Imaginary Unit

Imaginary unitFrom Wikipedia, the free encyclopediaContents1 Imaginary unit 11.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 i and i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Proper use. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4.1 Square roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4.2 Multiplication and division . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.4.3 Powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.4.4 Factorial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.4.5 Other operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.5 Alternative notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.6 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.10 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.11 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 Nested intervals 102.1 Higher dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 Nested radical 123.1 Denesting nested radicals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.1.1 Some identities of Ramanujan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.1.2 Landaus algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.2 In trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.3 In the solution of the cubic equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.4 Innitely nested radicals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.4.1 Square roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.4.2 Cube roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17iii CONTENTS3.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.6.1 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.7 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 183.7.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.7.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.7.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18Chapter 1Imaginary unit0 +i i 1+1 i in the complex or cartesian plane. Real numbers lie on the horizontal axis, and imaginary numbers lie on the vertical axisThe term imaginary unit or unit imaginary number refers to a solution to the equation x2= 1. By convention,the solution is usually denotedi. Since there is no real number with this property, it extends the real numbers,and under the assumption that the familiar properties of addition and multiplication (namely closure, associativity,12 CHAPTER 1. IMAGINARY UNITcommutativity and distributivity) continue to hold for this extension, the complex numbers are generated by includingit.Imaginary numbers are an important mathematical concept, which extends the real number system to the complexnumber system , which in turn provides at least one root for every nonconstant polynomial P(x). (See Algebraicclosure and Fundamental theorem of algebra.) The term "imaginary" is used because there is no real number havinga negative square.There are two complex square roots of 1, namely i and i, just as there are two complex square roots of every realnumber other than zero, which has one double square root.In contexts where i is ambiguous or problematic, j or the Greek is sometimes used (see Alternative notations).In the disciplines of electrical engineering and control systems engineering, the imaginary unit is often denoted by jinstead of i, because i is commonly used to denote electric current.For the history of the imaginary unit, see Complex number History.1.1 DenitionThe imaginary number i is dened solely by the property that its square is 1:i2= 1 .With i dened this way, it follows directly from algebra that i and i are both square roots of 1.Although the construction is called imaginary, and although the concept of an imaginary number may be intuitivelymore dicult to grasp than that of a real number, the construction is perfectly valid from a mathematical standpoint.Real number operations can be extended to imaginary and complex numbers by treating i as an unknown quantitywhile