In mathematics, the imaginary unit i (or sometimes the Latin j or the Greek iota, see below) allows the real number system to be extended to the complex number system . Its precise definition is dependent upon the particular method of extension. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... For programming language, see Iota and Jot. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ...

The primary motivation for this extension is the fact that not every polynomial equation with real coefficients f(x) = 0 has a solution in the real numbers. In particular, the equation x² + 1 = 0 has no real solution (see "Definition", below). However, if we allow complex numbers as solutions, then this equation, and indeed every polynomial equation f(x) = 0 does have a solution. (See algebraic closure and fundamental theorem of algebra.) In mathematics, a polynomial is an expression in which constants and variables are combined using only addition, subtraction, multiplication, and positive whole number exponents (raising to a power). ... Dissolving table salt (NaCl) in water This article is about a chemical solution; for other uses of the term solution, see solution (disambiguation). ... In mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed. ... In mathematics, the fundamental theorem of algebra states that every complex polynomial in one variable and of degree  â‰¥  has some complex root. ...

For a history of the imaginary unit, see the history of complex numbers. In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ...

The imaginary unit is often loosely referred to as the "square root of negative one" or the "square root of minus one", but see below for difficulties that may arise from a naive use of this idea. In mathematics, the imaginary unit (or sometimes the Latin or the Greek iota, see below) allows the real number system to be extended to the complex number system . ...

Definition

By definition, the imaginary unit i is one solution of the quadratic equation In mathematics, a quadratic equation is a polynomial equation of the second degree. ...

or equivalently

.

Since there is no real number that squares to any negative real number, we imagine such a number and assign to it the symbol i. It is important to realize, though, that i is just as well-defined a mathematical construct as the real numbers. Thus, despite its title as an "imaginary number", i is no more and no less real than any other class of numbers, despite being less intuitive to study.

Real number operations can be extended to imaginary and complex numbers by treating i as an unknown quantity while manipulating an expression, and then using the definition to replace occurrences of i ² with −1. Higher integral powers of i can also be replaced with −i , 1, i, or −1.

i and −i

Being a second order polynomial with no multiple real root, the above equation has two distinct solutions that are equally valid and that happen to be additive inverses of each other. More precisely, once a solution i of the equation has been fixed, the value −i ≠ 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 results as long as one of the solutions is chosen and fixed as the "positive i". This is because, although −i and i are not quantitatively equivalent (they are negatives of each other), there is no qualitative difference between i and −i (that cannot be said for −1 and +1). Both imaginary numbers have equal claim to square to −1. If all mathematical textbooks and published literature referring to imaginary or complex numbers were 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 = 0 with one of them as "positive" is purely a notational relic; neither root can be said to be more important than the other. This article is about the mathematical term; Multiplicity is also the title of a 1996 film. ... The additive inverse, or opposite, of a number n is the number which, when added to n, yields zero. ... In mathematics, the term well-defined is used to specify that a certain concept (a function, a property, a relation, etc. ...

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 2 field automorphisms of R[X]/ (X² + 1), the identity and the automorphism sending X to −X. (These are not the only field automorphisms of C, but are the only field automorphisms of C which keep each real number fixed.) See complex number, complex conjugation, field automorphism, and Galois group. In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication, and division (except division by zero) may be performed and the associative, commutative, and distributive rules hold, which are familiar from the arithmetic of ordinary numbers. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ... In predicate logic and technical fields that depend on it, uniqueness quantification, or unique existential quantification, is an attempt to formalise the notion of something being true for exactly one thing, or exactly one thing of a certain type. ... Look up Up to on Wiktionary, the free dictionary In mathematics, the phrase up to xxxx indicates that members of an equivalence class are to be regarded as a single entity for some purpose. ... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich, which shows a relation between two properties or operations. ... In mathematics, an automorphism is an isomorphism from a mathematical object to itself. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ... In mathematics, the complex conjugate of a complex number is given by changing the sign of the imaginary part. ... In mathematics, an automorphism is an isomorphism from a mathematical object to itself. ... In mathematics, a Galois group is a group associated with a certain type of field extension. ...

A similar issue arises if the complex numbers are interpreted as 2 × 2 real matrices (see complex number), because then both In mathematics, a matrix (plural matrices) is a rectangular table of numbers or, more generally, a table consisting of abstract quantities that can be added and multiplied. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ...

and

are 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, R) has exactly 2 elements — the identity and the automorphism which exchanges "CW" (clockwise) and "CCW" (counter-clockwise) rotations. See orthogonal group. Illustration of a unit circle. ... In mathematics, an automorphism is an isomorphism from a mathematical object to itself. ... In mathematics, the orthogonal group of degree n over a field F (written as O(n,F)) is the group of n-by-n orthogonal matrices with entries from F, with the group operation that of matrix multiplication. ... In mathematics, the orthogonal group of degree n over a field F (written as O(n,F)) is the group of n-by-n orthogonal matrices with entries from F, with the group operation that of matrix multiplication. ...

