query: who proved that square root of 2 is irrational

pos: Square root of 2 the name of Hippasus of Metapontum is often mentioned. For a while, the Pythagoreans treated as an official secret the discovery that the square root of two is irrational, and, according to legend, Hippasus was murdered for divulging it. The square root of two is occasionally called "Pythagoras' number" or "Pythagoras' constant", for example by . There are a number of algorithms for approximating , which in expressions as a ratio of integers or as a decimal can only be approximated. The most common algorithm for this, one used as a basis in many computers and calculators, is the Babylonian

pos: Proof by infinite descent the square root of 2 () is irrational (i.e. cannot be expressed as a fraction of two whole numbers) was discovered by the ancient Greeks, and is perhaps the earliest known example of a proof by infinite descent. Pythagoreans discovered that the diagonal of a square is incommensurable with its side, or in modern language, that the square root of two is irrational. Little is known with certainty about the time or circumstances of this discovery, but the name of Hippasus of Metapontum is often mentioned. For a while, the Pythagoreans treated as an official secret the discovery that the

pos: Ratio (quantities whose ratio, as value of a fraction, amounts to an irrational number). The earliest discovered example, found by the Pythagoreans, is the ratio of the length of the diagonal to the length of a side of a square, which is the square root of 2, formally formula_10 Another example is the ratio of a circle's circumference to its diameter, which is called , and is not just an algebraically irrational number, but a transcendental irrational. Also well-known is the golden ratio of two (mostly) lengths and , which is defined by the proportion Taking the ratios as fractions and

pos: Square root of 2 and this third by its own fourth less the thirty-fourth part of that fourth." That is, This approximation is the seventh in a sequence of increasingly accurate approximations based on the sequence of Pell numbers, which can be derived from the continued fraction expansion of . Despite having a smaller denominator, it is only slightly less accurate than the Babylonian approximation. Pythagoreans discovered that the diagonal of a square is incommensurable with its side, or in modern language, that the square root of two is irrational. Little is known with certainty about the time or circumstances of this discovery, but

pos: 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: Mathematical constant It is an important constant which appears in many expressions for the derivative of the Riemann zeta function. It has a numerical value of approximately 1.2824271291. Some constants, such as the square root of 2, Liouville's constant and Champernowne constant: are not important mathematical invariants but retain interest being simple representatives of special sets of numbers, the irrational numbers, the transcendental numbers and the normal numbers (in base 10) respectively. The discovery of the irrational numbers is usually attributed to the Pythagorean Hippasus of Metapontum who proved, most likely geometrically, the irrationality of the square root of 2. As for

neg: Square root of 2 if is a monic polynomial with integer coefficients, then any rational root of is necessarily an integer. Applying this to the polynomial , it follows that is either an integer or irrational. Because is not an integer (2 is not a perfect square), must therefore be irrational. This proof can be generalized to show that any root of any natural number which is not the square of a natural number is irrational. See quadratic irrational or infinite descent for a proof that the square root of any non-square natural number is irrational. One proof of the number's irrationality is the

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: 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: Irrational number to show that π is irrational, whence it follows immediately that π is irrational also. The existence of transcendental numbers was first established by Liouville (1844, 1851). Later, Georg Cantor (1873) proved their existence by a different method, which showed that every interval in the reals contains transcendental numbers. Charles Hermite (1873) first proved "e" transcendental, and Ferdinand von Lindemann (1882), starting from Hermite's conclusions, showed the same for π. Lindemann's proof was much simplified by Weierstrass (1885), still further by David Hilbert (1893), and was finally made elementary by Adolf Hurwitz and Paul Gordan. The square root of 2

neg: Square root of 3 Square root of 3 The square root of 3 is the positive real number that, when multiplied by itself, gives the number 3. It is more precisely called the principal square root of 3, to distinguish it from the negative number with the same property. It is denoted by . The square root of 3 is an irrational number. It is also known as Theodorus' constant, named after Theodorus of Cyrene, who proved its irrationality. The first sixty digits of its decimal expansion are: As of December 2013, its numerical value in decimal has been computed to at least ten

neg: Transcendental number transcendental numbers are irrational, since all rational numbers are algebraic. The converse is not true: not all irrational numbers are transcendental; e.g., the square root of 2 is irrational but not a transcendental number, since it is a solution of the polynomial equation . Another irrational number that is not transcendental is the golden ratio, formula_1 or formula_2, since it is a solution of the polynomial equation . The name "transcendental" comes from the Latin "transcendĕre" 'to climb over or beyond, surmount', and was first used for the mathematical concept in Leibniz's 1682 paper in which he proved that is

neg: Number 800 and 500 BC. The first existence proofs of irrational numbers is usually attributed to Pythagoras, more specifically to the Pythagorean Hippasus of Metapontum, who produced a (most likely geometrical) proof of the irrationality of the square root of 2. The story goes that Hippasus discovered irrational numbers when trying to represent the square root of 2 as a fraction. However Pythagoras believed in the absoluteness of numbers, and could not accept the existence of irrational numbers. He could not disprove their existence through logic, but he could not accept irrational numbers, and so, allegedly and frequently reported, he sentenced

neg: Quadratic irrational number be proved irrational. Theodorus of Cyrene proved the irrationality of the square roots of whole numbers up to 17 (except those few that are square numbers, such as 16), but stopped there, probably because the algebra he used could not be applied to the square root of numbers greater than 17. Euclid's Elements Book 10 is dedicated to classification of irrational magnitudes. The original proof of the irrationality of the non-square natural numbers depends on Euclid's lemma. Many proofs of the irrationality of the square roots of non-square natural numbers implicitly assume the fundamental theorem of arithmetic, which was first