manipulating an expression, and then using the denition to replace any occurrence of i2with 1. Higherintegral powers of i can also be replaced with i, 1, i, or 1:i3= i2i = (1)i = ii4= i3i = (i)i = (i2) = (1) = 1i5= i4i = (1)i = iSimilarly, as with any non-zero real number:i0= i11= i1i1= i11i= i1i=ii= 1As a complex number, i is represented in rectangular form as 0 + i, having a unit imaginary component and no realcomponent (i.e., the real component is zero).In polar form, i is represented as 1 ei/2, having an absolute value (ormagnitude) of 1 and an argument (or angle) of /2. In the complex plane (also known as the Cartesian plane), i is thepoint located one unit from the origin along the imaginary axis (which is at a right angle to the real axis).1.2 i and iBeing a quadratic polynomial with no multiple root, the dening equation x2= 1 has two distinct solutions, whichare equally valid and which happen to be additive and multiplicative inverses of each other. More precisely, once asolution i of the equation has been xed, the value i, which is distinct from i, is also a solution. Since the equationis the only denition of i, it appears that the denition is ambiguous (more precisely, not well-dened). However, noambiguity results as long as one or other of the solutions is chosen and labelled as i, with the other one then beinglabelled as i. This is because, although i and i are not quantitatively equivalent (they are negatives of each other),there is no algebraic dierence between i and i. Both imaginary numbers have equal claim to being the numberwhose square is 1. If all mathematical textbooks and published literature referring to imaginary or complex numbers1.3. PROPER USE 3were rewritten with i replacing every occurrence of +i (and therefore every occurrence of i replaced by (i) = +i),all facts and theorems would continue to be equivalently valid. The distinction between the two roots x of x2+ 1 = 0with one of them labelled with a minus sign is purely a notational relic; neither root can be said to be more primaryor fundamental than the other, and neither of them is positive or negative.The issue can be a subtle one. The most precise explanation is to say that although the complex eld, dened asR[x]/(x2+ 1), (see complex number) is unique up to isomorphism, it is not unique up to a unique isomorphism there are exactly 2 eld automorphisms of R[x]/(x2+ 1) which keep each real number xed: the identity and theautomorphism sending x to x. See also Complex conjugate and Galois group.A similar issue arises if the complex numbers are interpreted as 2 2 real matrices (see matrix representation ofcomplex numbers), because then bothX=(0 11 0)and X=(0 11 0)are solutions to the matrix equationX2= I= (1 00 1)=(1 00 1).In this case, the ambiguity results from the geometric choice of which direction around the unit circle is positiverotation. A more precise explanation is to say that the automorphism group of the special orthogonal group SO (2,R) has exactly 2 elements the identity and the automorphism which exchanges CW (clockwise) and CCW(counter-clockwise) rotations. See orthogonal group.All these ambiguities can be solved by adopting a more rigorous denition of complex number, and explicitly choosingone of the solutions to the equation to be the imaginary unit. For example, the ordered pair (0, 1), in the usualconstruction of the complex numbers with two-dimensional vectors.1.3 Proper useThe imaginary unit is sometimes written 1 in advanced mathematics contexts (as well as in less advanced populartexts). However, great care needs to be taken when manipulating formulas involving radicals. The notation is reservedeither for the principal square root function, which is only dened for real x 0, or for the principal branch of thecomplex square root function. Attempting to apply the calculation rules of the principal (real) square root functionto manipulate the principal branch of the complex square root function will produce false