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1. From Wikipedia, the free encyclopedia2. Lexicographical order

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  • Transcendental functionFrom Wikipedia, the free encyclopedia

  • Contents

    1 nth root 11.1 Etymology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

    1.1.1 Origin of the root symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.2 Etymology of surd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

    1.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Denition and notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

    1.3.1 Square roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3.2 Cube roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

    1.4 Identities and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.5 Simplied form of a radical expression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.6 Innite series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.7 Computing principal roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

    1.7.1 nth root algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.7.2 Digit-by-digit calculation of principal roots of decimal (base 10) numbers . . . . . . . . . 91.7.3 Logarithmic computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

    1.8 Geometric constructibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.9 Complex roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

    1.9.1 Square roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.9.2 Roots of unity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.9.3 nth roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

    1.10 Solving polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.11 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.12 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.13 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

    2 Transcendental function 152.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.4 Algebraic and transcendental functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.5 Transcendentally transcendental functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.6 Exceptional set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.7 Dimensional analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

    i

  • ii CONTENTS

    2.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.10 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.11 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 19

    2.11.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.11.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.11.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

  • Chapter 1

    nth root

    Roots of integer numbers from 0 to 10. Line labels = x. x-axis = n. y-axis = nth root of x.

    In mathematics, the nth root of a number x, where n is a positive integer, is a number r which, when raised to thepower n yields x

    rn = x;

    where n is the degree of the root. A root of degree 2 is called a square root and a root of degree 3, a cube root. Rootsof higher degree are referred by using ordinal numbers, as in fourth root, twentieth root, etc.For example:

    1

  • 2 CHAPTER 1. NTH ROOT

    2 is a square root of 4, since 22 = 4. 2 is also a square root of 4, since (2)2 = 4.

    A real number or complex number has n roots of degree n. While the roots of 0 are not distinct (all equaling 0), then nth roots of any other real or complex number are all distinct. If n is even and x is real and positive, one of its nthroots is positive, one is negative, and the rest are complex but not real; if n is even and x is real and negative, none ofthe nth roots is real. If n is odd and x is real, one nth root is real and has the same sign as x , while the other roots arenot real. Finally, if x is not real, then none of its nth roots is real.Roots are usually written using the radical symbol or radixp or p , with px or px denoting the square root, 3pxdenoting the cube root, 4px denoting the fourth root, and so on. In the expression npx , n is called the index, p isthe radical sign or radix, and x is called the radicand. Since the radical symbol denotes a function, when a numberis presented under the radical symbol it must return only one result, so a non-negative real root, called the principalnth root, is preferred rather than others; if the only real root is negative, as for the cube root of 8, again the real rootis considered the principal root. An unresolved root, especially one using the radical symbol, is often referred to as asurd[1] or a radical.[2] Any expression containing a radical, whether it is a square root, a cube root, or a higher root,is called a radical expression, and if it contains no transcendental functions or transcendental numbers it is called analgebraic expression.In calculus, roots are treated as special cases of exponentiation, where the exponent is a fraction:

    npx = x1/n

    Roots are particularly important in the theory of innite series; the root test determines the radius of convergence ofa power series. Nth roots can also be dened for complex numbers, and the complex roots of 1 (the roots of unity)play an important role in higher mathematics. Galois theory can be used to determine which algebraic numbers canbe expressed using roots, and to prove the Abel-Runi theorem, which states that a general polynomial equation ofdegree ve or higher cannot be solved using roots alone; this result is also known as the insolubility of the quintic.

    1.1 Etymology

    1.1.1 Origin of the root symbolThe origin of the root symbol is largely speculative. Some sources imply that the symbol was rst used by Arabicmathematicians. One of those mathematicians was Ab al-Hasan ibn Al al-Qalasd (14211486). Legend has itthat it was taken from the Arabic letter " " (m, /dim/), which is the rst letter in the Arabic word " " (jadhir,meaning root"; /dir/).[3] However, many scholars, including Leonhard Euler,[4] believe it originates from theletter r, the rst letter of the Latin word "radix" (meaning root), referring to the same mathematical operation.The symbol was rst seen in print without the vinculum (the horizontal bar over the numbers inside the radicalsymbol) in the year 1525 in Die Coss by Christo Rudol, a German mathematician.The Unicode and HTML character codes for the radical symbols are:

    1.1.2 Etymology of surdThe term surd traces back to al-Khwrizm (c. 825), who referred to rational and irrational numbers as audible andinaudible, respectively. This later led to the Arabic word " " (asamm, meaning deaf or dumb) for irrationalnumber 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.[5]

    1.2 HistoryMain articles: Square root History and Cube root History

  • 1.3. DEFINITION AND NOTATION 3

    1.3 Denition and notation

    +i

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    1 +1

    The four 4th roots of 1,none of which is real

    An nth root of a number x, where n is a positive integer, is any of the n real or complex numbers r whose nth poweris x:

    rn = x:

    Every positive real number x has a single positive nth root, called the principal nth root, which is written npx . Forn equal to 2 this is called the principal square root and the n is omitted. The nth root can also be represented usingexponentiation as x1/n.For even values of n, positive numbers also have a negative nth root, while negative numbers do not have a real nthroot. For odd values of n, every negative number x has a real negative nth root. For example, 2 has a real 5th root,5p2 = 1:148698354 : : : but 2 does not have any real 6th roots.Every non-zero number x, real or complex, has n dierent complex number nth roots including any positive or negativeroots. They are all distinct except in the case of x = 0, all of whose nth roots equal 0.The nth roots of almost all numbers (all integers except the nth powers, and all rationals except the quotients of two

  • 4 CHAPTER 1. NTH ROOT

    0

    +i

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    1 +1

    The three 3rd roots of 1,one of which is a negative real

    nth powers) are irrational. For example,

    p2 = 1:414213562 : : :

    All nth roots of integers, are algebraic numbers.

    1.3.1 Square rootsMa