query: square root of 9 belongs to what family of real numbers

pos: Real number Real number In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line. The adjective "real" in this context was introduced in the 17th century by René Descartes, who distinguished between real and imaginary roots of polynomials. The real numbers include all the rational numbers, such as the integer −5 and the fraction 4/3, and all the irrational numbers, such as (1.41421356..., the square root of 2, an irrational algebraic number). Included within the irrationals are the transcendental numbers, such as (3.14159265...). In addition to measuring distance, real numbers can be

pos: Square root number, and quadratic irrational for a proof for all non-square natural numbers.) The square root function maps rational numbers into algebraic numbers (a superset of the rational numbers). For all real numbers "x", For all nonnegative real numbers "x" and "y", and The square root function is continuous for all nonnegative "x" and differentiable for all positive "x". If "f" denotes the square root function, its derivative is given by: The Taylor series of about "x" = 0 converges for ≤ 1 and is given by The square root of a nonnegative number is used in the definition of Euclidean

pos: Discriminant discriminant The discriminant is zero if and only if at least two roots are equal. If the coefficients are real numbers, and the discriminant is not zero, the discriminant is positive if the roots are three distinct real numbers, and negative if there is one real root and two complex conjugate roots. The square root of the product of the discriminant by (and possibly also by the square of a rational number) appears in the formulas for the roots of a cubic polynomial. If the polynomial is irreducible and its coefficients are rational numbers (or belong to a number field),

pos: Complex number at least one complex solution "z", provided that at least one of the higher coefficients is nonzero. This is the statement of the "fundamental theorem of algebra", of Carl Friedrich Gauss and Jean le Rond d'Alembert. Because of this fact, C is called an algebraically closed field. This property does not hold for the field of rational numbers Q (the polynomial does not have a rational root, since is not a rational number) nor the real numbers R (the polynomial does not have a real root for , since the square of is positive for any real number ). There

pos: Discriminant sequence . The quadratic polynomial formula_17 has discriminant The square root of the discriminant appears in the quadratic formula for the roots of the quadratic polynomial: The discriminant is zero if and only if the two roots are equal. If are real numbers, the polynomial has two distinct real roots if the discriminant is positive, and two complex conjugate roots if it is negative. If are rational numbers, then the discriminant is the square of a rational number if and only if the two roots are rational numbers. The cubic polynomial formula_20 has discriminant In particular, the polynomial formula_22 has

pos: Number negative numbers, rational numbers such as and , real numbers such as and , and complex numbers, which extend the real numbers by adding a square root of. Calculations with numbers are done with arithmetical operations, the most familiar being addition, subtraction, multiplication, division, and exponentiation. Their study or usage is called arithmetic. The same term may also refer to number theory, the study of the properties of numbers. Besides their practical uses, numbers have cultural significance throughout the world. For example, in Western society, the number 13 is regarded as unlucky, and "a million" may signify "a lot." Though

pos: Square number length has area . Square numbers are non-negative. Another way of saying that a (non-negative) integer is a square number is that its square root is again an integer. For example, = 3, so 9 is a square number. A positive integer that has no perfect square divisors except 1 is called square-free. For a non-negative integer , the th square number is , with being the zeroth one. The concept of square can be extended to some other number systems. If rational numbers are included, then a square is the ratio of two square integers, and, conversely, the ratio

pos: Real number if for any there exists an integer "N" (possibly depending on ε) such that the distance is less than ε for "n" greater than "N". Every convergent sequence is a Cauchy sequence, and the converse is true for real numbers, and this means that the topological space of the real numbers is complete. The set of rational numbers is not complete. For example, the sequence (1; 1.4; 1.41; 1.414; 1.4142; 1.41421; ...), where each term adds a digit of the decimal expansion of the positive square root of 2, is Cauchy but it does not converge to a rational number

pos: Construction of the real numbers For convenience we may take the lower set formula_1 as the representative of any given Dedekind cut formula_2, since formula_3 completely determines formula_4. By doing this we may think intuitively of a real number as being represented by the set of all smaller rational numbers. In more detail, a real number formula_5 is any subset of the set formula_6 of rational numbers that fulfills the following conditions: As an example of a Dedekind cut representing an irrational number, we may take the positive square root of 2. This can be defined by the set formula_49. It can be seen from

pos: Spiral of Theodorus Theodorus and Theaetetus split the rational numbers and irrational numbers into different categories. Each of the triangles' hypotenuses "h" gives the square root of the corresponding natural number, with "h" = . Plato, tutored by Theodorus, questioned why Theodorus stopped at . The reason is commonly believed to be that the hypotenuse belongs to the last triangle that does not overlap the figure. In 1958, Erich Teuffel proved that no two hypotenuses will ever coincide, regardless of how far the spiral is continued. Also, if the sides of unit length are extended into a line, they will never pass through

pos: Richard Dedekind developed the notion now known as a Dedekind cut (German: "Schnitt"), now a standard definition of the real numbers. The idea of a cut is that an irrational number divides the rational numbers into two classes (sets), with all the numbers of one class (greater) being strictly greater than all the numbers of the other (lesser) class. For example, the square root of 2 defines all the nonnegative numbers whose squares are less than 2 and the negative numbers into the lesser class, and the positive numbers whose squares are greater than 2 into the greater class. Every location on

pos: Nested radical all three real solutions are written in terms of cube roots of complex numbers. On the other hand, consider the equation which has the rational solutions 1, 2, and —3. The general solution formula given above gives the solutions For any given choice of cube root and its conjugate, this contains nested radicals involving complex numbers, yet it is reducible (even though not obviously so) to one of the solutions 1, 2, or –3. Under certain conditions infinitely nested square roots such as represent rational numbers. This rational number can be found by realizing that "x" also appears under the

