Imaginary unit

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i in the complex or Cartesian plane. Real numbers lie on the horizontal axis, and imaginary numbers lie on the vertical axis. ImaginaryUnit5.svg
i in the complex or Cartesian plane. Real numbers lie on the horizontal axis, and imaginary numbers lie on the vertical axis.

The imaginary unit or unit imaginary number (i) 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.

Contents

Imaginary numbers are an important mathematical concept, which extend the real number system to the complex number system , in which at least one root for every nonconstant polynomial exists (see Algebraic closure and Fundamental theorem of algebra). Here, the term "imaginary" is used because there is no real number having a negative square.

There are two complex square roots of −1, namely i and i, just as there are two complex square roots of every real number other than zero (which has one double square root).

In contexts in which use of the letter i is ambiguous or problematic, the letter j or the Greek ι is sometimes used instead. [lower-alpha 1] For example, in electrical engineering and control systems engineering, the imaginary unit is normally denoted by j instead of i, because i is commonly used to denote electric current.

Definition

The powers of i
return cyclic values:
... (repeats the pattern
from bold 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
... (repeats the pattern
from the bold blue area)

The imaginary number i is defined solely by the property that its square is −1:

With i defined 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 intuitively more difficult 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 quantity while manipulating an expression (and using the definition to replace any occurrence of i2 with −1). Higher integral powers of i can also be replaced with i, 1, i, or −1:

or, equivalently,

Similarly, as with any non-zero real number:

As a complex number, i is represented in rectangular form as 0 + 1i, with a zero real component and a unit imaginary component. In polar form, i is represented as 1⋅e/2 (or just e/2), with an absolute value (or magnitude) of 1 and an argument (or angle) of π/2. In the complex plane (also known as the Argand plane), which is a special interpretation of a Cartesian plane, i is the point located one unit from the origin along the imaginary axis (which is orthogonal to the real axis).

i vs. −i

Being a quadratic polynomial with no multiple root, the defining equation x2 = −1 has two distinct solutions, which are equally valid and which happen to be additive and multiplicative inverses of each other. Once a solution i of the equation has been fixed, the value i, which is distinct from i, is also a solution. Since the equation is the only definition of i, it appears that the definition is ambiguous (more precisely, not well-defined). However, no ambiguity will result as long as one or other of the solutions is chosen and labelled as "i", with the other one then being labelled as i. [3] After all, although i and +i are not quantitatively equivalent (they are negatives of each other), there is no algebraic difference between +i and i, as both imaginary numbers have equal claim to being the number whose square is −1.

In fact, if all mathematical textbooks and published literature referring to imaginary or complex numbers were to be rewritten with i replacing every occurrence of +i (and therefore every occurrence of i replaced by −(−i) = +i), all facts and theorems would remain valid. The distinction between the two roots x of x2 + 1 = 0, with one of them labelled with a minus sign, is purely a notational relic; neither root can be said to be more primary or fundamental than the other, and neither of them is "positive" or "negative". [4]

The issue can be a subtle one: The most precise explanation is to say that although the complex field, defined as ℝ[x]/(x2 + 1) (see complex number), is unique up to isomorphism, it is not unique up to a unique isomorphism: There are exactly two field automorphisms of ℝ[x]/(x2 + 1) which keep each real number fixed: The identity and the automorphism sending x to x. For more, see complex conjugate and Galois group.

Matrices

( x, y ) is confined by hyperbola xy = -1 for an imaginary unit matrix. Dalveida negativa.jpg
( x, y ) is confined by hyperbola xy = –1 for an imaginary unit matrix.

A similar issue arises if the complex numbers are interpreted as 2 × 2 real matrices (see matrix representation of complex numbers), because then both

    and    

would be solutions to the matrix equation

In this case, the ambiguity results from the geometric choice of which "direction" around the unit circle is "positive" rotation. A more precise explanation is to say that the automorphism group of the special orthogonal group SO(2, ) has exactly two elements: The identity and the automorphism which exchanges "CW" (clockwise) and "CCW" (counter-clockwise) rotations. For more, see orthogonal group.

All these ambiguities can be solved by adopting a more rigorous definition of complex number, and by explicitly choosing one of the solutions to the equation to be the imaginary unit. For example, the ordered pair (0, 1), in the usual construction of the complex numbers with two-dimensional vectors.

Consider the matrix equation Here, z2 + xy = –1, so the product xy is negative because xy = –(1 + z2), thus the point (x, y) lies in quadrant II or IV. Furthermore,

so (x, y) is bounded by the hyperbola xy = –1.