results:1 = i i =1 1 =(1) (1) =1 = 1 (incorrect).Attempting to correct the calculation by specifying both the positive and negative roots only produces ambiguousresults:1 = i i = 1 1 = (1) (1) = 1 = 1 (ambiguous).Similarly:1i=11=11=11=1 = i (incorrect).The calculation rulesa b =a band4 CHAPTER 1. IMAGINARY UNITab=abare only valid for real, non-negative values of a and b.These problems are avoided by writing and manipulating i7, rather than expressions like 7. For a more thoroughdiscussion, see Square root and Branch point.1.4 Properties1.4.1 Square roots0 +i i 1+1 The two square roots of i in the complex planeThe square root of i can be expressed as either of two complex numbers[nb 1]i = (22+22i)= 22(1 + i).1.4. PROPERTIES 5Indeed, squaring the right-hand side gives(22(1 + i))2=(22)2(1 + i)2=12(1 + 2i + i2)=12(1 + 2i 1)= i.This result can also be derived with Eulers formulaeix= cos(x) + i sin(x)by substituting x = /2, givingei(/2)= cos(/2) + i sin(/2) = 0 + i1 = i.Taking the square root of both sides givesi = ei(/4),which, through application of Eulers formula to x = /4, givesi = (cos(/4) + i sin(/4))=12+i2=1 + i2= 22(1 + i).Similarly, the square root of i can be expressed as either of two complex numbers using Eulers formula:eix= cos(x) + i sin(x)by substituting x = 3/2, givingei(3/2)= cos(3/2) + i sin(3/2) = 0 i1 = i.Taking the square root of both sides givesi = ei(3/4),which, through application of Eulers formula to x = 3/4, givesi = (cos(3/4) + i sin(3/4))= 12+ i12= 1 + i2= 22(i 1).6 CHAPTER 1. IMAGINARY UNITMultiplying the square root of i by i also gives:i = (i) (12(1 + i))= 12(1i + i2)= 22(i 1)1.4.2 Multiplication and divisionMultiplying a complex number by i gives:i (a + bi) = ai + bi2= b + ai.(This is equivalent to a 90 counter-clockwise rotation of a vector about the origin in the complex plane.)Dividing by i is equivalent to multiplying by the reciprocal of i:1i=1i ii=ii2=i1= i.Using this identity to generalize division by i to all complex numbers gives:a + bii= i (a + bi) = ai bi2= b ai.(This is equivalent to a 90 clockwise rotation of a vector about the origin in the complex plane.)1.4.3 PowersThe powers of i repeat in a cycle expressible with the following pattern, where n is any integer:i4n= 1i4n+1= ii4n+2= 1i4n+3= i.This leads to the conclusion thatin= in mod4where mod represents the modulo operation. Equivalently:in= cos(n/2) + i sin(n/2)1.4. PROPERTIES 7i raised to the power of iMaking use of Eulers formula, iiisii=(ei(/2+2k))i= ei2(/2+2k)= e(/2+2k)where k Z , the set of integers.The principal value (for k = 0) is e/2 or approximately 0.207879576...[1]1.4.4 FactorialThe factorial of the imaginary unit i is most often given in terms of the gamma function evaluated at 1 + i:i! = (1 + i) 0.4980 0.1549i.Also,|i!| =sinh [2]1.4.5 Other operationsMany mathematical operations that can be carried out with real numbers can also be carried out with i, such asexponentiation, roots, logarithms, and trigonometric functions. However, it should be noted that all of the followingfunctions are complex multi-valued functions, and it should be clearly stated which branch of the Riemann surfacethe function is dened on in practice. Listed below are results for the most commonly chosen branch.A number raised to the ni power is:xni= cos(ln xn) + i sin(ln xn).The nith root of a number is:nix = cos(lnnx) i sin(lnnx).The imaginary-base logarithm of a number is:logi(x) =2 ln xi.As with any complex logarithm, the log base i is not uniquely dened.The cosine of i is a real number:cos(i) = cosh(1) =e + 1/e2=e2+ 12e 1.54308064....And the sine of i is purely imaginary:sin(i) = i sinh(1) =e 1/e2i =e212ei 1.17520119 i....8 CHAPTER 1. IMAGINARY UNIT1.5 Alternative notationsIn electrical engineering and related elds, the imaginary unit is often denoted by j to avoid confusion withelectric current as a function of time, traditionally denoted by i(t) or just i. The Python programming languagealso uses j to mark the imaginary part of a complex number. MATLAB associates both i and j with theimaginary unit, although 1i or 1j is preferable, for speed and improved robustness.