies, moths, and wasps, undergo holometabolism—complete metamorphism in four stages—from (1) embryo (ovum, egg), to (2) larva (such as grub, caterpillar), then (3) pupa (such as the chrysalis), and finally (4) the imago.^[72]
• In the common ABO blood group system, there are four blood types (A, B, O, AB).^[73]
• Humans have four canines and four wisdom teeth.^[74]
• The cow’s stomach is divided in four digestive compartments: reticulum, rumen, omasum and abomasum.^[75]

• Valency of carbon (that is basis of life on the Earth) is four. Also because of its tetrahedral crystal bond structure, diamond (one of the natural allotropes of carbon) is the hardest known naturally occurring material. It is also the valence of silicon, whose compounds form the majority of the mass of the Earth’s crust.^[76]
• The atomic number of beryllium^[77]
• There are four basic states of matter: solid, liquid, gas, and plasma.^[78]

• Special relativity and general relativity treat nature as four-dimensional: 3D regular space and one-dimensional time are treated together and called spacetime.^[79] Also, any event E has a light cone composed of four zones of possible communication and cause and effect (outside the light cone is strictly incommunicado).
• There are four fundamental forces (electromagnetism, gravitation, the weak nuclear force, and the strong nuclear force).^[80]
• In statistical mechanics, the four functions inequality is an inequality for four functions on a finite distributive lattice.^[81]

• The symbolic meanings of the number four are linked to those of the cross and the square. «Almost from prehistoric times, the number four was employed to signify what was solid, what could be touched and felt. Its relationship to the cross (four points) made it an outstanding symbol of wholeness and universality, a symbol which drew all to itself». Where lines of latitude and longitude intersect, they divide the earth into four proportions. Throughout the world kings and chieftains have been called «lord of the four suns» or «lord of the four quarters of the earth»,^[82] which is understood to refer to the extent of their powers both territorially and in terms of total control of their subjects’ doings.
• The Square of Opposition, in both its Aristotelian version and its Boolean version, consists of four forms: A («All S is R«), I («Some S is R«), E («No S is R«), and O («Some S is not R«).
• In regard to whether two given propositions can have the same truth value, there are four separate logical possibilities: the propositions are subalterns (possibly both are true, and possibly both are false); subcontraries (both may be true, but not that both are false); contraries (both may be false, but not that both are true); or contradictories (it is not possible that both are true, and it is not possible that both are false).
• Aristotle held that there are basically four causes in nature: the material, the formal, the efficient, and the final.^[83]
• The Stoics held with four basic categories, all viewed as bodies (substantial and insubstantial): (1) substance in the sense of substrate, primary formless matter; (2) quality, matter’s organization to differentiate and individualize something, and coming down to a physical ingredient such as pneuma, breath; (3) somehow holding (or disposed), as in a posture, state, shape, size, action, and (4) somehow holding (or disposed) toward something, as in relative location, familial relation, and so forth.
• Immanuel Kant expounded a table of judgments involving four three-way alternatives, in regard to (1) Quantity, (2) Quality, (3) Relation, (4) Modality, and, based thereupon, a table of four categories, named by the terms just listed, and each with three subcategories.
• Arthur Schopenhauer‘s doctoral thesis was On the Fourfold Root of the Principle of Sufficient Reason.
• Franz Brentano held that any major philosophical period has four phases: (1) Creative and rapidly progressing with scientific interest and results; then declining through the remaining phases, (2) practical, (3) increasingly skeptical, and (4) literary, mystical, and scientifically worthless—until philosophy is renewed through a new period’s first phase. (See Brentano’s essay «The Four Phases of Philosophy and Its Current State» 1895, tr. by Mezei and Smith 1998.)
• C. S. Peirce, usually a trichotomist, discussed four methods for overcoming troublesome uncertainties and achieving secure beliefs: (1) the method of tenacity (policy of sticking to initial belief), (2) the method of authority, (3) the method of congruity (following a fashionable paradigm), and (4) the fallibilistic, self-correcting method of science (see «The Fixation of Belief«, 1877); and four barriers to inquiry, barriers refused by the fallibilist: (1) assertion of absolute certainty; (2) maintaining that something is unknowable; (3) maintaining that something is inexplicable because absolutely basic or ultimate; (4) holding that perfect exactitude is possible, especially such as to quite preclude unusual and anomalous phenomena (see «F.R.L.» [First Rule of Logic], 1899).
• Paul Weiss built a system involving four modes of being: Actualities (substances in the sense of substantial, spatiotemporally finite beings), Ideality or Possibility (pure normative form), Existence (the dynamic field), and God (unity). (See Weiss’s Modes of Being, 1958).
• Karl Popper outlined a tetradic schema to describe the growth of theories and, via generalization, also the emergence of new behaviors and living organisms: (1) problem, (2) tentative theory, (3) (attempted) error-elimination (especially by way of critical discussion), and (4) new problem(s). (See Popper’s Objective Knowledge, 1972, revised 1979.)
• John Boyd (military strategist) made his key concept the decision cycle or OODA loop, consisting of four stages: (1) observation (data intake through the senses), (2) orientation (analysis and synthesis of data), (3) decision, and (4) action.^[84] Boyd held that his decision cycle has philosophical generality, though for strategists the point remains that, through swift decisions, one can disrupt an opponent’s decision cycle.
• Richard McKeon outlined four classes (each with four subclasses) of modes of philosophical inquiry: (1) Modes of Being (Being); (2) Modes of Thought (That which is); (3) Modes of Fact (Existence); (4) Modes of Simplicity (Experience)—and, corresponding to them, four classes (each with four subclasses) of philosophical semantics: Principles, Methods, Interpretations, and Selections. (See McKeon’s «Philosophic Semantics and Philosophic Inquiry» in Freedom and History and Other Essays, 1989.)
• Jonathan Lowe (E.J. Lowe) argues in The Four-Category Ontology, 2006, for four categories: kinds (substantial universals), attributes (relational universals and property-universals), objects (substantial particulars), and modes (relational particulars and property-particulars, also known as «tropes«). (See Lowe’s «Recent Advances in Metaphysics,» 2001, Eprint)
• Four opposed camps of the morality and nature of evil: moral absolutism, amoralism, moral relativism, and moral universalism.