Proper use

The imaginary unit is sometimes written −1  in advanced mathematics contexts [3] (as well as in less advanced popular texts). However, great care needs to be taken when manipulating formulas involving radicals. The radical sign notation is reserved either for the principal square root function, which is only defined for real x ≥ 0, or for the principal branch of the complex square root function. Attempting to apply the calculation rules of the principal (real) square root function to manipulate the principal branch of the complex square root function can produce false results: [5]

Similarly:

The calculation rules

and

are only valid for real, positive values of a and b. [6] [7] [8]

These problems can be avoided by writing and manipulating expressions like i7 , rather than −7 . For a more thorough discussion, see square root and branch point.

Properties

Square roots

The two square roots of i in the complex plane Imaginary2Root.svg
The two square roots of i in the complex plane
The three cube roots of i in the complex plane Imaginary3Root.svg
The three cube roots of i in the complex plane

Just like all nonzero complex numbers, i has two square roots: they are [lower-alpha 2]

Indeed, squaring both expressions yields:

Using the radical sign for the principal square root, we get:

Cube roots

The three cube roots of i are:

and

Similar to all the roots of 1, all the roots of i are the vertices of regular polygons, which are inscribed within the unit circle in the complex plane.

Multiplication and division

Multiplying a complex number by i gives:

(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:

Using this identity to generalize division by i to all complex numbers gives:

(This is equivalent to a 90° clockwise rotation of a vector about the origin in the complex plane.)

Powers

The powers of i repeat in a cycle expressible with the following pattern, where n is any integer:

This leads to the conclusion that

where mod represents the modulo operation. Equivalently:

i raised to the power of i

Making use of Euler's formula, ii is

where k ∈ ℤ, the set of integers.

The principal value (for k = 0) is eπ/2, or approximately 0.207879576 . [10]

Factorial

The factorial of the imaginary unit i is most often given in terms of the gamma function evaluated at 1 + i:

Also,

[11]

Other operations

Many mathematical operations that can be carried out with real numbers can also be carried out with i, such as exponentiation, roots, logarithms, and trigonometric functions. All of the following functions are complex multi-valued functions, and it should be clearly stated which branch of the Riemann surface the function is defined on in practice. Listed below are results for the most commonly chosen branch.

A number raised to the ni power is:

The nith root of a number is:

The imaginary-base logarithm of a number is:

As with any complex logarithm, the log base i is not uniquely defined.

The cosine of i is a real number:

And the sine of i is purely imaginary:

History

See also

Notes

  1. Some texts[ which? ] use the Greek letter iota (ι) for the imaginary unit to avoid confusion, especially with indices and subscripts.

    In electrical engineering and related fields, the imaginary unit is normally denoted by j to avoid confusion with electric current as a function of time, which is conventionally represented by i(t) or just i . [1]

    The Python programming language also uses j to mark the imaginary part of a complex number.

    MATLAB associates both i and j with the imaginary unit, although the input 1i or 1j is preferable, for speed and more robust expression parsing. [2]

    In the quaternions, Each of i, j, and k is a distinct imaginary unit.

    In bivectors and biquaternions, an additional imaginary unit h or is used.
  2. To find such a number, one can solve the equation
    (x + iy)2 = i
    where x and y are real parameters to be determined, or equivalently
    x2 + 2ixyy2 = i.
    Because the real and imaginary parts are always separate, we regroup the terms:
    x2y2 + 2ixy = 0 + i
    and by equating coefficients, real part and real coefficient of imaginary part separately, we get a system of two equations:
    x2y2 = 0
    2xy = 1 .
    Substituting y = ½ x into the first equation, we get
    x2 −¼ x2 = 0
    x2 = ¼ x2
    4x4 = 1
    Because x is a real number, this equation has two real solutions for x: x = 1/2  and x = −1/2 . Substituting either of these results into the equation 2xy = 1 in turn, we will get the corresponding result for y. Thus, the square roots of i are the numbers 1/2  + i/2  and −1/2 i/2 . [9]

Related Research Articles

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.

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:

Logarithm Inverse of the exponential function, which maps products to sums

In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a given number x is the exponent to which another fixed number, the base b, must be raised, to produce that number x. In the simplest case, the logarithm counts the number of occurrences of the same factor in repeated multiplication; e.g., since 1000 = 10 × 10 × 10 = 103, the "logarithm base 10" of 1000 is 3, or log10(1000) = 3. The logarithm of x to baseb is denoted as logb(x), or without parentheses, logbx, or even without the explicit base, log x, when no confusion is possible, or when the base does not matter such as in big O notation.

