Imaginary number

Last updated

... (repeats the pattern
from blue area)
i−3 = i
i−2 = −1
i−1 = −i
i0 = 1
i1 = i
i2 = −1
i3 = −i
i4 = 1
i5 = i
i6 = −1
in = im where m ≡ n mod 4

An imaginary number is a complex number that can be written as a real number multiplied by the imaginary unit i, [note 1] which is defined by its property i2 = −1. [1] [2] The square of an imaginary number bi is b2. For example, 5i is an imaginary number, and its square is −25. By definition, zero is considered to be both real and imaginary. [3]

Contents

Originally coined in the 17th century by René Descartes [4] as a derogatory term and regarded as fictitious or useless, the concept gained wide acceptance following the work of Leonhard Euler (in the 18th century) and Augustin-Louis Cauchy and Carl Friedrich Gauss (in the early 19th century).

An imaginary number bi can be added to a real number a to form a complex number of the form a + bi, where the real numbers a and b are called, respectively, the real part and the imaginary part of the complex number. [5] [note 2]

History

An illustration of the complex plane. The imaginary numbers are on the vertical coordinate axis. Complex conjugate picture.svg
An illustration of the complex plane. The imaginary numbers are on the vertical coordinate axis.

Although the Greek mathematician and engineer Hero of Alexandria is noted as the first to have conceived imaginary numbers, [6] [7] it was Rafael Bombelli who first set down the rules for multiplication of complex numbers in 1572. The concept had appeared in print earlier, such as in work by Gerolamo Cardano. At the time, imaginary numbers and negative numbers were poorly understood and were regarded by some as fictitious or useless much as zero once was. Many other mathematicians were slow to adopt the use of imaginary numbers, including René Descartes, who wrote about them in his La Géométrie in which the term imaginary was used and meant to be derogatory. [8] [9] The use of imaginary numbers was not widely accepted until the work of Leonhard Euler (1707–1783) and Carl Friedrich Gauss (1777–1855). The geometric significance of complex numbers as points in a plane was first described by Caspar Wessel (1745–1818). [10]

In 1843, William Rowan Hamilton extended the idea of an axis of imaginary numbers in the plane to a four-dimensional space of quaternion imaginaries in which three of the dimensions are analogous to the imaginary numbers in the complex field.

Geometric interpretation

90-degree rotations in the complex plane Rotations on the complex plane.svg
90-degree rotations in the complex plane

Geometrically, imaginary numbers are found on the vertical axis of the complex number plane, which allows them to be presented perpendicular to the real axis. One way of viewing imaginary numbers is to consider a standard number line positively increasing in magnitude to the right and negatively increasing in magnitude to the left. At 0 on the x-axis, a y-axis can be drawn with "positive" direction going up; "positive" imaginary numbers then increase in magnitude upwards, and "negative" imaginary numbers increase in magnitude downwards. This vertical axis is often called the "imaginary axis" and is denoted or .

In this representation, multiplication by –1 corresponds to a rotation of 180 degrees about the origin. Multiplication by i corresponds to a 90-degree rotation in the "positive" counterclockwise direction, and the equation i2 = −1 is interpreted as saying that, if we apply two 90-degree rotations about the origin, the net result is a single 180-degree rotation. Note that a 90-degree rotation in the "negative" (clockwise) direction also satisfies that interpretation, which reflects the fact that i also solves the equation x2 = −1. In general, multiplying by a complex number is the same as rotating around the origin by the complex number's argument, followed by a scaling by its magnitude.

Square roots of negative numbers

Care must be used when working with imaginary numbers that are expressed as the principal values of the square roots of negative numbers: [11]

That is sometimes written as:

The fallacy occurs as the equality fails when the variables are not suitably constrained. In that case, the equality fails to hold as the numbers are both negative, which can be demonstrated by:

where both x and y are non-negative real numbers.

See also

Notes

  1. j is usually used in engineering contexts where i has other meanings (such as electrical current)
  2. Both the real part and the imaginary part are defined as real numbers.