neg: HP 35s of functions able to handle complex numbers is limited and somewhat arbitrary. For example, directly taking the square root of a negative real number results in an error message instead of a complex number. This is strictly correct given that a nonnegative real number "a" has a unique nonnegative square root and this is called the "principal square root" which is denoted by . The symbol √ is called the "radical sign" or "radix". For example, the principal square root of 9 is 3, which is denoted by = 3, because and 3 is nonnegative. However raising x to the

neg: Square (algebra) is never less than the original number . Every positive real number is the square of exactly two numbers, one of which is strictly positive and the other of which is strictly negative. Zero is the square of only one number, itself. For this reason, it is possible to define the square root function, which associates with a non-negative real number the non-negative number whose square is the original number. No square root can be taken of a negative number within the system of real numbers, because squares of all real numbers are non-negative. The lack of real square roots

neg: Square root of a matrix above observation, that a positive-definite matrix has precisely one positive-definite square root: a positive definite matrix has only positive eigenvalues, and each of these eigenvalues has only one positive square root; and since the eigenvalues of the square root matrix are the diagonal elements of , for the square root matrix to be itself positive definite necessitates the use of only the unique positive square roots of the original eigenvalues. Just as with the real numbers, a real matrix may fail to have a real square root, but have a square root with complex-valued entries. Some matrices have "no square

neg: Algebraic function function determined by the equation Using the cubic formula, we get For formula_25 the square root is real and the cubic root is thus well defined, providing the unique real root. On the other hand, for formula_26 the square root is not real, and one has to choose, for the square root, either non-real square root. Thus the cubic root has to be chosen among three non-real numbers. If the same choices are done in the two terms of the formula, the three choices for the cubic root provide the three branches shown, in the accompanying image. It may be

neg: Number The real numbers are not, however, an algebraically closed field, because they do not include a solution (often called a square root of minus one) to the algebraic equation formula_13. Moving to a greater level of abstraction, the real numbers can be extended to the complex numbers. This set of numbers arose historically from trying to find closed formulas for the roots of cubic and quadratic polynomials. This led to expressions involving the square roots of negative numbers, and eventually to the definition of a new number: a square root of −1, denoted by "i", a symbol assigned by Leonhard

neg: Nth root real number is a positive real number, negative numbers do not have real square roots. However, for every negative real number there are two imaginary square roots. For example, the square roots of −25 are 5"i" and −5"i", where "i" represents a number whose square is . A cube root of a number "x" is a number "r" whose cube is "x": Every real number "x" has exactly one real cube root, written formula_4. For example, Every real number has two additional complex cube roots. Expressing the degree of an "n"th root in its exponent form, as in formula_37, makes

neg: Nth root positive "n"th root, called the principal "n"th root, which is written formula_6. For "n" equal to 2 this is called the principal square root and the "n" is omitted. The "n"th root can also be represented using exponentiation as "x". For even values of "n", positive numbers also have a negative "n"th root, while negative numbers do not have a real "n"th root. For odd values of "n", every negative number "x" has a real negative "n"th root. For example, −2 has a real 5th root, formula_29 but −2 does not have any real 6th roots. Every non-zero number "x",

neg: Square (algebra) for the negative numbers can be used to expand the real number system to the complex numbers, by postulating the imaginary unit , which is one of the square roots of −1. The property "every non negative real number is a square" has been generalized to the notion of a real closed field, which is an ordered field such that every non negative element is a square and every polynomial of odd degree has a root. The real closed fields cannot be distinguished from the field of real numbers by their algebraic properties: every property of the real numbers, which

neg: Square Root Day September 9, 1981 (9/9/81). Gordon remains the holiday's publicist, sending news releases to world media outlets. Gordon's daughter set up a Facebook group where people can share how they were celebrating the day. One suggested way of celebrating the holiday is by eating square radishes, or other root vegetables cut into shapes with square cross sections (thus creating a "square root"). Square Root Day occurs on the following dates each century: The number of years between consecutive Square Root Days in a century are consecutive odd numbers: 3, 5, 7, 9, 11, 13, 15, 17. This illustrates the fact that

neg: Square root 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 : the principal square root of "z" (defined below), and its negative. There are two complex numbers that square to a given arbitrary non-zero imaginary number formula_11 with real formula_12 where the two-digit pattern {3, 6} repeats over and over again in the partial denominators. Since , the above is also identical to the following generalized continued fractions: The

neg: Function (mathematics) continuity the function to a much larger domain. Frequently, for a starting point formula_168 there are several possible starting values for the function. For example, in defining the square root as the inverse function of the square function, for any positive real number formula_168 there are two choices for the value of the square root, one of which is positive and denoted formula_170 and another which is negative and denoted formula_171 These choices define two continuous functions, both having the nonnegative real numbers as a domain, and having either the nonnegative or the nonpositive real numbers as images. When looking

neg: Square root for the square root was first used in print in 1525 in Christoph Rudolff's "Coss". The principal square root function "f"("x") = (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

neg: Function (mathematics) reals), the square root function (mapping the non-negative reals to the non-negative reals), and the addition of real numbers. The whole set of real numbers may be taken as the domain of , even though the domain of the square root function is restricted to the non-negative real numbers; the image of consists of the reals that are not less than one. A computation that defines a function may often be described by an algorithm, and any kind of algorithm may be used. Sometimes, the definition of a function may involve elements or properties that can be defined, but not

neg: Square class numbers (with the multiplication operation) and formula_5 is the subgroup of positive numbers (as every positive number has a real square root). The quotient of these two groups is a group with two elements, corresponding to two cosets: the set of positive numbers and the set of negative numbers. Thus, the real numbers have two square classes, the positive numbers and the negative numbers. Square classes are frequently studied in relation to the theory of quadratic forms. The reason is that if formula_6 is an formula_1-vector space and formula_8 is a quadratic form and formula_9 is an element of formula_6