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.

Trigonometric functions Functions of an angle

In mathematics, the trigonometric functions are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all sciences that are related to geometry, such as navigation, solid mechanics, celestial mechanics, geodesy, and many others. They are among the simplest periodic functions, and as such are also widely used for studying periodic phenomena through Fourier analysis.

In mathematics, de Moivre's formula states that for any real number x and integer n it holds that

Box–Muller transform

The Box–Muller transform, by George Edward Pelham Box and Mervin Edgar Muller, is a random number sampling method for generating pairs of independent, standard, normally distributed random numbers, given a source of uniformly distributed random numbers. The method was in fact first mentioned explicitly by Raymond E. A. C. Paley and Norbert Wiener in 1934.

Exponentiation Mathematical operation

Exponentiation is a mathematical operation, written as bn, involving two numbers, the baseb and the exponent or powern, and pronounced as "b raised to the power of n". When n is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, bn is the product of multiplying n bases:

Root of unity Number that has an integer power equal to 1

In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that yields 1 when raised to some positive integer power n. Roots of unity are used in many branches of mathematics, and are especially important in number theory, the theory of group characters, and the discrete Fourier transform.

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.

Cube root The third power of the cube root of a number is the number itself

In mathematics, a cube root of a number x is a number y such that y3 = x. All nonzero real numbers, have exactly one real cube root and a pair of complex conjugate cube roots, and all nonzero complex numbers have three distinct complex cube roots. For example, the real cube root of 8, denoted , is 2, because 23 = 8, while the other cube roots of 8 are and . The three cube roots of −27i are

Inverse trigonometric functions arcsin, arccos, arctan, etc.

In mathematics, the inverse trigonometric functions are the inverse functions of the trigonometric functions. Specifically, they are the inverses of the sine, cosine, tangent, cotangent, secant, and cosecant functions, and are used to obtain an angle from any of the angle's trigonometric ratios. Inverse trigonometric functions are widely used in engineering, navigation, physics, and geometry.

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 mathematics, an algebraic function is a function that can be defined as the root of a polynomial equation. Quite often algebraic functions are algebraic expressions using a finite number of terms, involving only the algebraic operations addition, subtraction, multiplication, division, and raising to a fractional power. Examples of such functions are:

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

Sine trigonometric function of an angle

In mathematics, the sine is a trigonometric function of an angle. The sine of an acute angle is defined in the context of a right triangle: for the specified angle, it is the ratio of the length of the side that is opposite that angle, to the length of the longest side of the triangle. For an angle , the sine function is denoted simply as .

In algebra, casus irreducibilis is one of the cases that may arise in attempting to solve polynomials of degree 3 or higher with integer coefficients, to obtain roots that are expressed with radicals. It shows that many algebraic numbers are real-valued but cannot be expressed in radicals without introducing complex numbers. The most notable occurrence of casus irreducibilis is in the case of cubic polynomials that are irreducible over the rational numbers and have three real roots, which was proven by Pierre Wantzel in 1843. One can decide whether a given irreducible cubic polynomial is in casus irreducibilis using the discriminant Δ, via Cardano's formula. Let the cubic equation be given by

The Marsaglia polar method is a pseudo-random number sampling method for generating a pair of independent standard normal random variables. While it is superior to the Box–Muller transform, the Ziggurat algorithm is even more efficient.

In mathematics, a trigonometric number is an irrational number produced by taking the sine or cosine of a rational multiple of a full circle, or equivalently, the sine or cosine of an angle which in radians is a rational multiple of π, or the sine or cosine of a rational number of degrees. One of the simplest examples is

References

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  2. "MATLAB Product Documentation".
  3. 1 2 Weisstein, Eric W. "Imaginary Unit". mathworld.wolfram.com. Retrieved 10 August 2020.
  4. Doxiadēs, Apostolos K.; Mazur, Barry (2012). Circles Disturbed: The interplay of mathematics and narrative (illustrated ed.). Princeton University Press. p.  225. ISBN   978-0-691-14904-2 via Google Books.
  5. Bunch, Bryan (2012). Mathematical Fallacies and Paradoxes (illustrated ed.). Courier Corporation. p.  31-34. ISBN   978-0-486-13793-3 via Google Books.
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  10. Wells, David (1997) [1986]. The Penguin Dictionary of Curious and Interesting Numbers (revised ed.). UK: Penguin Books. p. 26. ISBN   0-14-026149-4.
  11. "abs(i!)". Wolfram Alpha.

Further reading