Warning

The imaginary unit is sometimes written in advanced mathematics contexts (as well as in less advanced popular texts), however great care needs to be taken when manipulating formulas involving radicals. The 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 will produce false results: In mathematics, an nth root of a number a is a number b, such that bn=a. ... In mathematics, a square root of a number x is a number r such that , or in words, a number r whose square (the result of multiplying the number by itself) is x. ...

The calculation rule

is only valid for real, non-negative values of a and b.

For a more thorough discussion of this phenomenon, see square root and branch. In mathematics, a square root of a number x is a number r such that , or in words, a number r whose square (the result of multiplying the number by itself) is x. ...

To avoid making such mistakes when manipulating complex numbers, a strategy is never to use a negative number under a square root sign. This means to avoid writing expressions like , one should write instead. That is the use for which the imaginary unit is intended.

Square root of the imaginary unit

One might assume that a further set of imaginary numbers need to be invented to account for the square root of i. However this is not necessary as it can be expressed as either of two complex numbers: ^[1]. This can be shown to be valid from:

Proof

One derive this number using de Moivre's formula. de Moivre's formula states that for any complex number x, and any integer n: de Moivres formula, named after Abraham de Moivre, states that for any complex number (and, in particular, for any real number) x and any integer n it holds that The formula is important because it connects complex numbers (i stands for the imaginary unit) and trigonometry. ... The integers are commonly denoted by the above symbol. ...

Because , we can use de Moivre's Formula to find .

Powers of i

The powers of i repeat in a cycle:

This can be expressed with the following pattern where n is any integer:

This leads to the conclusion that

i and

Using the semicircle equation

We can determine that

i and Euler's formula

Euler's formula is Eulers formula, named after Leonhard Euler, is a mathematical formula in complex analysis that shows a deep relationship between the trigonometric functions and the complex exponential function. ...

,

where x represents an angle expressed in radians. Substituting π / 2, for x,, one arrives at

If we raise each side to the power i, we get

or

.

In fact, has an infinite number of elements in the form of

where N is any integer. This is because in Euler's formula above, , the right side just represents the angle plus one full rotation around the circle. From the number theorist's point of view, is a quadratic irrational number, like , and by applying the Gelfond-Schneider theorem, we can conclude that all of the values we obtained above, and in particular, are transcendental. In mathematics, an irrational number is any real number that is not a rational number, i. ... In mathematics, the Gelfond-Schneider theorem is the following statement, originally proved by Aleksandr Gelfond: If is an algebraic number (with and ), and is an irrational algebraic number, then is a transcendental number. ... In mathematics, a transcendental number is any complex number that is not algebraic, that is, not the solution of a non-zero polynomial equation with integer (or, equivalently, rational) coefficients. ...

From the above identity , one arrives elegantly at Euler's identity For other meanings, see List of topics named after Leonhard Euler In mathematical analysis, Eulers identity, named after Leonhard Euler, is the equation where is Eulers number, the base of the natural logarithm, is the imaginary unit, one of the two complex numbers whose square is negative one...

This identity remarkably relates the five most significant mathematical entities, along with the principle of equality and the operations of addition, multiplication, and exponentiation, in one simple expression.

Operations with i

Many mathematical operations that can be carried out with real numbers can also be carried out with i, such as exponentation, roots and logarithms.

A number raised to the ni power is:

The ni^th root of a number is:

The log base i of a number is: An imaginary-base logarithm occurs when one has a logarithm with base i. ...

Alternative notation

In electrical engineering and related fields, the imaginary unit is often written as j to avoid confusion with electrical current as a function of time, traditionally denoted by i(t) or just i. The Python programming language also uses j to denote the imaginary unit. Electrical Engineers design power systems… … and complex electronic circuits. ... In electricity, current refers to electric current, which is the flow of electric charge. ... Python is a high-level programming language first released by Guido van Rossum in 1991. ...

Some extra care needs to be taken in certain textbooks which define j = −i, in particular to traveling waves (e.g. a right traveling plane wave in the x direction ).

Some texts use the Greek letter iota to write the imaginary unit to avoid confusion. For example: Biquaternion. For programming language, see Iota and Jot. ... In mathematics, a biquaternion (or complex quaternion) is an element of the (unique) quaternion algebra over the complex numbers. ...

References

1. ^ University of Toronto Mathematics Network: What is the square root of i? URL retrieved March 26, 2007.

See also

• Imaginary number
• Complex number
• Complex plane
• Roots of unity