neg: Gelfond–Schneider constant Gelfond–Schneider constant The Gelfond–Schneider constant or Hilbert number is two to the power of the square root of two: which was proved to be a transcendental number by Rodion Kuzmin in 1930. In 1934, Aleksandr Gelfond and Theodor Schneider independently proved the more general "Gelfond–Schneider theorem", which solved the part of Hilbert's seventh problem described below. The square root of the Gelfond–Schneider constant is the transcendental number This same constant can be used to prove that "an irrational elevated to an irrational power may be rational", even without first proving its transcendence. The proof proceeds as follows: either is rational,

neg: Gelfond–Schneider constant by Gelfond and by Schneider. Gelfond–Schneider constant The Gelfond–Schneider constant or Hilbert number is two to the power of the square root of two: which was proved to be a transcendental number by Rodion Kuzmin in 1930. In 1934, Aleksandr Gelfond and Theodor Schneider independently proved the more general "Gelfond–Schneider theorem", which solved the part of Hilbert's seventh problem described below. The square root of the Gelfond–Schneider constant is the transcendental number This same constant can be used to prove that "an irrational elevated to an irrational power may be rational", even without first proving its transcendence. The proof proceeds

neg: Irrational number was the first number proved irrational, and that article contains a number of proofs. The golden ratio is another famous quadratic irrational and there is a simple proof of its irrationality in its article. The square roots of all natural numbers which are not perfect squares are irrational and a proof may be found in quadratic irrationals. The proof above for the square root of two can be generalized using the fundamental theorem of arithmetic. This asserts that every integer has a unique factorization into primes. Using it we can show that if a rational number is not an integer

neg: Irrational number rely on deductive reasoning for proof.” This method of exhaustion is the first step in the creation of calculus. Theodorus of Cyrene proved the irrationality of the surds of whole numbers up to 17, but stopped there probably because the algebra he used could not be applied to the square root of 17. It was not until Eudoxus developed a theory of proportion that took into account irrational as well as rational ratios that a strong mathematical foundation of irrational numbers was created. Geometrical and mathematical problems involving irrational numbers such as square roots were addressed very early during the

neg: Square root of 2 Square root of 2 The square root of 2, or the (1/2)th power of 2, written in mathematics as or , is the positive algebraic number that, when multiplied by itself, gives the number 2. Technically, it is called the principal square root of 2, to distinguish it from the negative number with the same property. Geometrically the square root of 2 is the length of a diagonal across a square with sides of one unit of length; this follows from the Pythagorean theorem. It was probably the first number known to be irrational. As a good rational approximation for

neg: Theodorus of Cyrene "Elements" as a test for incommensurability. In modern terms, the theorem is that a real number with an "infinite" continued fraction expansion is irrational. Irrational square roots have periodic expansions. The period of the square root of 19 has length 6, which is greater than the period of the square root of any smaller number. The period of √17 has length one (so does √18; but the irrationality of √18 follows from that of √2). The so-called Spiral of Theodorus is composed of contiguous right triangles with hypotenuse lengths equal √2, √3, √4, …, √17; additional triangles cause the diagram

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: Square root of a matrix a "positive-semidefinite matrix" has precisely one positive-semidefinite square root, which can be called its "principal square root". While the square root of a nonnegative integer is either again an integer or an irrational number, in contrast an "integer matrix" can have a square root whose entries are rational, yet not integer. For example, the matrix formula_8 has the non-integer square root formula_9 as well as the integer square root matrix formula_10. The 2×2 identity matrix is another example. A 2×2 matrix with two distinct nonzero eigenvalues has four square roots. More generally, an matrix with "distinct nonzero eigenvalues" has 2

neg: Spiral of Theodorus in the sequence is a right triangle with side lengths and 1, and with hypotenuse . For example, the 16th triangle has sides measuring 4 (=), 1 and hypotenuse of Although all of Theodorus' work has been lost, Plato put Theodorus into his dialogue "Theaetetus", which tells of his work. It is assumed that Theodorus had proved that all of the square roots of non-square integers from 3 to 17 are irrational by means of the Spiral of Theodorus. Plato does not attribute the irrationality of the square root of 2 to Theodorus, because it was well known before him.

neg: Proof of impossibility Euclidean. A good example is the irrational number the square root of 2. It is simply the length of the hypotenuse of a right triangle with legs both one unit in length, and it can be constructed with straightedge and compass. But it was proved centuries after Euclid that Euclidean numbers cannot involve any operations other than addition, subtraction, multiplication, division, and the extraction of square roots. Both trisecting the general angle and doubling the cube require taking cube roots, which are not constructible numbers by compass and straightedge. formula_2 is not a Euclidean number ... and therefore it is

neg: Irreducible fraction an irreducible fraction is utilized in various proofs of the irrationality of the square root of 2 and of other irrational numbers. For example, one proof notes that if the square root of 2 could be represented as a ratio of integers, then it would have in particular the fully reduced representation formula_11 where "a" and "b" are the smallest possible; but given that formula_11 equals the square root of 2, so does formula_13 (since cross-multiplying this with formula_11 shows that they are equal). Since the latter is a ratio of smaller integers, this is a contradiction, so the premise

