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Imaginary unit
The term imaginary unit or unit imaginary number refers to a solution to the equation x^{2} = −1. As a convention, the solution is usually denoted i. Since there is no real number with this property, it extends the real numbers, and under the assumption that the familiar properties of addition and multiplication (namely closure, associativity, commutativity and distributivity) continue to hold for this extension, the complex number system is generated by including it.
Imaginary numbers are an important 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 nonconstant polynomial P(x) (see Algebraic closure and Fundamental theorem of algebra). 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 where i is ambiguous or problematic, j or the Greek ι is sometimes used (see § Alternative notations). In the disciplines of electrical engineering and control systems engineering, the imaginary unit is often denoted by j instead of i, because i is commonly used to denote electric current.
For the history of the imaginary unit, see Complex number § History.
Contents
Definition
The powers of i 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 i is defined solely by the property that its square is −1:
- <math>i^2 = -1 \ . </math>
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 then 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:
- <math>i^3 = i^2 i = (-1) i = -i \,</math>
- <math>i^4 = i^3 i = (-i) i = -(i^2) = -(-1) = 1 \,</math>
- <math>i^5 = i^4 i = (1) i = i \,</math>
Similarly, as with any non-zero real number:
- <math>i^0 = i^{1-1} = i^1i^{-1} = i^1\frac{1}{i} = i\frac{1}{i} = \frac{i}{i} = 1 \,</math>
As a complex number, i is equal to 0 + i, having a unit imaginary component and no real component (i.e., the real component is zero). In polar form, i 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), i is the point located one unit from the origin along the imaginary axis (which is at a right angle to the real axis).
i 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 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 results as long as one or other of the solutions is chosen and labelled as "i", with the other one then being labelled as −i. This is because, although −i and i are not quantitatively equivalent (they are negatives of each other), there is no algebraic difference between i 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 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".
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 x 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
- <math>X = \begin{pmatrix}
0 & -1 \\ 1 & \;\;0 \end{pmatrix} </math> and <math>X = \begin{pmatrix} \;\;0 & 1 \\ -1 & 0 \end{pmatrix}
</math>
are solutions to the matrix equation
- <math> X^2 = -I = - \begin{pmatrix}
1 & 0 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} -1 & \;\;0 \\ \;\;0 & -1 \end{pmatrix}. \ </math>
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 √−1 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:
- <math>-1 = i \cdot i = \sqrt{-1} \cdot \sqrt{-1} = \sqrt{(-1) \cdot (-1)} = \sqrt{1} = 1</math> (incorrect).
Attempting to correct the calculation by specifying both the positive and negative roots only produces ambiguous results:
- <math>-1 = i \cdot i = \pm \sqrt{-1} \cdot \pm \sqrt{-1} = \pm \sqrt{(-1) \cdot (-1)} = \pm \sqrt{1} = \pm 1</math> (ambiguous).
Similarly:
- <math>\frac{1}{i} = \frac{\sqrt{1}}{\sqrt{-1}} = \sqrt{\frac{1}{-1}} = \sqrt{\frac{-1}{1}} = \sqrt{-1} = i</math> (incorrect).
The calculation rules
- <math>\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}</math>
and
- <math>\frac{\sqrt{a}} {\sqrt{b}} = \sqrt{\frac{a}{b}}</math>
are only valid for real, non-negative values of a and b.
These problems are avoided by writing and manipulating i√7, rather than expressions like √−7. For a more thorough discussion, see Square root and Branch point.