neg: Square root 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 name of the square root function varies from programming language to programming language, with codice_1 (often pronounced "squirt" ) being common, used in C, C++, and derived languages like JavaScript, PHP, and Python. 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

neg: Deductive lambda calculus not part of the Boolean domain but it exists in the lambda calculus domain. Using division and signed numbers, the "Y" combinator may be used to define an expression representing a whole number square root. The Church encoding may also be extended further to rational and real numbers, so that a real square root may be defined. The Church-Turing thesis implies that any computable operator (and its operands) can be represented in lambda calculus. Using such an encoding, Using the implementation of divide then, represents two values in the domain of the signed numbers, if "n" is not equal to

neg: Imaginary unit real, non-negative values of and . These problems are avoided by writing and manipulating expressions like , rather than . For a more thorough discussion, see Square root and Branch point. has two square roots, just like all complex numbers (except zero, which has a double root). These two roots can be expressed as the complex numbers: Indeed, squaring both expressions: Using the radical sign for the principal square root gives: The three cube roots of are: Similar to all of the roots of 1, all of the roots of are the vertices of regular polygons inscribed within the unit

neg: Square root square root of numbers having many digits. It was known to the ancient Greeks that square roots of positive whole numbers that are not perfect squares are always irrational numbers: numbers not expressible as a ratio of two integers (that is to say they cannot be written exactly as "m/n", where "m" and "n" are integers). This is the theorem "Euclid X, 9" almost certainly due to Theaetetus dating back to circa 380 BC. The particular case is assumed to date back earlier to the Pythagoreans and is traditionally attributed to Hippasus. It is exactly the length of the diagonal

neg: Heptagonal number Heptagonal number A heptagonal number is a figurate number that represents a heptagon. The "n"-th heptagonal number is given by the formula The first few heptagonal numbers are: The parity of heptagonal numbers follows the pattern odd-odd-even-even. Like square numbers, the digital root in base 10 of a heptagonal number can only be 1, 4, 7 or 9. Five times a heptagonal number, plus 1 equals a triangular number. A formula for the sum of the reciprocals of the heptagonal numbers is given by: In analogy to the square root of "x, "one can calculate the heptagonal root of "x",

neg: Napier's bones even numbers 2 through 18, and the last column just has the numbers 1 through 9. Let's find the square root of 46785399 with the bones. First, group its digits in twos starting from the right so it looks like this: Start with the leftmost group 46. Pick the largest square on the square root bone less than 46, which is 36 from the sixth row. Because we picked the sixth row, the first digit of the solution is 6. Now read the second column from the sixth row on the square root bone, 12, and set 12 on the

neg: Square root of a 2 by 2 matrix pairs: If "R" is a square root of "M", then –"R" is also a square root of "M", since (–"R")(–"R") = (–1)(–1)("RR") = "R" = "M". Let where "A", "B", "C", and "D" may be real or complex numbers. Furthermore, let "τ = A + D" be the trace of "M", and "δ = AD – BC" be its determinant. Let "s" be such that "s" = "δ", and "t" be such that "t" = "τ" + 2"s". That is, Then, if "t" ≠ 0, a square root of "M" is Indeed, the square of "R" is Note that "R"

neg: Two New Sciences The book also contains a discussion of infinity. Galileo considers the example of numbers and their squares. He starts by noting that: It cannot be denied that there are as many [squares] as there are numbers because every number is a [square] root of some square: 1 ↔ 1, 2 ↔ 4, 3 ↔ 9, 4 ↔ 16, and so on. (In modern language, there is a bijection between the elements of the set of positive integers N and the set of squares S, and S is a proper subset of density zero.) But he notes what appears to be

neg: Square root Square root In mathematics, a square root of a number "a" is a number "y" such that ; in other words, a number "y" whose "square" (the result of multiplying the number by itself, or ) is "a". For example, 4 and −4 are square roots of 16 because . Every nonnegative real number "a" has a unique nonnegative square root, called the "principal square root", which is denoted by , where √ is called the "radical sign" or "radix". For example, the principal square root of 9 is 3, which is denoted by = 3, because and 3 is

neg: Totally real number field whether the square root of a positive or negative number is adjoined to Q. In the case of cubic fields, a cubic integer polynomial "P" irreducible over Q will have at least one real root. If it has one real and two complex roots the corresponding cubic extension of Q defined by adjoining the real root will "not" be totally real, although it is a field of real numbers. The totally real number fields play a significant special role in algebraic number theory. An abelian extension of Q is either totally real, or contains a totally real subfield over which

neg: Inverse hyperbolic functions minimum. For all inverse hyperbolic functions, the principal value may be defined in terms of principal values of the square root and the logarithm function. However, in some cases, the formulas of do not give a correct principal value, as giving a domain of definition which is too small and, in one case non-connected. The principal value of the inverse hyperbolic sine is given by The argument of the square root is a non-positive real number if and only if belongs to one of the intervals and of the imaginary axis. If the argument of the logarithm is real, then

neg: Magic square different magic squares, respectively. As another example, the construction of 5×5 magic square is given. Numbers are directly written in place of alphabets. The numbered squares are referred to as "primary square" or "root square" if they are filled with primary numbers or root numbers, respectively. The numbers are placed about the skew diagonal in the root square such that the middle column of the resulting root square has 0, 5, 10, 15, 20 (from bottom to top). The primary square is obtained by rotating the root square counter-clockwise by 90 degrees, and replacing the numbers. The resulting square is

neg: Quadratic irrational number they must satisfy a quadratic equation and they must be irrational. The solutions to the quadratic equation "ax" + "bx" + "c" = 0 are Thus quadratic irrationals are precisely those real numbers in this form that are not rational. Since "b" and 2"a" are both integers, asking when the above quantity is irrational is the same as asking when the square root of an integer is irrational. The answer to this is that the square root of any natural number that is not a square number is irrational. The square root of 2 was the first such number to