neg: Periodic continued fraction fraction "x" is said to be "purely periodic". For example, the regular continued fraction for the golden ratio φ – given by [1; 1, 1, 1, …] – is purely periodic, while the regular continued fraction for the square root of two – [1; 2, 2, 2, …] – is periodic, but not purely periodic. A quadratic irrational number is an irrational real root of the quadratic equation where the coefficients "a", "b", and "c" are integers, and the discriminant, "b" − 4"ac", is greater than zero. By the quadratic formula every quadratic irrational can be written in the form

neg: 2 the sum of the reciprocals of the powers of "x" equals itself. In symbols This comes from the fact that: Powers of two are central to the concept of Mersenne primes, and important to computer science. Two is the first Mersenne prime exponent. Taking the square root of a number is such a common mathematical operation, that the spot on the root sign where the exponent would normally be written for cubic and other roots, may simply be left blank for square roots, as it is tacitly understood. The square root of 2 was the first known irrational number. The

neg: Square root of a 2 by 2 matrix use the simplified formula where "a" = ±√"A" and "d" = ±√"D"; which, for the various sign choices, gives four, two, or one distinct matrices, if none of, only one of, or both "A" and "D" are zero, respectively. Because it has duplicate eigenvalues, the 2×2 identity matrix formula_10 has infinitely many symmetric rational square roots given by where is any Pythagorean triple—that is, any set of positive integers such that formula_12 In addition, any non-integer, irrational, or complex values of "r, s, t" satisfying formula_13 give square root matrices. The identity matrix also has infinitely many non-symmetric square roots.

neg: Square root of 2 following proof by infinite descent. It is also a proof by contradiction, also known as an indirect proof, in that the proposition is proved by assuming that the opposite of the proposition is true and showing that this assumption is false, thereby implying that the proposition must be true. Because there is a contradiction, the assumption (1) that is a rational number must be false. This means that is not a rational number; i.e., is irrational. This proof was hinted at by Aristotle, in his "Analytica Priora", §I.23. It appeared first as a full proof in Euclid's "Elements", as proposition

neg: Contraposition the original statement, is also true. Therefore, if it can be proven that formula_40 cannot be expressed as an irreducible fraction, then it must be the case that formula_40 is not a rational number. The latter can be proved by contradiction. The previous example employed the contrapositive of a definition to prove a theorem. One can also prove a theorem by proving the contrapositive of the theorem's statement. To prove that "if a positive integer "N" is a non-square number, its square root is irrational", we can equivalently prove its contrapositive, that "if a positive integer "N" has a square

neg: Nth root for "irrational number" being translated into Latin as "surdus" (meaning "deaf" or "mute"). Gerard of Cremona (c. 1150), Fibonacci (1202), and then Robert Recorde (1551) all used the term to refer to "unresolved irrational roots". A square root of a number "x" is a number "r" which, when squared, becomes "x": Every positive real number has two square roots, one positive and one negative. For example, the two square roots of 25 are 5 and −5. The positive square root is also known as the principal square root, and is denoted with a radical sign: Since the square of every

neg: Wolfgang von Wersin without exception root figures and are all irrational numbers. The calculations for measure relations of the Orthogons are based, without exception, on the Pythagorean doctrine." Examples of these root figure relations are: the Diagon relation is 1: square root of 2, the Sixton is 1: square root of 3 and the Doppelquadrat is 1: square root of 4. Mathematical ratios for all twelve Orthogons: Ratios for all twelve Orthogons: Quadrat 1:1 – Hemidiagon 1:1.118 – Trion 1:1.154 – Quadriagon 1:1.207 – Biauron 1:1.236 – Penton 1:1.376 – Diagon 1:1.414 – Bipenton 1:1.46 – Hemiolion 1:1.5 – Auron 1:1.618 – Sixton

neg: Periodic continued fraction where "P", "D", and "Q" are integers, "D" > 0 is not a perfect square (but not necessarily square-free), and "Q" divides the quantity "P" − "D" (for example (6+)/4). Such a quadratic irrational may also be written in another form with a square-root of a square-free number (for example (3+)/2) as explained for quadratic irrationals. By considering the complete quotients of periodic continued fractions, Euler was able to prove that if "x" is a regular periodic continued fraction, then "x" is a quadratic irrational number. The proof is straightforward. From the fraction itself, one can construct the quadratic equation

neg: Constructive proof it merely gives a number of possibilities (in this case, two mutually exclusive possibilities) and shows that one of them—but does not show "which" one—must yield the desired example. It turns out that formula_21 is irrational because of the Gelfond–Schneider theorem, but this fact is irrelevant to the correctness of the non-constructive proof. A "constructive" proof of the above theorem on irrational powers of irrationals would give an actual example, such as: The square root of 2 is irrational, and 3 is rational. formula_27 is also irrational: if it were equal to formula_28, then, by the properties of logarithms, 9

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 2 117 of Book X. However, since the early 19th century historians have agreed that this proof is an interpolation and not attributable to Euclid. An alternative proof uses the same approach with the fundamental theorem of arithmetic which says every integer greater than 1 has a unique factorization into powers of primes. This proof can be generalized to show that if an integer is not an exact th power of another integer then its th root is irrational. For a proof of the same result which does not rely on the fundamental theorem of arithmetic, see: quadratic irrational. The following

neg: 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

neg: Fraction (mathematics) a/b, where a and b are integers and b is not zero, is called the set of rational numbers and is represented by the symbol Q, which stands for quotient. The test for a number being a rational number is that it can be written in that form (i.e., as a common fraction). However, the word "fraction" is also used to describe mathematical expressions that are not rational numbers, for example algebraic fractions (quotients of algebraic expressions), and expressions that contain irrational numbers, such as /2 (see square root of 2) and π/4 (see proof that π is irrational). In

