Imaginary unit
The imaginary unit or unit imaginary number, denoted as Template:Mvar, is a mathematical concept which extends the real number system ℝ to the complex number system ℂ, which in turn provides at least one root for every polynomial P(x) (see algebraic closure and fundamental theorem of algebra). The imaginary unit's core property is that i^{2} = −1. The term "imaginary" is used because there is no real number having a negative square.
There are in fact two complex square roots of −1, namely Template:Mvar and −i, just as there are two complex square roots of every other real number, except zero, which has one double square root.
In contexts where Template:Mvar is ambiguous or problematic, Template:Mvar or the Greek ι (see alternative notations) is sometimes used. In the disciplines of electrical engineering and control systems engineering, the imaginary unit is often denoted by Template:Mvar instead of Template:Mvar, because Template:Mvar is commonly used to denote electric current.
For the history of the imaginary unit, see Complex number: History.
Contents
Definition
The powers of Template:Mvar return cyclic values: |
---|
... (repeats the pattern from 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 blue area) |
The imaginary number Template:Mvar is defined solely by the property that its square is −1:
With Template:Mvar defined this way, it follows directly from algebra that Template:Mvar 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 Template:Mvar as an unknown quantity while manipulating an expression, and then using the definition to replace any occurrence of i^{2} with −1. Higher integral powers of Template:Mvar can also be replaced with −i, 1, Template:Mvar, or −1:
Similarly, as with any non-zero real number:
As a complex number, Template:Mvar is equal to 0 + i, having a unit imaginary component and no real component (i.e., the real component is zero). In polar form, Template:Mvar is 1 cis ^{π}/_{2}, having an absolute value (or magnitude) of 1 and an argument (or angle) of ^{π}/_{2}. In the complex plane (also known as the Cartesian plane), Template:Mvar is the point located one unit from the origin along the imaginary axis (which is at a right angle to the real axis).
{{safesubst:#invoke:anchor|main}} Template:Mvar and −i
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. More precisely, once a solution Template:Mvar of the equation has been fixed, the value −i, which is distinct from Template:Mvar, is also a solution. Since the equation is the only definition of Template:Mvar, it appears that the definition is ambiguous (more precisely, not well-defined). However, no ambiguity results as long as one or other of the solutions is chosen and labelled as "Template:Mvar", with the other one then being labelled as −i. This is because, although −i and Template:Mvar are not quantitatively equivalent (they are negatives of each other), there is no algebraic difference between Template:Mvar and −i. Both imaginary numbers have equal claim to being the number whose square is −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 Template:Mvar 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".
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 2 field automorphisms of ℝ[x]/(x^{2} + 1) which keep each real number fixed: the identity and the automorphism sending Template:Mvar 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 of complex numbers), because then both
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.
All these ambiguities can be solved by adopting a more rigorous definition of complex number, and 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.
Proper use
The imaginary unit is sometimes written Template:Sqrt 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:
Attempting to correct the calculation by specifying both the positive and negative roots only produces ambiguous results:
Similarly:
The calculation rules
and
are only valid for real, non-negative values of Template:Mvar and Template:Mvar.
These problems are avoided by writing and manipulating iTemplate:Sqrt, rather than expressions like Template:Sqrt. For a more thorough discussion, see Square root and Branch point.
Properties
Square roots
The square root of Template:Mvar can be expressed as either of two complex numbers^{[nb 1]}
Indeed, squaring the right-hand side gives
This result can also be derived with Euler's formula
by substituting x = π/2, giving
Taking the square root of both sides gives
which, through application of Euler's formula to x = π/4, gives
Similarly, the square root of −i can be expressed as either of two complex numbers using Euler's formula:
by substituting x = 3π/2, giving
Taking the square root of both sides gives
which, through application of Euler's formula to x = 3π/4, gives
Multiplying the square root of Template:Mvar by Template:Mvar also gives:
Multiplication and division
Multiplying a complex number by Template:Mvar gives:
(This is equivalent to a 90° counter-clockwise rotation of a vector about the origin in the complex plane.)
Dividing by Template:Mvar is equivalent to multiplying by the reciprocal of Template:Mvar:
Using this identity to generalize division by Template:Mvar 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 Template:Mvar 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:
Template:Mvar raised to the power of Template:Mvar
Making use of Euler's formula, i^{i} is
where , the set of integers.
The principal value (for k = 0) is e^{−π/2} or approximately 0.207879576...^{[1]}
Factorial
The factorial of the imaginary unit Template:Mvar is most often given in terms of the gamma function evaluated at 1 + i:
Also,
Other operations
Many mathematical operations that can be carried out with real numbers can also be carried out with Template:Mvar, such as exponentiation, roots, logarithms, and trigonometric functions. However, it should be noted that 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 Template:Mvar is not uniquely defined.
The cosine of Template:Mvar is a real number:
And the sine of Template:Mvar is purely imaginary:
Alternative notations
- In electrical engineering and related fields, the imaginary unit is often denoted by Template:Mvar to avoid confusion with electrical current as a function of time, traditionally denoted by i(t) or just Template:Mvar. The Python programming language also uses Template:Mvar to mark the imaginary part of a complex number. MATLAB associates both Template:Mvar and Template:Mvar with the imaginary unit, although 1i or 1j is preferable, for speed and improved robustness.^{[3]}
- Some texts use the Greek letter iota (Template:Mvar) for the imaginary unit, to avoid confusion, esp. with index and subscripts.
- Each of Template:Mvar, Template:Mvar, and Template:Mvar is an imaginary unit in the quaternions. In bivectors and biquaternions an additional imaginary unit Template:Mvar is used.
Matrices
When 2 × 2 real matrices m are used for a source, and the number one (1) is identified with the identity matrix, and minus one (−1) with the negative of the identity matrix, then there are many solutions to m^{2} = −1. In fact, there are many solutions to m^{2} = +1 and m^{2} = 0 also. Any such m can be taken as a basis vector, along with 1, to form a planar algebra.
See also
Notes
- ↑ To find such a number, one can solve the equations
- (x + iy)^{2} = i
- x^{2} + 2ixy − y^{2} = i
- x^{2} − y^{2} + 2ixy = 0 + i
- x^{2} − y^{2} = 0
- 2xy = 1
- x^{2} − 1/4x^{2} = 0
- x^{2} = 1/4x^{2}
- 4x^{4} = 1
References
- ↑ "The Penguin Dictionary of Curious and Interesting Numbers" by David Wells, Page 26.
- ↑ "abs(i!)", WolframAlpha.
- ↑ Template:Cite web
Further reading
- {{#invoke:citation/CS1|citation
|CitationClass=book }}