neg: Absolute value the absolute value of a real number is that number's distance from zero along the real number line, and more generally the absolute value of the difference of two real numbers is the distance between them. Indeed, the notion of an abstract distance function in mathematics can be seen to be a generalisation of the absolute value of the difference (see "Distance" below). Since the square root symbol represents the unique "positive" square root (when applied to a positive number), it follows that is equivalent to the definition above, and may be used as an alternative definition of the absolute

neg: Inverse hyperbolic functions it is positive. Thus this formula defines a principal value for arsinh with branch cuts and . This is optimal, as the branch cuts must connect the singular points and to the infinity. The formula for the inverse hyperbolic cosine given in is not convenient, as, with principal values of the logarithm and the square root, the principal value of arcosh would not be defined for imaginary . Thus the square root has to be factorized, leading to The principal values of the square roots are both defined except if belongs to the real interval . If the argument of

neg: Number the circumference of any circle to its diameter. When pi is written as as it sometimes is, the ellipsis does not mean that the decimals repeat (they do not), but rather that there is no end to them. It has been proved that is irrational. Another well known number, proven to be an irrational real number, is the square root of 2, that is, the unique positive real number whose square is 2. Both these numbers have been approximated (by computer) to trillions of digits. Not only these prominent examples, but almost all real numbers are irrational and therefore have

neg: Inverse hyperbolic functions root that has a positive real part. This defines a single valued analytic function, which is defined everywhere except for non-positive real values of the variables (where the two square roots have a zero real part). This principal value of the square root function is denoted formula_29 in what follows. Similarly, the principal value of the logarithm, denoted formula_30 in what follows, is defined as the value for which the imaginary part as the smallest absolute value. It is defined everywhere except for non positive real values of the variable, for which two different values of the logarithm reach the

neg: U-bit U-bit In quantum mechanics, the u-bit or ubit is a proposed theoretical entity which arises in attempts to reformulate wave functions using only real numbers instead of the complex numbers conventionally used. In order to discover the real probability of a given quantum event occurring, the conventional calculation carries out an operation, analogous to squaring, on an associated set of complex numbers. A complex number involves the use of the square root of minus one, a number which is described as "imaginary" in contrast to the familiar "real" numbers used for counting and describing real physical objects. Because the computed

neg: Square root of 3 1. On the complex plane, this distance is expressed as mentioned below. It is the length of the space diagonal of a unit cube. The vesica piscis has a major axis to minor axis ratio equal to 1:, this can be shown by constructing two equilateral triangles within it. Multiplication of by the imaginary unit gives a square root of -3, an imaginary number. More exactly, (see square root of negative numbers). It is an Eisenstein integer. Namely, it is expressed as the difference between two non-real cubic roots of 1 (which are Eisenstein integers). In power engineering, the voltage

neg: Cube root solution expressions involve the real cube root of a real number, while if all three solutions are real numbers then they may be expressed in terms of the complex cube root of a complex number. Quartic equations can also be solved in terms of cube roots and square roots. The calculation of cube roots can be to traced back to Babylonian mathematicians from as early as 1800 BCE. In the fourth century BCE Plato posed the problem of doubling the cube, which required a compass-and-straightedge construction of the edge of a cube with twice the volume of a given cube;

neg: Definable real number Definable real number Informally, a definable real number is a real number that can be uniquely specified by its description. The description may be expressed as a construction or as a formula of a formal language. For example, the positive square root of 2, formula_1, can be defined as the unique positive solution to the equation formula_2, and it can be constructed with a compass and straightedge. Different choices of a formal language or its interpretation can give rise to different notions of definability. Specific varieties of definable numbers include the constructible numbers of geometry, the algebraic numbers, and the

neg: Inverse hyperbolic functions the real interval . Therefore, these formulas define convenient principal values, for which the branch cuts are and for the inverse hyperbolic tangent, and for the inverse hyperbolic cotangent. In view of a better numerical evaluation near the branch cuts, some authors use the following equivalent definition of the principal values, although the second one introduces a removable singularity at . For the inverse hyperbolic cosecant, the principal value is defined as It is defined when the arguments of the logarithm and the square root are not non-positive real numbers. The principal value of the square root is thus defined

neg: Subtract a square some threshold formula_1 must be at least proportional to the square root of formula_1, for otherwise there would not be enough of them to provide winning moves from all the hot numbers. Cold numbers tend to end in 0, 2, 4, 5, 7, or 9. Cold values that end with other digits are quite uncommon. This holds in particular for cold numbers ending in 6. Out of all the over 180,000 cold numbers less than 40 million, only one ends in a 6: 11,356. No two cold numbers can differ by a square, because if they did then a move

neg: Rafael Bombelli formula_2 which is another way of stating that the discriminant of the cubic is negative. The solution of this kind of equation requires taking the cube root of the sum of one number and the square root of some negative number. Before Bombelli delves into using imaginary numbers practically, he goes into a detailed explanation of the properties of complex numbers. Right away, he makes it clear that the rules of arithmetic for imaginary numbers are not the same as for real numbers. This was a big accomplishment, as even numerous subsequent mathematicians were extremely confused on the topic. Bombelli

neg: Least-upper-bound property square root of two is irrational). The construction of the real numbers using Dedekind cuts takes advantage of this failure by defining the irrational numbers as the least upper bounds of certain subsets of the rationals. The least-upper-bound property is equivalent to other forms of the completeness axiom, such as the convergence of Cauchy sequences or the nested intervals theorem. The logical status of the property depends on the construction of the real numbers used: in the synthetic approach, the property is usually taken as an axiom for the real numbers (see least upper bound axiom); in a constructive approach,