neg: Irrational number the coefficients formula_7 are integers and formula_8. Any rational root of this polynomial equation must be of the form "r" /"s", where "r" is a divisor of "a" and "s" is a divisor of "a". If a real root formula_9 of a polynomial formula_10 is not among these finitely many possibilities, it must be an irrational algebraic number. An exemplary proof for the existence of such algebraic irrationals is by showing that "x" = (2 + 1) is an irrational root of a polynomial with integer coefficients: it satisfies ("x" − 1) = 2 and hence "x" − 2"x" −

neg: Golden ratio is a contradiction that follows from the assumption that is rational. Another short proof—perhaps more commonly known—of the irrationality of the golden ratio makes use of the closure of rational numbers under addition and multiplication. If formula_15 is rational, then formula_16 is also rational, which is a contradiction if it is already known that the square root of a non-square natural number is irrational. The golden ratio is also an algebraic number and even an algebraic integer. It has minimal polynomial Having degree 2, this polynomial actually has two roots, the other being the golden ratio conjugate. The conjugate root

neg: Proof by infinite descent square root of two is irrational, and, according to legend, Hippasus was murdered for divulging it. The square root of two is occasionally called "Pythagoras' number" or "Pythagoras' Constant", for example . The ancient Greeks, not having algebra, worked out a geometric proof by infinite descent (John Horton Conway presented another geometric proof (no. 8 ' ' ' ) by infinite descent that may be more accessible). The following is an algebraic proof along similar lines: Suppose that were rational. Then it could be written as for two natural numbers, and . Then squaring would give so 2 must divide

neg: Square root of 5 Square root of 5 The square root of 5 is the positive real number that, when multiplied by itself, gives the prime number 5. It is more precisely called the principal square root of 5, to distinguish it from the negative number with the same property. This number appears in the fractional expression for the golden ratio. It can be denoted in surd form as: It is an irrational algebraic number. The first sixty significant digits of its decimal expansion are: which can be rounded down to 2.236 to within 99.99% accuracy. The approximation (≈ 2.23611) for the square root

neg: Square root of 3 between two phases in a three-phase system equals times the line to neutral voltage. This is because any two phases are 120° apart, and two points on a circle 120 degrees apart are separated by times the radius (see geometry examples above). Square root of 3 The square root of 3 is the positive real number that, when multiplied by itself, gives the number 3. It is more precisely called the principal square root of 3, to distinguish it from the negative number with the same property. It is denoted by . The square root of 3 is an irrational

neg: Fashionable Nonsense nevertheless agrees with Sokal and Bricmont that the "square root of −1" which Lacan discusses (and for which Plotnitsky introduces the symbol formula_1) is not, in spite of its identical name, "identical, directly linked, or even metaphorized via the mathematical square root of −1," and that the latter "is "not" the erectile organ." However, with regard to the second sense, which Plotnisky describes by stating that "all imaginary and complex numbers are, by definition, irrational," mathematicians agree with Sokal and Bricmont in not taking complex numbers as irrational. Indeed, the concept of rational numbers has been extended into the complex

neg: Mathematical constant number with the same property. Geometrically the square root of 2 is the length of a diagonal across a square with sides of one unit of length; this follows from the Pythagorean theorem. It was probably the first number known to be irrational. Its numerical value truncated to 65 decimal places is: The quick approximation 99/70 (≈ 1.41429) for the square root of two is frequently used. Despite having a denominator of only 70, it differs from the correct value by less than 1/10,000 (approx. 7.2 × 10). The imaginary unit or unit imaginary number, denoted as , is a

neg: Dedekind cut partition formula_14 with This cut represents the irrational number in Dedekind's construction. To establish this truly, one must show that this really is a cut and that it is the square root of two. However, neither claim is immediate. Showing that it is a cut requires showing that for any positive rational "x" with , there is a rational "y" with and . The choice formula_17 works. Then we have a cut and it has a square no larger than 2, but to show equality requires showing that if "r" is any rational number less than 2, then there is

neg: 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

neg: 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

neg: Proof by contradiction mathematics is the proof that the square root of 2 is irrational. If it were rational, it could be expressed as a fraction "a"/"b" in lowest terms, where "a" and "b" are integers, at least one of which is odd. But if "a"/"b" = , then "a" = 2"b". Therefore, "a" must be even. Because the square of an odd number is odd, that in turn implies that "a" is even. This means that "b" must be odd because a/b is in lowest terms. On the other hand, if "a" is even, then "a" is a multiple of 4. If

neg: Square the square is no more than 1/2. Squaring the circle is the problem, proposed by ancient geometers, of constructing a square with the same area as a given circle by using only a finite number of steps with compass and straightedge. In 1882, the task was proven to be impossible, as a consequence of the Lindemann–Weierstrass theorem which proves that pi () is a transcendental number, rather than an algebraic irrational number; that is, it is not the root of any polynomial with rational coefficients. In non-Euclidean geometry, squares are more generally polygons with 4 equal sides and equal angles.