[3]Some texts use the Greek letter iota () for the imaginary unit, to avoid confusion, especially with index andsubscripts.Each of i, j, and k is an imaginary unit in the quaternions. In bivectors and biquaternions an additional imaginaryunit h is used.1.6 MatricesWhen 2 2 real matrices m are used for a source, and the number one (1) is identied with the identity matrix, andminus one (1) with the negative of the identity matrix, then there are many solutions to m2= 1. In fact, there aremany solutions to m2= +1 and m2= 0 also. Any such m can be taken as a basis vector, along with 1, to form a planaralgebra.1.7 See alsoComplex planeImaginary numberMultiplicity (mathematics)Root of unityUnit complex number1.8 Notes[1] To nd such a number, one can solve the equation(x + iy)2= iwhere x and y are real parameters to be determined, or equivalentlyx2+ 2ixy y2= i.Because the real and imaginary parts are always separate, we regroup the terms:x2 y2+ 2ixy = 0 + iand by equating coecients, real part and real coecient of imaginary part separately, we get a system of two equations:x2 y2= 02xy = 1.Substituting y = 1/2x into the rst equation, we getx2 1/4x2= 0x2= 1/4x24x4= 1Because x is a real number, this equation has two real solutions for x: x = 1/2 and x = 1/2. Substituting either of theseresults into the equation 2xy = 1 in turn, we will get the corresponding result for y. Thus, the square roots of i are thenumbers 1/2 + i/2 and 1/2 i/2. (University of Toronto Mathematics Network: What is the square root of i? URLretrieved March 26, 2007.)1.9. REFERENCES 91.9 References[1] The Penguin Dictionary of Curious and Interesting Numbers by David Wells, Page 26.[2] "abs(i!)", WolframAlpha.[3] MATLAB Product Documentation.1.10 Further readingNahin, Paul J. (1998). An Imaginary Tale:The Story of 1. Chichester: Princeton University Press. ISBN0-691-02795-1.1.11 External linksEulers work on Imaginary Roots of Polynomials at ConvergenceChapter 2Nested intervals0000In mathematics, a sequence of nested intervals is understood as a collection of sets of real numbersInsuch that each set In is an interval of the real line, for n = 1, 2, 3, ..., and that furtherIn is a subset of Infor all n. In other words, the intervals diminish, with the left-hand end moving only towards the right, and the right-hand end only to the left.The main question to be posed is the nature of the intersection of all the In. Without any further information, all thatcan be said is that the intersection J of all the In, i.e. the set of all points common to the intervals, is either the emptyset, a point, or some interval.The possibility of an empty intersection can be illustrated by the intersection when In is the open interval(0, 2n).Here the intersection is empty, because no number x is both greater than 0 and less than every fraction 2n.The situation is dierent for closed intervals. The nested intervals theoremstates that if each In is a closed and boundedinterval, sayIn = [an, bn]with102.1. HIGHER DIMENSIONS 11an bnthen under the assumption of nesting, the intersection of the In is not empty. It may be a singleton set {c}, or anotherclosed interval [a, b]. More explicitly, the requirement of nesting means thatan an andbn bn .Moreover, if the length of the intervals converges to 0, then the intersection of the In is a singleton.One can consider the complement of each interval, written as(, an) (bn, ) . By De Morgans laws, thecomplement of the intersection is a union of two disjoint open sets. By the connectedness of the real line there mustbe something between them. This shows that the intersection of (even an uncountable number of) nested, closed, andbounded intervals is nonempty.2.1 Higher dimensionsIn two dimensions there is a similar result: nested closed disks in the plane must have a common intersection. Thisresult