Properties
Square roots
The square root of i can be expressed as either of two complex numbers^{[nb 1]}
- <math> \sqrt{i} = \pm \left( \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2}i \right) = \pm \frac{\sqrt{2}}2 (1 + i). </math>
Indeed, squaring the right-hand side gives
- <math>
\begin{align} \left( \pm \frac{\sqrt{2}}2 (1 + i) \right)^2 \ & = \left( \pm \frac{\sqrt{2}}2 \right)^2 (1 + i)^2 \ \\
& = \frac{1}{2} (1 + 2i + i^2) \\ & = \frac{1}{2} (1 + 2i - 1) \ \\ & = i. \ \\
\end{align} </math>
This result can also be derived with Euler's formula
- <math>e^{ix} = \cos(x) + i\sin(x) \,</math>
by substituting x = π/2, giving
- <math>e^{i(\pi/2)} = \cos(\pi/2) + i\sin(\pi/2) = 0 + i1 = i\,\! .</math>
Taking the square root of both sides gives
- <math> \sqrt{i} = \pm e^{i(\pi/4)} \,\! ,</math>
which, through application of Euler's formula to x = π/4, gives
- <math>
\begin{align}
\sqrt{i} & = \pm (\cos(\pi/4) + i\sin(\pi/4)) \\ & = \frac{1}{\pm \sqrt{2}} + \frac{i}{\pm \sqrt{2}}\\ & = \frac{1+i}{\pm \sqrt{2}}\\ & = \pm \frac{\sqrt{2}}2 (1 + i).\\
\end{align} </math>
Similarly, the square root of −i can be expressed as either of two complex numbers using Euler's formula:
- <math>e^{ix} = \cos(x) + i\sin(x) \,</math>
by substituting x = 3π/2, giving
- <math>e^{i(3\pi/2)} = \cos(3\pi/2) + i\sin(3\pi/2) = 0 - i1 = -i\,\! .</math>
Taking the square root of both sides gives
- <math> \sqrt{-i} = \pm e^{i(3\pi/4)} \,\! ,</math>
which, through application of Euler's formula to x = 3π/4, gives
- <math>
\begin{align}
\sqrt{-i} & = \pm (\cos(3\pi/4) + i\sin(3\pi/4)) \\ & = -\frac{1}{\pm \sqrt{2}} + i\frac{1}{\pm \sqrt{2}}\\ & = \frac{-1 + i}{\pm \sqrt{2}}\\ & = \pm \frac{\sqrt{2}}2 (i - 1).\\
\end{align} </math>
Multiplying the square root of i by i also gives:
- <math>
\begin{align}
\sqrt{-i} = (i)\cdot (\pm\frac{1}{\sqrt{2}}(1 + i)) \\ & = \pm\frac{1}{\sqrt{2}}(1i + i^{2})\\ & = \pm\frac{\sqrt{2}}{2}(i - 1)\\
\end{align} </math>
Multiplication and division
Multiplying a complex number by i gives:
- <math>i\,(a + bi) = ai + bi^2 = -b + ai.</math>
(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:
- <math>\frac{1}{i} = \frac{1}{i} \cdot \frac{i}{i} = \frac{i}{i^2} = \frac{i}{-1} = -i.</math>
Using this identity to generalize division by i to all complex numbers gives:
- <math>\frac{a + bi}{i} = -i\,(a + bi) = -ai - bi^2 = b - ai. </math>
(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:
- <math>i^{4n} = 1\,</math>
- <math>i^{4n+1} = i\,</math>
- <math>i^{4n+2} = -1\,</math>
- <math>i^{4n+3} = -i.\,</math>
This leads to the conclusion that
- <math>i^n = i^{n \bmod 4}\,</math>
where mod represents the modulo operation. Equivalently:
- <math>i^n = \cos(n\pi/2)+i\sin(n\pi/2)</math>
i raised to the power of i
Making use of Euler's formula, i^{i} is
- <math>i^i = \left( e^{i (\pi/2 + 2k \pi)} \right)^i = e^{i^2 (\pi/2 + 2k \pi)} = e^{- (\pi/2 + 2k \pi)}</math>
where <math>k \in \mathbb{Z}</math>, 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 i is most often given in terms of the gamma function evaluated at 1 + i:
- <math>i! = \Gamma(1+i) \approx 0.4980 - 0.1549i.</math>
Also,
- <math>|i!| = \sqrt{\pi \over \sinh \pi} </math>^{[2]}
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. 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:
- <math> \!\ x^{ni} = \cos(\ln x^n) + i \sin(\ln x^n ).</math>
The ni^{th} root of a number is:
- <math> \!\ \sqrt[ni]{x} = \cos(\ln \sqrt[n]{x} ) - i \sin(\ln \sqrt[n]{x} ).</math>
The imaginary-base logarithm of a number is:
- <math> \log_i(x) = {{2 \ln x } \over i\pi}.</math>
As with any complex logarithm, the log base i is not uniquely defined.
The cosine of i is a real number:
- <math> \cos(i) = \cosh(1) = {{e + 1/e} \over 2} = {{e^2 + 1} \over 2e} \approx 1.54308064... .</math>
And the sine of i is purely imaginary:
- <math> \sin(i) = i\sinh(1) \, = {{e - 1/e} \over 2} \, i = {{e^2 - 1} \over 2e} \, i \approx 1.17520119 \, i... .</math>
Alternative notations
- In electrical engineering and related fields, the imaginary unit is often denoted by 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 mark the imaginary part of a complex number. MATLAB associates both i and j with the imaginary unit, although 1i or 1j is preferable, for speed and improved robustness.^{[3]}
- Some texts use the Greek letter iota (ι) for the imaginary unit, to avoid confusion, especially with index and subscripts.
- Each of i, j, and k is an imaginary unit in the quaternions. In bivectors and biquaternions an additional imaginary unit h 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 equation
- (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.
- ^ "MATLAB Product Documentation".
Further reading
- Nahin, Paul J. (1998). An Imaginary Tale: The Story of √−1. Chichester: Princeton University Press. ISBN 0-691-02795-1.