neg: Singly and doubly even a row: D12, D6, and D3. Generally, with a score of , one has such game shots. This is why is such a desirable score: it splits 5 times. The classic proof that the square root of 2 is irrational operates by infinite descent. Usually, the descent part of the proof is abstracted away by assuming (or proving) the existence of irreducible representations of rational numbers. An alternate approach is to exploit the existence of the ν operator. Assume by contradiction that where "a" and "b" are non-zero natural numbers. Square both sides of the equality and apply the 2-order

neg: Square root of 2 produce another whole-number square, even when the equation is not in its simplest form. "Proof": Assume formula_40 is rational. Therefore, But the lemma proves that the sum of two identical whole-number squares cannot produce another whole-number square. Therefore, the assumption that formula_40 is rational is contradicted. formula_40 is irrational. Q. E. D. One-half of , also the reciprocal of , approximately , is a common quantity in geometry and trigonometry because the unit vector that makes a 45° angle with the axes in a plane has the coordinates This number satisfies One interesting property of is as follows: since This

neg: Dynamic rectangle a root rectangle. Its diagonal equals φ times the length of the shorter side. If a root-φ rectangle is divided by a diagonal, the result is two congruent Kepler triangles. Jay Hambidge, as part of his theory of dynamic symmetry, includes the root rectangles in what he calls "dynamic rectangles", which have irrational and geometric fractions as ratios, such as the golden ratio or square roots. Hambidge distinguishes these from rectangles with rational proportions, which he terms "static rectangles". According to him, root-2, 3, 4 and 5 rectangles are often found in Gothic and Classical Greek and Roman art, objects

neg: Sexagesimal fractions. For example: The fact in arithmetic that the two numbers that are adjacent to sixty, namely 59 and 61, are both prime numbers implies that simple repeating fractions that repeat with a period of one or two sexagesimal digits can only have 59 or 61 as their denominators ( = 0;; = 0;), and that other non-regular primes have fractions that repeat with a longer period. The square root of 2, the length of the diagonal of a unit square, was approximated by the Babylonians of the Old Babylonian Period () as Because ≈ ... is an irrational number,

neg: Square root of 2 as shown in Figure 1. The square overlap region in the middle () must equal the sum of the two uncovered squares (). But these squares on the diagonal have positive integer sides that are smaller than the original squares. Repeating this process we can find arbitrarily small squares one twice the area of the other, yet both having positive integer sides, which is impossible since positive integers cannot be less than 1. Another geometric reductio ad absurdum argument showing that is irrational appeared in 2000 in the American Mathematical Monthly. It is also an example of proof by infinite

neg: T. K. Seung are rigorous enough. To be sure, they cannot be as rigorous as the definition of mathematical Forms. The difference of these two types of definition is like the difference between the square root of a square number such as 4 or 9 and that of an oblong number such as 2 or 3. The former is a rational number; the latter is an irrational number. Unlike the former, the computation of the latter can never be completed. Although it is only an approximation of its true value, it is qualitatively different from a mere guess or opinion. Since its method

neg: Quadratic irrational number Quadratic irrational number In mathematics, a quadratic irrational number (also known as a quadratic irrational, a quadratic irrationality or quadratic surd) is an irrational number that is the solution to some quadratic equation with rational coefficients which is irreducible over the set of rational numbers. Since fractions in the coefficients of a quadratic equation can be cleared by multiplying both sides by their common denominator, a quadratic irrational is an irrational root of some quadratic equation whose coefficients are integers. The quadratic irrational numbers, a subset of the complex numbers, are algebraic numbers of degree 2, and can therefore be

neg: General number field sieve norm of "a" − "r""b" and hence that the product of the corresponding factors "a" − "r""b" is a square in Z["r"], with a "square root" which can be determined (as a product of known factors in Z["r"])—it will typically be represented as an irrational algebraic number. Similarly, the product of the factors "a" − "r""b" is a square in Z["r"], with a "square root" which also can be computed. It should be remarked that the use of Gaussian elimination does not give the optimal run time of the algorithm. Instead, sparse matrix solving algorithms such as Block Lanczos or

neg: Irrational number circumference to its diameter, Euler's number "e", the golden ratio "φ", and the square root of two; in fact all square roots of natural numbers, other than of perfect squares, are irrational. It can be shown that irrational numbers, when expressed in a positional numeral system (e.g. as decimal numbers, or with any other natural basis), do not terminate, nor do they repeat, i.e., do not contain a subsequence of digits, the repetition of which makes up the tail of the representation. For example, the decimal representation of the number starts with 3.14159, but no finite number of digits can

neg: Liu Hui's π algorithm of the next–order polygon bisected from . The same calculation is done repeatedly, each step requiring only one addition and one square root extraction. Calculation of square roots of irrational numbers was not an easy task in the third century with counting rods. Liu Hui discovered a shortcut by comparing the area differentials of polygons, and found that the proportion of the difference in area of successive order polygons was approximately 1/4. Let denote the difference in areas of -gon and (/2)-gon He found: Hence: Area of unit radius circle = In which That is all the subsequent excess areas

neg: Quadratic irrational number proven by Carl Friedrich Gauss in his Disquisitiones Arithmeticae. This asserts that every integer has a unique factorization into primes. For any rational non-integer in lowest terms there must be a prime in the denominator which does not divide into the numerator. When the numerator is squared that prime will still not divide into it because of the unique factorization. Therefore, the square of a rational non-integer is always a non-integer; by contrapositive, the square root of an integer is always either another integer, or irrational. Euclid used a restricted version of the fundamental theorem and some careful argument to

neg: Prover9 until 2009. Prover9 is free software, and therefore, open source software; it is released under GPL version 2 or later. The traditional "all men are mortal", "Socrates is a man", prove "Socrates is mortal" can be expressed this way in Prover9: This will be automatically converted into clausal form (which Prover9 also accepts): A proof that the square root of 2 is irrational can be expressed this way: Prover9 Prover9 is an automated theorem prover for First-order and equational logic developed by William McCune. Prover9 is the successor of the Otter theorem prover also developed by MCCune.. Prover9 is noted