Related Research Articles

Absolute value Magnitude of a possibly negative number

In mathematics, the absolute value or modulus of a real number x, denoted |x|, is the non-negative value of x without regard to its sign. Namely, |x| = x if x is positive, and |x| = −x if x is negative, and |0| = 0. For example, the absolute value of 3 is 3, and the absolute value of −3 is also 3. The absolute value of a number may be thought of as its distance from zero.

Analytic geometry Study of geometry using a coordinate system

In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry.

Complex number Element of a number system in which –1 has a square root

In mathematics, a complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is a symbol called the imaginary unit, and satisfying the equation i2 = −1. Because no "real" number satisfies this equation, i was called an imaginary number by René Descartes. For the complex number a + bi, a is called the real part and b is called the imaginary part. The set of complex numbers is denoted by either of the symbols or C. Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers and are fundamental in many aspects of the scientific description of the natural world.

Cartesian coordinate system Coordinate system

A Cartesian coordinate system in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in the same unit of length. Each reference line is called a coordinate axis or just axis of the system, and the point where they meet is its origin, at ordered pair (0, 0). The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as signed distances from the origin.

Eulers formula Expression of the complex exponential in terms of sine and cosine

Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that for any real number x:

Polynomial In mathematics, sum of products of variables, power of variables, and coefficients

In mathematics, a polynomial is an expression consisting of variables and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponentiation of variables. An example of a polynomial of a single indeterminate x is x2 − 4x + 7. An example in three variables is x3 + 2xyz2yz + 1.

In algebra, a quadratic equation is any equation that can be rearranged in standard form as

Square root Number whose square is a given number

In mathematics, a square root of a number x is a number y such that y2 = x; in other words, a number y whose square (the result of multiplying the number by itself, or y ⋅ y) is x. For example, 4 and −4 are square roots of 16, because 42 = (−4)2 = 16. Every nonnegative real number x has a unique nonnegative square root, called the principal square root, which is denoted by where the symbol is called the radical sign or radix. For example, the principal square root of 9 is 3, which is denoted by because 32 = 3 ⋅ 3 = 9 and 3 is nonnegative. The term (or number) whose square root is being considered is known as the radicand. The radicand is the number or expression underneath the radical sign, in this case 9.

Imaginary unit Principal square root of −1

The imaginary unit or unit imaginary number is a solution to the quadratic equation x2 + 1 = 0. Although there is no real number with this property, i can be used to extend the real numbers to what are called complex numbers, using addition and multiplication. A simple example of the use of i in a complex number is 2 + 3i.

Complex plane Geometric representation of the complex numbers

In mathematics, the complex plane or z-plane is the plane associated with complex coordinate system, formed or established by the real axis and the perpendicular imaginary axis. It can be thought of as a modified Cartesian plane, with the real part of a complex number represented by a displacement along the x-axis, and the imaginary part by a displacement along the y-axis.

In mathematics, an nth root of a number x is a number r which, when raised to the power n, yields x:

In mathematics, certain kinds of mistaken proof are often exhibited, and sometimes collected, as illustrations of a concept called mathematical fallacy. There is a distinction between a simple mistake and a mathematical fallacy in a proof, in that a mistake in a proof leads to an invalid proof while in the best-known examples of mathematical fallacies there is some element of concealment or deception in the presentation of the proof.

In mathematics, an algebraic equation or polynomial equation is an equation of the form

Tetration Repeated or iterated exponentiation

In mathematics, tetration is an operation based on iterated, or repeated, exponentiation. It is the next hyperoperation after exponentiation, but before pentation. The word was coined by Reuben Louis Goodstein from tetra- (four) and iteration.

In linear algebra, a rotation matrix is a transformation matrix that is used to perform a rotation in Euclidean space. For example, using the convention below, the matrix

Mohrs circle Geometric civil engineering calculation technique

Mohr's circle is a two-dimensional graphical representation of the transformation law for the Cauchy stress tensor.