neg: Heptagonal number meaning the number of terms in the sequence up to and including "x". The heptagonal root of "x " is given by the formula which is obtained by using the quadratic formula to solve formula_4 for its unique positive root "n". Heptagonal number A heptagonal number is a figurate number that represents a heptagon. The "n"-th heptagonal number is given by the formula The first few heptagonal numbers are: The parity of heptagonal numbers follows the pattern odd-odd-even-even. Like square numbers, the digital root in base 10 of a heptagonal number can only be 1, 4, 7 or 9. Five

neg: Real number uses the metric topology presentation. The reals are a contractible (hence connected and simply connected), separable and complete metric space of Hausdorff dimension 1. The real numbers are locally compact but not compact. There are various properties that uniquely specify them; for instance, all unbounded, connected, and separable order topologies are necessarily homeomorphic to the reals. Every nonnegative real number has a square root in R, although no negative number does. This shows that the order on R is determined by its algebraic structure. Also, every polynomial of odd degree admits at least one real root: these two properties make

neg: 2 × 2 real matrices does the square root function described above: half or three-quarters of P must be excluded in the cases "mm" = 0 or "mm" = . Further function theory can be seen in the article complex functions for the C structure, or in the article motor variable for the split-complex structure. Every real matrix can be interpreted as one of three types of (generalized) complex numbers: standard complex numbers, dual numbers, and split-complex numbers. Above, the algebra of matrices is profiled as a union of complex planes, all sharing the same real axis. These planes are presented as commutative subrings "P".

neg: Definable real number computable numbers. One way of specifying a real number uses geometric techniques. A real number "r" is a constructible number if there is a method to construct a line segment of length "r" using a compass and straightedge, beginning with a fixed line segment of length 1. Each positive integer, and each positive rational number, is constructible. The positive square root of 2 is constructible. However, the cube root of 2 is not constructible; this is related to the impossibility of doubling the cube. A real number "r" is called an algebraic number if there is a polynomial "p"("x"), with

neg: Real number the square roots of certain numbers such as 2 and 61 could not be exactly determined. Around 500 BC, the Greek mathematicians led by Pythagoras realized the need for irrational numbers, in particular the irrationality of the square root of 2. The Middle Ages brought the acceptance of zero, negative, integral, and fractional numbers, first by Indian and Chinese mathematicians, and then by Arabic mathematicians, who were also the first to treat irrational numbers as algebraic objects, which was made possible by the development of algebra. Arabic mathematicians merged the concepts of "number" and "magnitude" into a more general idea

neg: Square root of a matrix root". An example is the matrix formula_11. In general, a complex matrix with positive real eigenvalues has a unique square root with positive eigenvalues called the "principal square root". Moreover, the operation of taking the principal square root is continuous on this set of matrices. If the matrix has real entries, then the square root also has real entries. These properties are consequences of the holomorphic functional calculus applied to matrices. The existence and uniqueness of the principal square root can be deduced directly from the Jordan normal form (see below). For a 2 × 2 matrix, there are explicit

neg: Definable real number on the reals, etc. are definable . Definable real number Informally, a definable real number is a real number that can be uniquely specified by its description. The description may be expressed as a construction or as a formula of a formal language. For example, the positive square root of 2, formula_1, can be defined as the unique positive solution to the equation formula_2, and it can be constructed with a compass and straightedge. Different choices of a formal language or its interpretation can give rise to different notions of definability. Specific varieties of definable numbers include the constructible numbers

neg: Radical symbol are both imaginary numbers, and the square root symbol refers to the principal square root, the one with positive imaginary part. For the definition of the principal square root of other complex numbers, see Square root#Principal square root of a complex number. The origin of the root symbol √ is largely speculative. Some sources imply that the symbol was first used by Arab mathematicians. One of those mathematicians was Abū al-Hasan ibn Alī al-Qalasādī (1421–1486). Legend has it that it was taken from the Arabic letter "" ("ǧīm", which is the first letter in the Arabic word "" ("jadhir", meaning

neg: Powerful number powerful numbers in the interval [1,"x"]. Then "k"("x") is proportional to the square root of "x". More precisely, (Golomb, 1970). The two smallest consecutive powerful numbers are 8 and 9. Since Pell's equation has infinitely many integral solutions, there are infinitely many pairs of consecutive powerful numbers (Golomb, 1970); more generally, one can find consecutive powerful numbers by solving a similar Pell equation for any perfect cube . However, one of the two powerful numbers in a pair formed in this way must be a square. According to Guy, Erdős has asked whether there are infinitely many pairs of consecutive

neg: Imaginary unit The cosine of is a real number: And the sine of is purely imaginary: Imaginary unit The imaginary unit or unit imaginary number () is a solution to the quadratic equation . Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition and multiplication. A simple example of the use of in a complex number is . Imaginary numbers are an important mathematical concept, which extend the real number system to the complex number system , which in turn provides at least one root for

neg: Square root 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 "j", especially in the context of electricity where ""i"" traditionally represents electric current) and called the imaginary unit, which is "defined" such that . Using this notation, we can think of "i" as the square root of −1, but notice that we also have 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

neg: Methods of computing square roots that formula_211 is as small as possible. A completely different method for computing the square root is based on the CORDIC algorithm, which uses only very simple operations (addition, subtraction, with bitshift and table lookup to implement multiplication). The square root of "S" may be obtained as the output formula_197 using the hyperbolic coordinate system in vectoring mode, with the following initialization: Quadratic irrationals (numbers of the form formula_216, where "a", "b" and "c" are integers), and in particular, square roots of integers, have periodic continued fractions. Sometimes what is desired is finding not the numerical value of a square

neg: Square root of 2 fraction representation: The convergents formed by truncating this representation form a sequence of fractions that approximate the square root of two to increasing accuracy, and that are described by the Pell numbers (known as side and diameter numbers to the ancient Greeks because of their use in approximating the ratio between the sides and diagonal of a square). The first convergents are: . The convergent differs from by almost exactly and then the next convergent is . The following nested square expressions converge to formula_79: The reciprocal of the square root of two (the square root of ) is a