neg: 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

neg: Quadratic irrational number prove the theorem. His proof is in Euclid's Elements Book X Proposition 9. The fundamental theorem of arithmetic is not actually required to prove the result, however. There are self-contained proofs by Richard Dedekind, among others. The following proof was adapted by Colin Richard Hughes from a proof of the irrationality of the square root of two found by Theodor Estermann in 1975. Assume "D" is a non-square natural number, then there is a number "n" such that: so in particular Assume the square root of "D" is a rational number "p"/"q", assume the "q" here is the smallest for

neg: Nth root which is also negative. This can be found by first multiplying both sides of the defining equation by −1 to obtain formula_83 then proceeding as before to find |"r"|, and using . The ancient Greek mathematicians knew how to use compass and straightedge to construct a length equal to the square root of a given length. In 1837 Pierre Wantzel proved that an "n"th root of a given length cannot be constructed if "n" is not a power of 2. Every complex number other than 0 has "n" different "n"th roots. The two square roots of a complex number are

neg: Square root of 2 is a right angle and is half a right angle, is also a right isosceles triangle. Hence implies . By symmetry, , and is also a right isosceles triangle. It also follows that . Hence we have an even smaller right isosceles triangle, with hypotenuse length and legs . These values are integers even smaller than and and in the same ratio, contradicting the hypothesis that is in lowest terms. Therefore, and cannot be both integers, hence is irrational. This is another proof by contradiction, supposing that is rational. In a constructive approach, one distinguishes between on the one hand

neg: Reduction (complexity) can only use the squaring function one time, and only at the end. In this case, even if we're allowed to use all the basic arithmetic operations, including multiplication, no reduction exists in general, because in order to get the desired result as a square we have to compute its square root first, and this square root could be an irrational number like formula_2 that cannot be constructed by arithmetic operations on rational numbers. Going in the other direction, however, we can certainly square a number with just one multiplication, only at the end. Using this limited form of reduction,

neg: Mathematics and art P. King describes mathematics as an art, stating that "the keys to mathematics are beauty and elegance and not dullness and technicality", and that beauty is the motivating force for mathematical research. King cites the mathematician G. H. Hardy's 1940 essay "A Mathematician's Apology". In it, Hardy discusses why he finds two theorems of classical times as first rate, namely Euclid's proof there are infinitely many prime numbers, and the proof that the square root of 2 is irrational. King evaluates this last against Hardy's criteria for mathematical elegance: ""seriousness, depth, generality, unexpectedness, inevitability", and "economy"" (King's italics), and describes

neg: Square root of 5 theorem in Diophantine approximations states that every irrational number can be approximated by infinitely many rational numbers in lowest terms in such a way that and that is best possible, in the sense that for any larger constant than , there are some irrational numbers for which only finitely many such approximations exist. Closely related to this is the theorem that of any three consecutive convergents , , , of a number , at least one of the three inequalities holds: And the in the denominator is the best bound possible since the convergents of the golden ratio make the

neg: Proof that e is irrational a proof that "e" is not a root of a second degree polynomial with rational coefficients. This last fact implies that "e" is irrational. His proofs are similar to Fourier's proof of the irrationality of "e". In 1891, Hurwitz explained how it is possible to prove along the same line of ideas that "e" is not a root of a third degree polynomial with rational coefficients. In particular, "e" is irrational. More generally, "e" is irrational for any non-zero rational "q". Proof that e is irrational The number "e" was introduced by Jacob Bernoulli in 1683. More than half a

neg: Proof of impossibility one of two possible counterexamples must be a valid counterexample, without showing which one it is. The proof by Pythagoras (or more likely one of his students) about 500 BCE has had a profound effect on mathematics. It shows that the square root of 2 cannot be expressed as the ratio of two integers (counting numbers). The proof bifurcated "the numbers" into two non-overlapping collections—the rational numbers and the irrational numbers. This bifurcation was used by Cantor in his diagonal method, which in turn was used by Turing in his proof that the "Entscheidungsproblem" (the decision problem of Hilbert) is

neg: Mathematical proof also covers number theory, including a proof that the square root of two is irrational and that there are infinitely many prime numbers. Further advances took place in medieval Islamic mathematics. While earlier Greek proofs were largely geometric demonstrations, the development of arithmetic and algebra by Islamic mathematicians allowed more general proofs that no longer depended on geometry. In the 10th century CE, the Iraqi mathematician Al-Hashimi provided general proofs for numbers (rather than geometric demonstrations) as he considered multiplication, division, etc. for "lines." He used this method to provide a proof of the existence of irrational numbers. An inductive

neg: Square root of 2 not being rational, and on the other hand being irrational (i.e., being quantifiably apart from every rational), the latter being a stronger property. Given positive integers and , because the valuation (i.e., highest power of 2 dividing a number) of is odd, while the valuation of is even, they must be distinct integers; thus . Then the latter inequality being true because we assume (otherwise the quantitative apartness can be trivially established). This gives a lower bound of for the difference , yielding a direct proof of irrationality not relying on the law of excluded middle; see Errett Bishop (1985,

neg: Square root of a 2 by 2 matrix Square root of a 2 by 2 matrix A square root of a 2×2 matrix "M" is another 2 by 2 matrix "R" such that "M" = "R", where "R" stands for the matrix product of "R" with itself. In general there can be no, two, four or even an infinitude of square root matrices. In many cases such a matrix "R" can be obtained by an explicit formula. A 2×2 matrix with two distinct nonzero eigenvalues has four square roots. A positive-definite matrix has precisely one positive-definite square root. Square roots of a matrix of any dimension come in