was shown by Hermann Weyl to classify the singular behaviour of certain dierential equations.2.2 See alsoBisectionCantors Intersection Theorem2.3 ReferencesFridy, J. A. (2000), 3.3 The Nested Intervals Theorem, Introductory Analysis: The Theory of Calculus,Academic Press, p. 29, ISBN 9780122676550.Shilov, Georgi E. (2012), 1.8 The Principle of Nested Intervals, Elementary Real and Complex Analysis,Dover Books on Mathematics, Courier Dover Publications, pp. 2122, ISBN 9780486135007.Sohrab, Houshang H. (2003), Theorem 2.1.5 (Nested Intervals Theorem)", Basic Real Analysis, Springer, p.45, ISBN 9780817642112.Chapter 3Nested radicalIn algebra, a nested radical is a radical expression (one containing a square root sign, cube root sign, etc.) thatcontains (nests) another radical expression. Examples include:5 25which arises in discussing the regular pentagon;5 + 26 ,or more complicated ones such as:32 +3 +34 .3.1 Denesting nested radicalsSome nested radicals can be rewritten in a form that is not nested. For example,3 + 22 = 1 +2 ,5 + 26 =2 +3,332 1 =1 32 +3439.Rewriting a nested radical in this way is called denesting. This process is generally considered a dicult problem,although a special class of nested radical can be denested by assuming it denests into a sum of two surds:a bc =d e.Squaring both sides of this equation yields:a bc = d + e 2de.123.1. DENESTING NESTED RADICALS 13This can be solved by nding two numbers such that their sum is equal to a and their product is b2c/4, or by equatingcoecients of like termssetting rational and irrational parts on both sides of the equation equal to each other. Thesolutions for e and d can be obtained by rst equating the rational parts:a = d + e,which givesd = a e,e = a d.For the irrational parts note thatbc = 2de,and squaring both sides yieldsb2c = 4de.By plugging in a d for e one obtainsb2c = 4(a d)d = 4ad 4d2.Rearranging terms will give a quadratic equation which can be solved for d using the quadratic formula:4d24ad + b2c = 0,d =a a2b2c2.Since a = d+e, the solution e is the algebraic conjugate of d. If we setd =a +a2b2c2,thene =a a2b2c2.However, this approach works for nested radicals of the form a bc if and only if a2b2c is a rationalnumber, in which case the nested radical can be denested into a sum of surds.In some cases, higher-power radicals may be needed to denest the nested radical.3.1.1 Some identities of RamanujanSrinivasa Ramanujan demonstrated a number of curious identities involving denesting of radicals. Among them arethe following:[1]14 CHAPTER 3. NESTED RADICAL43 + 2453 245=45 + 145 1=12(3 +45 +5 +4125),328 327 =13(398 328 1),353255275=5125+5325 5925,332 1 =319 329+349. [2]Other odd-looking radicals inspired by Ramanujan include:449 + 206 +449 206 = 23,3(2 +3)(5 6)+ 3(23 + 32)=10 13 565 +6.3.1.2 Landaus algorithmIn 1989 Susan Landau introduced the rst algorithm for deciding which nested radicals can be denested.[3] Earlieralgorithms worked in some cases but not others.3.2 In trigonometryMain article: Exact trigonometric constantsIn trigonometry, the sines and cosines of many angles can be expressed in terms of nested radicals. For example,sin60= sin 3=116[2(1 3)5 +5 +2(5 1)(3 + 1)]andsin24= sin 7.5=122 2 +3 =122 1+32.3.3 In the solution of the cubic equationNested radicals appear in the algebraic solution of the cubic equation. Any cubic equation can be written in simpliedform without a quadratic term, asx3+ px + q= 0,whose general solution for one of the roots is3.4. INFINITELY NESTED RADICALS 15x =3q2+q24+p327+3q2 q24+p327;here the rst cube root is dened to be any specic cube root of the radicand, and the second cube root is dened tobe the complex conjugate of the rst one. The nested radicals in this solution cannot in general be simplied unlessthe cubic equation has at least one rational solution. Indeed, if the cubic has three irrational but real solutions, wehave the casus irreducibilis, in which all three real solutions are written in terms