neg: −1 numbers. Although there are no real square roots of -1, the complex number "i" satisfies "i" = −1, and as such can be considered as a square root of −1. The only other complex number who's square is 1 is −"i". In the algebra of quaternions, which contain the complex plane, the equation "x" = −1 has infinite solutions. Exponentiation of a non-zero real number can be extended to negative integers. We make the definition that "x" = , meaning that we define raising a number to the power −1 to have the same effect as taking its reciprocal. This

neg: Complex number usual (positive) th root of the positive real number . While the th root of a positive real number is chosen to be the "positive" real number satisfying , there is no natural way of distinguishing one particular complex th root of a complex number. Therefore, the th root of is considered as a multivalued function (in ), as opposed to a usual function , for which is a uniquely defined number. Formulas such as (which holds for positive real numbers), do in general not hold for complex numbers. The set C of complex numbers is a field. Briefly, this

neg: Cube root the cube root of numbers having many digits in the "Aryabhatiya" (section 2.5). Cube root In mathematics, a cube root of a number "x" is a number "y" such that "y" = "x". All real numbers (except zero) have exactly one real cube root and a pair of complex conjugate cube roots, and all nonzero complex numbers have three distinct complex cube roots. For example, the real cube root of 8, denoted , is 2, because 2 = 8, while the other cube roots of 8 are −1 + "i" and −1 − "i". The three cube roots of −27"i"

neg: Cube root of a number "x" are the numbers "y" which satisfy the equation For any real number "x", there is "one" real number "y" such that "y" = "x". The cube function is increasing, so does not give the same result for two different inputs, plus it covers all real numbers. In other words, it is a bijection, or one-to-one. Then we can define an inverse function that is also one-to-one. For real numbers, we can define a unique cube root of all real numbers. If this definition is used, the cube root of a negative number is a negative number.

neg: Root-finding algorithm method to compute this factorization is Yun's algorithm. Notes Sources Root-finding algorithm In mathematics and computing, a root-finding algorithm is an algorithm for finding roots of continuous functions. A root of a function , from the real numbers to real numbers or from the complex numbers to the complex numbers, is a number such that . As, generally, the roots of a function cannot be computed exactly, nor expressed in closed form, root-finding algorithms provide approximations to roots, expressed either as floating point numbers or as small isolating intervals, or disks for complex roots (an interval or disk output being

neg: Root-finding algorithm Root-finding algorithm In mathematics and computing, a root-finding algorithm is an algorithm for finding roots of continuous functions. A root of a function , from the real numbers to real numbers or from the complex numbers to the complex numbers, is a number such that . As, generally, the roots of a function cannot be computed exactly, nor expressed in closed form, root-finding algorithms provide approximations to roots, expressed either as floating point numbers or as small isolating intervals, or disks for complex roots (an interval or disk output being equivalent to an approximate output together with an error bound).

neg: Natural number rationals; the complex numbers, by including with the real numbers the unresolved square root of minus one (and also the sums and products thereof); and so on. These chains of extensions make the natural numbers canonically embedded (identified) in the other number systems. Properties of the natural numbers, such as divisibility and the distribution of prime numbers, are studied in number theory. Problems concerning counting and ordering, such as partitioning and enumerations, are studied in combinatorics. In common language, for example in primary school, natural numbers may be called counting numbers both to intuitively exclude the negative integers and zero,

neg: Imaginary unit Imaginary unit The imaginary unit or unit imaginary number () is a solution to the quadratic equation . Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition and multiplication. A simple example of the use of in a complex number is . Imaginary numbers are an important mathematical concept, which extend the real number system to the complex number system , which in turn provides at least one root for every nonconstant polynomial . (See Algebraic closure and Fundamental theorem of algebra.) The term

neg: Methods of computing square roots is a simple case where the answer is a perfect square root XY, the algorithm stops here. The same idea can be extended to any arbitrary square root computation next. Suppose we are able to find the square root of "N" by expressing it as a sum of "n" positive numbers such that By repeatedly applying the basic identity the right-hand-side term can be expanded as This expression allows us to find the square root by sequentially guessing the values of formula_61s. Suppose that the numbers formula_62 have already been guessed, then the m-th term of the right-hand-side of above

neg: Three-phase together as complex numbers and then converting from rectangular to polar co-ordinates. If the three-phase root mean square (RMS) currents are formula_37, formula_38, and formula_39, the neutral RMS current is: which resolves to The polar magnitude of this is the square root of the sum of the squares of the real and imaginary parts, which reduces to With linear loads, the neutral only carries the current due to imbalance between the phases. Devices that utilize rectifier-capacitor front ends (such as switch-mode power supplies for computers, office equipment and the like) introduce third order harmonics. Third harmonic currents are in-phase on

neg: Matrix analysis "size" of a particular matrix. The norm of a matrix is a positive real number. For all matrices A and B in "M"("F"), and all numbers "α" in "F", a matrix norm, delimited by double vertical bars || ... ||, fulfills: The Frobenius norm is analogous to the dot product of Euclidean vectors; multiply matrix elements entry-wise, add up the results, then take the positive square root: It is defined for matrices of any dimension (i.e. no restriction to square matrices). Matrix elements are not restricted to constant numbers, they can be mathematical variables. A functions of a matrix takes

neg: Rational function the degree of formula_29 and both are real polynomials. The rational function is not defined at It is asymptotic to formula_32 as formula_33 The rational function is defined for all real numbers, but not for all complex numbers, since if "x" were a square root of formula_35 (i.e. the imaginary unit or its negative), then formal evaluation would lead to division by zero: which is undefined. A constant function such as "f"("x") = π is a rational function since constants are polynomials. Note that the function itself is rational, even though the value of "f"("x") is irrational for all "x".