neg: Nested radical the cube roots being the real cube roots. In the case of three real roots, the square root expression is an imaginary number; here any real root is expressed by defining the first cube root to be any specific complex cube root of the complex radicand, and by defining the second cube root to be the complex conjugate of the first one. The nested radicals in this solution cannot in general be simplified unless the cubic equation has at least one rational solution. Indeed, if the cubic has three irrational but real solutions, we have the "casus irreducibilis", in which

neg: History of algebra is found in "Al-Jabr", and in one case the same example as found in "Al-Jabr", and even goes beyond "Al-Jabr" by giving a geometric proof that if the discriminant is negative then the quadratic equation has no solution. The similarity between these two works has led some historians to conclude that Arabic algebra may have been well developed by the time of al-Khwarizmi and 'Abd al-Hamid. Arabic mathematicians treated irrational numbers as algebraic objects. The Egyptian mathematician Abū Kāmil Shujā ibn Aslam (c. 850–930) was the first to accept irrational numbers (often in the form of a square root, cube

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: 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: Proof that π is irrational In 1761, Lambert proved that is irrational by first showing that this continued fraction expansion holds: Then Lambert proved that if "x" is non-zero and rational then this expression must be irrational. Since tan(/4) = 1, it follows that /4 is irrational and therefore that is irrational. A simplification of Lambert's proof is given below. This proof uses the characterization of as the smallest positive number whose half is a zero of the cosine function and it actually proves that is irrational. As in many proofs of irrationality, the argument proceeds by reductio ad absurdum. Consider the sequences ("A") and

neg: 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

neg: Silver ratio Silver ratio In mathematics, two quantities are in the silver ratio (also silver mean or silver constant) if the ratio of the sum of the smaller and twice the larger of those quantities, to the larger quantity, is the same as the ratio of the larger one to the smaller one (see below). This defines the silver ratio as an irrational mathematical constant, whose value of one plus the square root of 2 is approximately 2.4142135623. Its name is an allusion to the golden ratio; analogously to the way the golden ratio is the limiting ratio of consecutive Fibonacci numbers,

neg: Nth root the number is irrational. Since for positive real numbers and the equality formula_55 holds, the above property can be extended to positive rational numbers. Let formula_56, with and coprime and positive integers, be a rational number, has a rational "n"th root, if both positive have an integer "n"th root, i.e., formula_57 is the product of "n"th powers of rational numbers. If one or both "n"th roots of or are irrational, the quotient is irrational, too. The "n"th root of a number "A" can be computed by the "n"th root algorithm, a special case of Newton's method. Start with an initial

neg: Number roughly 300 BC. Of the Indian texts, the most relevant is the Sthananga Sutra, which also covers number theory as part of a general study of mathematics. The concept of decimal fractions is closely linked with decimal place-value notation; the two seem to have developed in tandem. For example, it is common for the Jain math sutra to include calculations of decimal-fraction approximations to pi or the square root of 2. Similarly, Babylonian math texts had always used sexagesimal (base 60) fractions with great frequency. The earliest known use of irrational numbers was in the Indian Sulba Sutras composed between

neg: Integer square root formula_21 is not a perfect square. If formula_21 is a perfect square, the sequence ends up in a period-two cycle between formula_15 and formula_24 instead of converging. For termination, it suffices to check that either the number has converged or it has increased by exactly one from the previous step, in which case the new result is discarded. Although formula_5 is irrational for many formula_26, the sequence formula_9 contains only rational terms when formula_28 is rational. Thus, with this method it is unnecessary to exit the field of rational numbers in order to calculate formula_6, a fact which has some

neg: Repeating decimal repetition. Examples of such irrational numbers are the square root of 2 and . There are several notational conventions for representing repeating decimals. None of them are accepted universally. In English, there are various ways to read repeating decimals aloud. For example, formula_1 may be read "one point three repeating", "one point three repeated", "one point three recurring", or "one point three into infinity". In order to convert a rational number represented as a fraction into decimal form, one may use long division. For example, consider the rational number 5/74: etc. Observe that at each step we have a remainder;

neg: Proof that π is irrational Laczkovich to find a new and simpler proof of the fact that the tangent function has the continued fraction expansion that Lambert had discovered. Laczkovich's result can also be expressed in . In fact, Γ("k")"J"(2"x") = "x""f"("x"). So Laczkovich's result is equivalent to: If "x" ≠ 0 and if "x" is rational, then Proof that π is irrational In the 1760's, Johann Heinrich Lambert proved that the number (pi) is irrational: that is, it cannot be expressed as a fraction "a"/"b", where "a" is an integer and "b" is a non-zero integer. In the 19th century, Charles Hermite found a

neg: Apotome (mathematics) Apotome (mathematics) In the historical study of mathematics, an apotome is a line segment formed from a longer line segment by breaking it into two parts, one of which is commensurable only in power to the whole; the other part is the apotome. In this definition, two line segments are said to be "commensurable only in power" when the ratio of their lengths is an irrational number but the ratio of their squared lengths is rational. Translated into modern algebraic language, an apotome can be interpreted as a quadratic irrational number formed by subtracting one square root of a rational

neg: Rational root theorem divides 1, limiting the possibilities to ±1, ±2, ±3, and ±6. Of these, 1, 2, and –3 equate the polynomial to zero, and hence are its rational roots. (In fact these are its only roots since a cubic has only three roots; in general, a polynomial could have some rational and some irrational roots.) Every rational root of the polynomial must be among the numbers symbolically indicated by which gives the list of 8 possible answers: These root candidates can be tested using Horner's method (for instance). In this particular case there is exactly one rational root. If a root