of cube roots of complex numbers.On the other hand, consider the equationx37x + 6 = 0,which has the rational solutions 1, 2, and 3. The general solution formula given above gives the solutionsx =33 +103i9+33 103i9.For any given choice of cube root and its conjugate, this contains nested radicals involving complex numbers, yet itis reducible (even though not obviously so) to one of the solutions 1, 2, or 3.3.4 Innitely nested radicals3.4.1 Square rootsUnder certain conditions innitely nested square roots such asx =2 +2 +2 +2 + represent rational numbers. This rational number can be found by realizing that x also appears under the radical sign,which gives the equationx =2 + x.If we solve this equation, we nd that x = 2 (the second solution x = 1 doesn't apply, under the convention that thepositive square root is meant). This approach can also be used to show that generally, if n > 0, then:n +n +n +n + =12(1 +1 + 4n)and is the real root of the equation x2 x n = 0. For n = 1, this root is the golden ratio , approximately equal to1.618. The same procedure also works to get thatn n n n =12(1 +1 + 4n).and is the real root of the equation x2+ x n = 0. For n = 1, this root is the reciprocal of the golden ratio , whichis equal to 1. This method will give a rational x value for all values of n such that16 CHAPTER 3. NESTED RADICALn = x2+ x.Ramanujan posed this problem to the 'Journal of Indian Mathematical Society':? =1 + 21 + 31 + .This can be solved by noting a more general formulation:? =ax + (n + a)2+ xa(x + n) + (n + a)2+ (x + n) Setting this to F(x) and squaring both sides gives us:F(x)2= ax + (n + a)2+ xa(x + n) + (n + a)2+ (x + n) Which can be simplied to:F(x)2= ax + (n + a)2+ xF(x + n)It can then be shown that:F(x) = x + n + aSo, setting a =0, n = 1, and x = 2:3 =1 + 21 + 31 + .Ramanujan stated this radical in his lost notebook

5 +

5 +

5 5 +5 +5 +5 =2 +5 +15 652The repeating pattern of the signs is (+, +, , +)In Vites expression for piVites formula for pi, the ratio of a circles circumference to its diameter, is2=222 +222 +2 +22 .3.5. SEE ALSO 173.4.2 Cube rootsIn certain cases, innitely nested cube roots such asx =36 +36 +36 +36 + can represent rational numbers as well. Again, by realizing that the whole expression appears inside itself, we are leftwith the equationx =36 + x.If we solve this equation, we nd that x = 2. More generally, we nd that3n +3n +3n +3n + is the real root of the equation x3 x n = 0 for all n > 0. For n = 1, this root is the plastic number , approximatelyequal to 1.3247.The same procedure also works to get3n 3n 3n 3n as the real root of the equation x3+ x n = 0 for all n and x where n > 0 and |x| 1.3.5 See alsoSum of radicalsSpiral of Theodorus3.6 References[1] Landau, Susan. A note on 'Zippel Denesting'". CiteSeerX: 10 .1 .1 .35 .5512.[2] Landau, Susan. RADICALS AND UNITS IN RAMANUJANS WORK (POSTSCRIPT).[3] Landau, Susan (1992). Simplication of Nested Radicals. Journal of Computation (SIAM) 21: 85110. doi:10.1109/SFCS.1989.63496.CiteSeerX: 10 .1 .1 .34 .2003.3.6.1 Further readingLandau, Susan (1994). How to Tangle with a Nested Radical. Mathematical Intelligencer16: 4955.doi:10.1007/bf03024284.Decreasing the Nesting Depth of Expressions Involving Square RootsSimplifying Square Roots of Square RootsWeisstein, Eric W., Square Root, MathWorld.Weisstein, Eric W., Nested Radical, MathWorld.18 CHAPTER 3. NESTED RADICAL3.7 Text and image sources, contributors, and licenses3.7.1 Text Imaginary unit Source: https://en.wikipedia.org/wiki/Imaginary_unit?oldid=674755584 Contributors: AxelBoldt, Michael Hardy, Pnm,Delirium, Theresa knott, Andres, Mxn, Agtx, Revolver, Charles Matthews, Nohat, Dysprosia, Furrykef, SirJective, Fredrik, Bkell, Prime-Fan, Lzur, Jleedev, Alan Liefting, Giftlite, Gene Ward Smith, Philwelch, Lethe, Fropu, Guanaco, Nayuki, Noe, Bob.v.R, Gauss, Si-moneau, TreyHarris, Mschlindwein, Gazpacho, Mormegil, Ardonik, Paul August, Bender235, Andrejj, Plugwash, Rgdboer, EmilJ, Vi-ames, Cje~enwiki, Rbj, DG~enwiki, Haham