The **imaginary unit** or **unit imaginary number** (**i**) is a solution to the quadratic equation *x*^{2} + 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 + 3*i*.

- Definition
- i vs. −i
- Matrices
- Proper use
- Properties
- Square roots
- Cube roots
- Multiplication and division
- Powers
- Factorial
- Other operations
- History
- See also
- Notes
- References
- Further reading
- External links

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.

The powers of i return cyclic values: |
---|

... (repeats the pattern from bold blue area) |

i^{−3} = i |

i^{−2} = −1 |

i^{−1} = −i |

i^{0} = 1 |

i^{1} = i |

i^{2} = −1 |

i^{3} = −i |

i^{4} = 1 |

i^{5} = i |

i^{6} = −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 *i*^{2} 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 + 1*i*, with a zero real component and a unit imaginary component. In polar form, i is represented as 1⋅*e*^{iπ/2} (or just *e*^{iπ/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).

Being a quadratic polynomial with no multiple root, the defining equation *x*^{2} = −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 *x*^{2} + 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*]/(*x*^{2} + 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*]/(*x*^{2} + 1) which keep each real number fixed: The identity and the automorphism sending *x* to −*x*. For more, see complex conjugate and Galois group.

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, ℝ) 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, *z*^{2} + *xy* = –1, so the product xy is negative because *xy* = –(1 + *z*^{2}), thus the point (*x*, *y*) lies in quadrant II or IV. Furthermore,

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

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 *i*√7 , rather than √−7 . For a more thorough discussion, see square root and branch point.

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:

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.

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.)

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:

Making use of Euler's formula, *i*^{i} is

where *k* ∈ ℤ, the set of integers.

The principal value (for *k* = 0) is *e*^{−π/2}, or approximately 0.207879576 .^{ [10] }

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

Also,

^{ [11] }

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 *ni*^{th} 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:

- ↑ 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 1*i*or 1*j*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 ℓ is used. - ↑ To find such a number, one can solve the equation
- (
*x*+*iy*)^{2}=*i*

*x*^{2}+ 2*ixy*−*y*^{2}=*i*.

*x*^{2}−*y*^{2}+ 2*ixy*= 0 +*i*

*x*^{2}−*y*^{2}= 0- 2
*xy*= 1 .

*y*= ½*x*into the first equation, we get*x*^{2}−¼*x*^{2}= 0*x*^{2}= ¼*x*^{2}- 4
*x*^{4}= 1

*x*= 1/√2 and*x*= −1/√2 . Substituting either of these results into the equation 2*xy*= 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] } - (

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 *i*^{2} = −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.

**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:

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 = 10^{3}, the "logarithm base 10" of 1000 is 3, or log_{10}(1000) = 3. The logarithm of x to *base*b is denoted as log_{b}(*x*), or without parentheses, log_{b} *x*, 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.

In mathematics, a **square root** of a number *x* is a number *y* such that *y*^{2} = *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 4^{2} = (−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 3^{2} = 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.

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

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** is a mathematical operation, written as *b*^{n}, involving two numbers, the *base*b and the *exponent* or *power*n, 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, *b*^{n} is the product of multiplying n bases:

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

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, a **cube root** of a number x is a number y such that *y*^{3} = *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 2^{3} = 8, while the other cube roots of 8 are and . The three cube roots of −27*i* are

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.

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

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

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

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*Mathematical Methods in the Physical Sciences*(3rd ed.). New York [u.a.]: Wiley. p. 49. ISBN 0-471-19826-9. - "MATLAB Product Documentation".
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*mathworld.wolfram.com*. Retrieved 10 August 2020. - ↑ 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. - ↑ Bunch, Bryan (2012).
*Mathematical Fallacies and Paradoxes*(illustrated ed.). Courier Corporation. p. 31-34. ISBN 978-0-486-13793-3 – via Google Books. - ↑ Kramer, Arthur (2012).
*Math for Electricity & Electronics*(4th ed.). Cengage Learning. p. 81. ISBN 978-1-133-70753-0 – via Google Books. - ↑ Picciotto, Henri; Wah, Anita (1994).
*Algebra: Themes, tools, concepts*(Teachers’ ed.). Henri Picciotto. p. 424. ISBN 978-1-56107-252-1 – via Google Books. - ↑ 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 – via Google Books. - "What is the square root of i ?".
*University of Toronto Mathematics Network*. Retrieved 26 March 2007. - ↑ Wells, David (1997) [1986].
*The Penguin Dictionary of Curious and Interesting Numbers*(revised ed.). UK: Penguin Books. p. 26. ISBN 0-14-026149-4. - ↑ "abs(i!)".
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- Euler, Leonhard. "Imaginary Roots of Polynomials". at "Convergence".
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