neg: Square root of a matrix , then the Jordan decomposition into generalized eigenspaces of can be deduced from the partial fraction expansion of . The corresponding projections onto the generalized eigenspaces are given by real polynomials in . On each eigenspace, has the form as above. The power series expression for the square root on the eigenspace show that the principal square root of has the form "q"("A") where "q"("t") is a polynomial with real coefficients. Another way to find the square root of an "n" × "n" matrix "A" is the Denman–Beavers square root iteration. Let "Y" = "A" and "Z" = "I", where

neg: Hilbert space series of real numbers: Just as with a series of scalars, a series of vectors that converges absolutely also converges to some limit vector in the Euclidean space, in the sense that This property expresses the "completeness" of Euclidean space: that a series that converges absolutely also converges in the ordinary sense. Hilbert spaces are often taken over the complex numbers. The complex plane denoted by is equipped with a notion of magnitude, the complex modulus which is defined as the square root of the product of with its complex conjugate: If is a decomposition of into its real and

neg: Exponentiation −2. The principal value of 4 is 2, but −2 is also a valid square root. If the definition of exponentiation of real numbers is extended to allow negative results then the result is no longer well-behaved. Neither the logarithm method nor the rational exponent method can be used to define "b" as a real number for a negative real number "b" and an arbitrary real number "r". Indeed, "e" is positive for every real number "r", so ln("b") is not defined as a real number for . The rational exponent method cannot be used for negative values of "b"

neg: Biquaternion basis for the (real) quaternions , and let be complex numbers, then is a "biquaternion". To distinguish square roots of minus one in the biquaternions, Hamilton and Arthur W. Conway used the convention of representing the square root of minus one in the scalar field ℂ by "h" to avoid confusion with the in the quaternion group. Commutativity of the scalar field with the quaternion group is assumed: Hamilton introduced the terms bivector, "biconjugate, bitensor", and "biversor" to extend notions used with real quaternions . Hamilton's primary exposition on biquaternions came in 1853 in his "Lectures on Quaternions". The editions

neg: Triangular number Hence every triangular number is either divisible by three or has a remainder of 1 when divided by 9: The digital root pattern for triangular numbers, repeating every nine terms, as shown above, is "1, 3, 6, 1, 6, 3, 1, 9, 9". The converse of the statement above is, however, not always true. For example, the digital root of 12, which is not a triangular number, is 3 and divisible by three. If is a triangular number, then is also a triangular number, given is an odd square and The first several pairs of this form (not counting )

neg: Trigonometric constants expressed in real radicals number of sides). This is because the sum of the angles of any "n"-gon is 180° × ("n" – 2) and so the measure of each angle of any regular "n"-gon is 180° × ("n" – 2) ÷ "n". Thus for example the entry "45°: square" means that, with "n" = 4, 180° ÷ "n" = 45°, and the number of degrees in each angle of a square is ("n" – 2) × 45° = 90°. For cube roots of non-real numbers that appear in this table, one has to take the principal value, that is the cube root with

neg: Cube root Cube root In mathematics, a cube root of a number "x" is a number "y" such that "y" = "x". All real numbers (except zero) have exactly one real cube root and a pair of complex conjugate cube roots, and all nonzero complex numbers have three distinct complex cube roots. For example, the real cube root of 8, denoted , is 2, because 2 = 8, while the other cube roots of 8 are −1 + "i" and −1 − "i". The three cube roots of −27"i" are The cube root operation is not distributive with addition or subtraction. In

neg: Cube root some contexts, particularly when the number whose cube root is to be taken is a real number, one of the cube roots (in this particular case the real one) is referred to as the "principal cube root", denoted with the radical sign . The cube root operation is associative with exponentiation and distributive with multiplication and division if considering only real numbers, but not always if considering complex numbers: for example, the cube of any cube root of 8 is 8, but the three cube roots of 8 are 8, −4 + 4"i", and −4 − 4"i". The cube roots

neg: Nth root always negatives of each other. For example, the square roots of are and , and the square roots of are If we express a complex number in polar form, then the square root can be obtained by taking the square root of the radius and halving the angle: A "principal" root of a complex number may be chosen in various ways, for example which introduces a branch cut in the complex plane along the positive real axis with the condition , or along the negative real axis with . Using the first(last) branch cut the principal square root formula_87 maps

neg: Square root of a matrix the state error covariance matrix is required for the unscented transform which is the statistical linearization method used. A comparison between different matrix square root calculation methods within a UKF application of GPS/INS sensor fusion was presented, which indicated that the Cholesky decomposition method was best suited for UKF applications. Square root of a matrix In mathematics, the square root of a matrix extends the notion of square root from numbers to matrices. Matrix is said to be a square root of if the matrix product is equal to . In general, a matrix can have several square roots. For

neg: Mathematical fallacy invalid proof that formula_41: When the square root was extracted, it was the "negative" root −2, rather than the "positive" root, that was relevant for the particular solution in the problem. Invalid proofs utilizing powers and roots are often of the following kind: The fallacy is that the rule formula_47 is generally valid only if both "x" and "y" are positive (when dealing with real numbers), which is not the case here. Alternatively, imaginary roots are obfuscated in the following: The error here lies in the last equality, where we are ignoring the other fourth roots of 1, which are

neg: Complex polygon Complex polygon The term complex polygon can mean two different things: In geometry, a complex polygon is a polygon in the complex Hilbert plane, which has two complex dimensions. A complex number may be represented in the form formula_1, where formula_2 and formula_3 are real numbers, and formula_4 is the square root of formula_5. Multiples of formula_4 such as formula_7 are called "imaginary numbers". A complex number lies in a complex plane having one real and one imaginary dimension, which may be represented as an Argand diagram. So a single complex dimension comprises two spatial dimensions, but of different kinds