hanuka, Ricky81682, RoySmith, Burn, EvenT, Dirac1933, MIT Trekkie, Algocu, OlegAlexandrov, Joriki, Mindmatrix, Justinlebar, LeonWhite, Lkjhgfdsa, MFH, Wurzel~enwiki, Philbarker, Bobmilkman, Sj, Bob A, Volfy,Antimatt, RobertG, Mathbot, Intersoa, Born2cycle, Kri, Chobot, Nylex, DVdm, Algebraist, EamonnPKeane, UkPaolo, Wavelength,PiAndWhippedCream, Prometheus235, RussBot, Kauner, Rick Norwood, Trovatore, Amakuha, Kyle Barbour, Light current, ArthurRubin, StealthFox, Gesslein, Ghazer~enwiki, Finell, Matikkapoika~enwiki, That Guy, From That Show!, Eykanal, Dash77, SmackBot,PizzaMargherita, DTM, Skizzik, Silly rabbit, Octahedron80, Bob K, Hgrosser, Modest Genius, Can't sleep, clown will eat me, Tamfang,Taggart Transcontinental, Nmnogueira, Zchenyu, Lambiam, Loadmaster, Xiaphias, Mets501, Elb2000, JDAWiseman, Macwiki, Quaeler,Chris53516, Majora4, Phoenixrod, Courcelles, Bstepp99, CmdrObot, Wafulz, FilipeS, Yaris678, Cydebot, Benzi455, KarolS, Dadofsam,Cario24, Vinyanov, Janlo, Mungomba, Ati7, Duncan McB, AntiVandalBot, Majorly, Luna Santin, Edokter, JAnDbot, Carmicheal99,Andonic, Fourchannel, Penubag, Magioladitis, Vanish2, JamesBWatson, SHCarter, Usien6, Sullivan.t.j, Rajpaj, Gwern, MartinBot, The-mania, Paulnwatts, M samadi, Prokoev2, Icefall5, Gene Ray (Cubic), Pope Nigel the porter, KylieTastic, Adamd1008, 28bytes, Je G.,JohnBlackburne, Fxrbds, Thurth, Philip Trueman, Reagar, SveinHarris, Anonymous Dissident, Lou.weird, Maxno, Synthebot, Dmcq,MaCRoEco, Jorge C.Al, SieBot, X-Fi6, This, that and the other, MikeGogulski, Fan Railer, Essap, Masgatotkaca, 888Xristos, Vhomet,Classicalmasta, Drjjgonzalez, Nic bor, Kortaggio, SallyForth123, Twelvepack, ClueBot, Maymay, Alexbot, Frozen4322, Bentu, Dark-icebot, MystBot, DominoSonic, Kbdankbot, Addbot, Jkasd, Fgnievinski, Ronhjones, SamatBot, Tide rolls, Yobot, AnomieBOT, Materi-alscientist, Beerockxs, ArthurBot, Xqbot, Sionus, The Evil IP address, Isheden, Jeremymwest, Phillofantiock, Shadowjams, Datakid1100,X7q, Majopius, Prosaicpat, William915, DrilBot, Goalsonly23, 10metreh, Number Googol, , Robo Cop, Willmea, Konstantin Pest, Duo-duoduo, Aa42john, Qaedtgujol, Netheril96, K6ka, JSquish, 1e+a2e, StringTheory11, Quondum, Hunocsi, ChuispastonBot, ClueBot NG,LutherVinci, Jpenedones, Helpful Pixie Bot, Theoldsparkle, HMman, B3stb33stm4n, SarahLZ, SergeantHippyZombie, ChrisGualtieri,Collingwood, Pokajanje, Thirteenthreefourteen, Ambyjkl, SkateTier, Originalmiles, BrianPansky, Loraof, Vmwwiikkiivm and Anony-mous: 212 Nested intervals Source: https://en.wikipedia.org/wiki/Nested_intervals?oldid=631120148 Contributors: Zundark, Charles Matthews,Giftlite, EmilJ, HannsEwald, SmackBot, Hydrogen Iodide, JCSantos, Radagast83, Jim.belk, Vanisaac, David Eppstein, Myrkkyhammas,VolkovBot, Geometry guy, Le Pied-bot~enwiki, Thehotelambush, Addbot, Erik9bot, ZroBot, ChuispastonBot, Brad7777, DeathOfBal-ance, SeriouslySmart and Anonymous: 6 Nested radical Source: https://en.wikipedia.org/wiki/Nested_radical?oldid=662336582 Contributors: Michael Hardy, Silversh, CharlesMatthews, Joy, Phil Boswell, Tobias Bergemann, TheMaestro, Tomruen, Lemontea, Blotwell, Woohookitty, Rjwilmsi, KFP, Tetracube,Mets501, Cryptic C62, CBM, Doctormatt, Us38, Futurebird, David Eppstein, Warut, VolkovBot, Pasixxxx, Anonymous Dissident,Mike4ty4, SieBot, Phe-bot, T-rithy, Addbot, AnomieBOT, Twri, LilHelpa, Isheden, Sspawarwiki, E-sinhe, Mikhail Ryazanov, Ts123ize,Gfhjhf, Vahid alpha, Loraof and Anonymous: 273.7.2 Images File:Illustration_nested_intervals.svg Source: https://upload.wikimedia.org/wikipedia/commons/4/48/Illustration_nested_intervals.svgLicense: CC BY-SA 3.0 Contributors: Own work Original artist: Stephan Kulla (User:Stephan Kulla) File:Imaginary2Root.svgSource: https://upload.wikimedia.org/wikipedia/commons/3/32/Imaginary2Root.svgLicense: CCBY-SA3.0 Contributors: Own work Original artist: Loadmaster (David R. 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