In algebra, a nested radical is a radical expression that contains (nests) another radical expression. Examples include

In abstract algebra, a bicomplex number is a pair (w, z) of complex numbers constructed by the Cayley–Dickson process that defines the bicomplex conjugate , and the product of two bicomplex numbers as

Sign (mathematics)

In mathematics, the concept of sign originates from the property that every real number is either positive, negative or zero. Depending on local conventions, zero is either considered as being neither a positive number, nor a negative number, or as belonging to both negative and positive numbers. Whenever not specifically mentioned, this article adheres to the first convention.

Argument (complex analysis)

In mathematics, the argument of a complex number z, denoted arg(z), is the angle between the positive real axis and the line joining the origin and z, represented as a point in the complex plane, shown as in Figure 1. It is a multi-valued function operating on the nonzero complex numbers. To define a single-valued function, the principal value of the argument is used. It is often chosen to be the unique value of the argument that lies within the interval (–π, π].

References

  1. Uno Ingard, K. (1988). "Chapter 2". Fundamentals of Waves and Oscillations. Cambridge University Press. p. 38. ISBN   0-521-33957-X.
  2. Weisstein, Eric W. "Imaginary Number". mathworld.wolfram.com. Retrieved 2020-08-10.
  3. Sinha, K.C. (2008). A Text Book of Mathematics Class XI (Second ed.). Rastogi Publications. p. 11.2. ISBN   978-81-7133-912-9.
  4. Giaquinta, Mariano; Modica, Giuseppe (2004). Mathematical Analysis: Approximation and Discrete Processes (illustrated ed.). Springer Science & Business Media. p. 121. ISBN   978-0-8176-4337-9. Extract of page 121
  5. Aufmann, Richard; Barker, Vernon C.; Nation, Richard (2009). College Algebra: Enhanced Edition (6th ed.). Cengage Learning. p. 66. ISBN   1-4390-4379-5.
  6. Hargittai, István (1992). Fivefold symmetry (2nd ed.). World Scientific. p. 153. ISBN   981-02-0600-3.
  7. Roy, Stephen Campbell (2007). Complex numbers: lattice simulation and zeta function applications. Horwood. p. 1. ISBN   1-904275-25-7.
  8. Descartes, René, Discourse de la Méthode ... (Leiden, (Netherlands): Jan Maire, 1637), appended book: La Géométrie, book three, p. 380. From page 380: "Au reste tant les vrayes racines que les fausses ne sont pas tousjours reelles; mais quelquefois seulement imaginaires; c'est a dire qu'on peut bien tousjours en imaginer autant que jay dit en chasque Equation; mais qu'il n'y a quelquefois aucune quantité, qui corresponde a celles qu'on imagine, comme encore qu'on en puisse imaginer trois en celle cy, x3 – 6xx + 13x – 10 = 0, il n'y en a toutefois qu'une reelle, qui est 2, & pour les deux autres, quoy qu'on les augmente, ou diminue, ou multiplie en la façon que je viens d'expliquer, on ne sçauroit les rendre autres qu'imaginaires." (Moreover, the true roots as well as the false [roots] are not always real; but sometimes only imaginary [quantities]; that is to say, one can always imagine as many of them in each equation as I said; but there is sometimes no quantity that corresponds to what one imagines, just as although one can imagine three of them in this [equation], x3 – 6xx + 13x – 10 = 0, only one of them however is real, which is 2, and regarding the other two, although one increase, or decrease, or multiply them in the manner that I just explained, one would not be able to make them other than imaginary [quantities].)
  9. Martinez, Albert A. (2006), Negative Math: How Mathematical Rules Can Be Positively Bent, Princeton: Princeton University Press, ISBN   0-691-12309-8 , discusses ambiguities of meaning in imaginary expressions in historical context.
  10. Rozenfeld, Boris Abramovich (1988). "Chapter 10". A History of Non-Euclidean Geometry: Evolution of the Concept of a Geometric Space. Springer. p. 382. ISBN   0-387-96458-4.
  11. Nahin, Paul J. (2010). An Imaginary Tale: The Story of "i" [the square root of minus one]. Princeton University Press. p. 12. ISBN   978-1-4008-3029-9. Extract of page 12

Bibliography