• The resin identification code used in recycling to identify low-density polyethylene.^[85]
• Most furniture has four legs – tables, chairs, etc.
• The four color process (CMYK) is used for printing.^[86]
• Wide use of rectangles (with four angles and four sides) because they have effective form and capability for close adjacency to each other (houses, rooms, tables, bricks, sheets of paper, screens, film frames).
• In the Rich Text Format specification, language code 4 is for the Chinese language. Codes for regional variants of Chinese are congruent to 4 mod 256.
• Credit card machines have four-twelve function keys.
• On most phones, the 4 key is associated with the letters G, H, and I,^[87] but on the BlackBerry Pearl, it is the key for D and F.
• On many computer keyboards, the «4» key may also be used to type the dollar sign ($) if the shift key is held down.
• It is the number of bits in a nibble, equivalent to half a byte^[88]
• In internet slang, «4» can replace the word «for» (as «four» and «for» are pronounced similarly). For example, typing «4u» instead of «for you».
• In Leetspeak, «4» may be used to replace the letter «A».
• The TCP/IP stack consists of four layers.^[89]

• Many internal combustion engines are called four-stroke engines because they complete one thermodynamic cycle in four distinct steps: Intake, compression, power, and exhaust.
• Most vehicles, including motor vehicles, and particularly cars/automobiles and light commercial vehicles have four road wheels.
• «Quattro«, meaning four in the Italian language, is used by Audi as a trademark to indicate that all-wheel drive (AWD) technologies are used on Audi-branded cars.^[90] The word «Quattro» was initially used by Audi in 1980 in its original 4WD coupé, the Audi Quattro. Audi also has a privately held subsidiary company called quattro GmbH.
• List of highways numbered 4