neg: Trigonometric constants expressed in real radicals function values are expressible in, for example, cube roots of complex numbers that cannot be rewritten in terms of roots of real numbers. For example, the trigonometric function values of any angle that is one-third of an angle "θ" considered in this article can be expressed in cube roots and square roots by using the cubic equation formula to solve but in general the solution for the cosine of the one-third angle involves the cube root of a complex number (giving "casus irreducibilis"). In practice, all values of sines, cosines, and tangents not found in this article are approximated using

neg: Cayley–Dickson construction involution known as conjugation. The product of an element and its conjugate (or sometimes the square root of this product) is called the norm. The symmetries of the real field disappear as the Cayley–Dickson construction is repeatedly applied: first losing order, then commutativity of multiplication, associativity of multiplication, and next alternativity. More generally, the Cayley–Dickson construction takes any algebra with involution to another algebra with involution of twice the dimension. The complex numbers can be written as ordered pairs ("a", "b") of real numbers "a" and "b", with the addition operator being component-by-component and with multiplication defined by A complex

neg: Pseudo-Euclidean space replaced with . The quadratic form corresponds to the square of a vector in Euclidean case. To define the vector norm (and distance) in an invariant manner, one has to get square roots of magnitudes, which leads to possibly imaginary distances; see square root of negative numbers. But even for a triangle with positive magnitudes of all three sides (whose square roots are real and positive), the triangle inequality does not hold in general. Hence terms "norm" and "distance" are avoided in pseudo-Euclidean geometry, replaced with "magnitude" and "interval" respectively. Though, for a curve whose tangent vectors all have the

neg: Square root of a 2 by 2 matrix may have complex entries even if "M" is a real matrix; this will be the case, in particular, if the determinant "δ" is negative. When a matrix can be expressed as real multiple of the exponent of a matrix, formula_5 then the square root is formula_6 In this case the square root is real, and can be interpreted as the square root of a type of complex number. If "M" is an idempotent matrix, meaning that "MM" = "M", then if it is not the identity matrix its determinant is zero, and its trace equals its rank which (excluding the

neg: Square root of a matrix Square root of a matrix In mathematics, the square root of a matrix extends the notion of square root from numbers to matrices. Matrix is said to be a square root of if the matrix product is equal to . In general, a matrix can have several square roots. For example, the matrix formula_1 has square roots formula_2 and formula_3, as well as their additive inverses. Another example is the 2×2 identity matrix formula_4 which has infinitely many symmetric rational square roots given by where formula_6 is any Pythagorean triple—that is, any set of positive integers such that formula_7. However,

neg: Quadratic equation at multiple points, graphing calculators generally require one to identify the desired root by positioning a cursor at a "guessed" value for the root. (Some graphing calculators require bracketing the root on both sides of the zero.) The calculator then proceeds, by an iterative algorithm, to refine the estimated position of the root to the limit of calculator accuracy. Although the quadratic formula provides an exact solution, the result is not exact if real numbers are approximated during the computation, as usual in numerical analysis, where real numbers are approximated by floating point numbers (called "reals" in many programming languages).

neg: Integer-valued function function with natural, or non-negative integer values is a partial case of integer-valued function. Integer-valued functions defined on the domain of all real numbers include the floor and ceiling functions, the Dirichlet function, the sign function and the Heaviside step function (except possibly at 0). Integer-valued functions defined on the domain of non-negative real numbers include the integer square root function and the prime-counting function. On an arbitrary set , integer-valued functions form a ring with pointwise operations of addition and multiplication, and also an algebra over the ring of integers. Since the latter is an ordered ring, the functions

neg: Square number of two square integers is a square, e.g., formula_1. Starting with 1, there are square numbers up to and including , where the expression represents the floor of the number . The squares smaller than 60 = 3600 are: The difference between any perfect square and its predecessor is given by the identity . Equivalently, it is possible to count up square numbers by adding together the last square, the last square's root, and the current root, that is, . The number is a square number if and only if one can arrange points in a square: The expression for

neg: Infinitesimal basic infinitesimal x has a square root. This field is rich enough to allow a significant amount of analysis to be done, but its elements can still be represented on a computer in the same sense that real numbers can be represented in floating point. The field of transseries is larger than the Levi-Civita field. An example of a transseries is: where for purposes of ordering "x" is considered infinite. Conway's surreal numbers fall into category 2. They are a system designed to be as rich as possible in different sizes of numbers, but not necessarily for convenience in doing

neg: Imaginary unit planar subalgebra The imaginary unit is sometimes written in advanced mathematics contexts (as well as in less advanced popular texts). However, great care needs to be taken when manipulating formulas involving radicals. The radical sign notation is reserved either for the principal square root function, which is "only" defined for real , or for the principal branch of the complex square root function. Attempting to apply the calculation rules of the principal (real) square root function to manipulate the principal branch of the complex square root function can produce false results: Similarly: The calculation rules and are only valid for

neg: Nth root algorithm Nth root algorithm The principal "n"th root formula_1 of a positive real number "A", is the positive real solution of the equation (for integer "n" there are "n" distinct complex solutions to this equation if formula_3, but only one is positive and real). There is a very fast-converging "n"th root algorithm for finding formula_1: A special case is the familiar square-root algorithm. By setting "n" = 2, the "iteration rule" in step 2 becomes the square root iteration rule: Several different derivations of this algorithm are possible. One derivation shows it is a special case of Newton's method (also called

neg: Radical symbol Radical symbol In mathematics, the radical sign or radical symbol or root symbol is a symbol for the square root or higher-order root of a number. The square root of a number formula_1 is written as while the formula_3th root of formula_1 is written as It is also used for other meanings in more advanced mathematics, such as the radical of an ideal. Each positive real number has two square roots, one positive and the other negative. The square root symbol refers to the principal square root, which is the positive one. The two square roots of a negative number