neg: Abū Kāmil Shujāʿ ibn Aslam life and career of Abu Kamil except that he was a successor of al-Khwarizmi, whom he never personally met. The "Algebra" is perhaps Abu Kamil's most influential work, which he intended to supersede and expand upon that of Al-Khwarizmi. Whereas the "Algebra" of al-Khwarizmi was geared towards the general public, Abu Kamil was addressing other mathematicians, or readers familiar with Euclid's "Elements". In this book Abu Kamil solves systems of equations whose solutions are whole numbers and fractions, and accepted irrational numbers (in the form of a square root or fourth root) as solutions and coefficients to quadratic equations. The

neg: Schizophrenic number Schizophrenic number A schizophrenic number (also known as mock rational number) is an irrational number that displays certain characteristics of rational numbers. "The Universal Book of Mathematics" defines "schizophrenic number" as: The sequence of numbers generated by the recurrence relation f(n) = 10 f(n − 1) + n described above is: The integer parts of their square roots, alternate between numbers with irregular digits and numbers with repeating digits, in a similar way to the alternations appearing within the fractional part of each square root. Clifford A. Pickover has said that the schizophrenic numbers were discovered by Kevin Brown. In

neg: Proof that e is irrational Proof that e is irrational The number "e" was introduced by Jacob Bernoulli in 1683. More than half a century later, Euler, who had been a student of Jacob's younger brother Johann, proved that "e" is irrational; that is, that it cannot be expressed as the quotient of two integers. Euler wrote the first proof of the fact that "e" is irrational in 1737 (but the text was only published seven years later). He computed the representation of "e" as a simple continued fraction, which is Since this continued fraction is infinite and every rational number has a terminating continued

neg: Apéry's theorem a positive integer then for some rational number "p"/"q". Specifically, writing the infinite series on the left as ζ(2"n") he showed where the "B" are the rational Bernoulli numbers. Once it was proved that π is always irrational this showed that ζ(2"n") is irrational for all positive integers "n". No such representation in terms of π is known for the so-called zeta constants for odd arguments, the values ζ(2"n"+1) for positive integers "n". It has been conjectured that the ratios of these quantities are transcendental for every integer "n" ≥ 1. Because of this, no proof could be found to

neg: Extraneous and missing solutions If we take the positive square root of both sides, we get: We are not taking the square root of any negative values here, since both "x" and 4 are necessarily positive. But we have lost the solution "x" = −2. The reason is that "x" is actually not in general the "positive" square root of "x". If "x" is negative, the positive square root of "x" is "-x". If the step is taken correctly, it leads instead to the equation: This equation has the same two solutions as the original one: "x" = 2, and "x" = −2. Extraneous

neg: Area polygons). Swiss scientist Johann Heinrich Lambert in 1761 proved that π, the ratio of a circle's area to its squared radius, is irrational, meaning it is not equal to the quotient of any two whole numbers. In 1794 French mathematician Adrien-Marie Legendre proved that π is irrational; this also proves that π is irrational. In 1882, German mathematician Ferdinand von Lindemann proved that π is transcendental (not the solution of any polynomial equation with rational coefficients), confirming a conjecture made by both Legendre and Euler. Heron (or Hero) of Alexandria found what is known as Heron's formula for the area

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: Proof that π is irrational Proof that π is irrational In the 1760's, Johann Heinrich Lambert proved that the number (pi) is irrational: that is, it cannot be expressed as a fraction "a"/"b", where "a" is an integer and "b" is a non-zero integer. In the 19th century, Charles Hermite found a proof that requires no prerequisite knowledge beyond basic calculus. Three simplifications of Hermite's proof are due to Mary Cartwright, Ivan Niven, and Nicolas Bourbaki. Another proof, which is a simplification of Lambert's proof, is due to Miklós Laczkovich. In 1882, Ferdinand von Lindemann proved that is not just irrational, but transcendental as well.

neg: Schizophrenic number his book "Wonders of Numbers" he has so described the history of schizophrenic numbers: Schizophrenic number A schizophrenic number (also known as mock rational number) is an irrational number that displays certain characteristics of rational numbers. "The Universal Book of Mathematics" defines "schizophrenic number" as: The sequence of numbers generated by the recurrence relation f(n) = 10 f(n − 1) + n described above is: The integer parts of their square roots, alternate between numbers with irregular digits and numbers with repeating digits, in a similar way to the alternations appearing within the fractional part of each square root. Clifford

neg: Dynamic rectangle in which the ratio of the longer side to the shorter is the square root of an integer, such as , , etc. The root-2 rectangle (ACDK in Fig. 10) is constructed by extending two opposite sides of a square to the length of the square's diagonal. The root-3 rectangle is constructed by extending the two longer sides of a root-2 rectangle to the length of the root-2 rectangle's diagonal. Each successive root rectangle is produced by extending a root rectangle's longer sides to equal the length of that rectangle's diagonal. The root-φ rectangle is a dynamic rectangle but not

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: Square root of a 2 by 2 matrix If "B" is zero but "A" and "D" are not both zero, one can use This formula will provide two solutions if "A" = "D" or "A" = 0 or "D" = 0, and four otherwise. A similar formula can be used when "C" is zero but "A" and "D" are not both zero. Square root of a 2 by 2 matrix A square root of a 2×2 matrix "M" is another 2 by 2 matrix "R" such that "M" = "R", where "R" stands for the matrix product of "R" with itself. In general there can be no, two,