• The phrase «four-letter word» is used to describe many swear words in the English language.^[92]
• Four is the only number whose name in English has the same number of letters as its value.
• Four (, formal writing: , pinyin sì) is considered an unlucky number in Chinese, Korean, Vietnamese and Japanese cultures mostly in Eastern Asia because it sounds like the word «death» (, pinyin sǐ). To avoid complaints from people with tetraphobia, many numbered product lines skip the «four»: e.g. Nokia cell phones (there was no series beginning with a 4 until the Nokia 4.2), Palm PDAs, etc. Some buildings skip floor 4 or replace the number with the letter «F», particularly in heavily Asian areas. See tetraphobia and Numbers in Chinese culture.
• In Pythagorean numerology (a pseudocience) the number 4 represents security and stability.
• The number of characters in a canonical four-character idiom.
• In the NATO phonetic alphabet, the digit 4 is called «fower».^[93]
• In astrology, Cancer is the 4th astrological sign of the Zodiac.^[94]

• In Tarot, The Emperor is the fourth trump or Major Arcana card.^[95]
• In Tetris, a game named for the Greek word for 4, every shape in the game is formed of 4 blocks each.^[96]
• 4 represents the number of Justices on the Supreme Court of the United States necessary to grant a writ of certiorari (i.e., agree to hear a case; it is one less than the number necessary to render a majority decision) at the court’s current size.^[97]
• Number Four is a character in the book series Lorien Legacies.^[98]
• In the performing arts, the fourth wall is an imaginary barrier which separates the audience from the performers, and is «broken» when performers communicate directly to the audience.^[99]

• In written music, common time is constructed of four beats per measure and a quarter note receives one beat.^[100]
• In popular or modern music, the most common time signature is also founded on four beats, i.e., 4/4 having four quarter note beats.
• The common major scale is built on two sets of four notes (e.g., CDEF, GABC), where the first and last notes create an octave interval (a pair-of-four relationship).
• The interval of a perfect fourth is a foundational element of many genres of music, represented in music theory as the tonic and subdominant relationship. Four is also embodied within the circle of fifths (also known as circle of fourths), which reveals the interval of four in more active harmonic contexts.
• The typical number of movements in a symphony.^[101]
• The number of completed, numbered symphonies by Johannes Brahms.^[102]
• The number of strings on a violin, a viola, a cello, double bass, a cuatro, a typical bass guitar, and a ukulele, and the number of string pairs on a mandolin.
• «Four calling birds» is the gift on the fourth day of Christmas in the carol «The Twelve Days of Christmas«.^[103]

Methods of computing square roots are numerical analysis algorithms for approximating the principal, or non-negative, square root (usually denoted
,
, or
) of a real number. Arithmetically, it means given
, a procedure for finding a number which when multiplied by itself, yields
; algebraically, it means a procedure for finding the non-negative root of the equation
; geometrically, it means given two line segments, a procedure for constructing their geometric mean.

The continued fraction representation of a real number can be used instead of its decimal or binary expansion and this representation has the property that the square root of any rational number (which is not already a perfect square) has a periodic, repeating expansion, similar to how rational numbers have repeating expansions in the decimal notation system.

The method employed depends on what the result is to be used for (i.e. how accurate it has to be), how much effort one is willing to put into the procedure, and what tools are at hand. The methods may be roughly classified as those suitable for mental calculation, those usually requiring at least paper and pencil, and those which are implemented as programs to be executed on a digital electronic computer or other computing device. Algorithms may take into account convergence (how many iterations are required to achieve a specified precision), computational complexity of individual operations (i.e. division) or iterations, and error propagation (the accuracy of the final result).

Procedures for finding square roots (particularly the square root of 2) have been known since at least the period of ancient Babylon in the 17th century BCE. Heron’s method from first century Egypt was the first ascertainable algorithm for computing square root. Modern analytic methods began to be developed after introduction of the Arabic numeral system to western Europe in the early Renaissance. Today, nearly all computing devices have a fast and accurate square root function, either as a programming language construct, a compiler intrinsic or library function, or as a hardware operator, based on one of the described procedures.