- i is the complex number with i^2 = -1; in polar form it is e^{iπ/2} and its power cycle has a period of 4.
- Operating with complex numbers: addition, multiplication, quotient with conjugate and polar form (De Moivre and roots).
- Multiplying/dividing by i rotates 90°; beware of radicals and branches in multivalued functions.
- Notable results: i^i = e^{-π/2} (main value) ei! = Γ(1+i) ≈ 0,4980 − 0,1549i.
The imaginary unit, represented by i , is one of those concepts that seem mysterious the first time they appear, yet prove incredibly practical once properly understood. Its defining characteristic is that it satisfies i² = -1 , something impossible in the world of real numbers but perfectly consistent within the complex numbers.
Beyond the anecdote, treating i as an algebraic symbol with the rule i² = -1 allows us to extend all the usual operations of the real numbers to a richer universe: that of the complex numbers. This extension opens the door to solving equations, modeling rotations, and working with signals, among many other applications.
What exactly is the imaginary unit?
The definition is concrete: the imaginary unit i is a complex number whose square is -1 . If we write a complex number in rectangular form as a + bi, then i is the complex number 0 + 1·i: its real part is 0 and its imaginary part is 1.
In polar form, i has modulus 1 and argument π/2 , so it can be written as e iπ/2 . In the complex plane (Argand plane), the point corresponding to i is one unit from the origin on the imaginary axis, perpendicular to the real axis.
The equation x² = -1 has two distinct solutions: + i and -i . There is no one "more correct" than the other; choosing which we call i and which -i is a matter of labeling, since algebraically they are symmetric.
Basic properties and powers of i
From the rule i 2 = -1, elementary powers and properties are immediately deduced: i 0 = 1, i 1 = i, i 2 = -1, i 3 = -i, ei 4 = 1, starting the cycle again.
This pattern is cyclic with period 4. For any integer n, i n depends on n mod 4 : simply divide n by 4 and keep the remainder r ∈ {0,1,2,3}; thus i n = i r . For negative exponents, i -n = 1/i n , and since 1/i = -i, it simplifies easily.
Multiplying by i “rotates” the pattern: i·i = -1, i·(-1) = -i, i·(-i) = 1, i·1 = i . This periodically alternating behavior is the basis of many fast calculations with powers of i.
How it is represented and interpreted on the complex plane
Any complex number is written as a + bi, with a,b ∈ ℝ . If b = 0 we recover a pure real number; if a = 0 we obtain a pure imaginary number bi.
Geometrically, a + bi corresponds to the vector (a,b) in the plane . The horizontal axis is the real axis and the vertical axis is the imaginary axis. The magnitude is the distance to the origin, |a + bi| = √(a 2 + b 2 ), and the argument is the angle φ such that cos φ = a/|z| and sin φ = b/|z|, taking care to place it in the correct quadrant.
The polar or trigonometric form is especially convenient for multiplying, dividing, raising to powers and extracting roots: z = r·e iφ = r(cos φ + i sin φ) , where r ≥ 0 is the modulus and φ is the argument.
Sums, products, quotients, and conjugates in binomial form
Operations in the form a + bi are direct: (a + bi) + (c + di) = (a + c) + (b + d)i , and subtraction is analogous. Real parts are added to real parts and imaginary parts to imaginary parts.
To multiply, (a + bi)(c + di) = (ac – bd) + (ad + bc)i , using i 2 = -1. It is the same expansion as a product of binomials, with the exception of the term i 2.
The complex conjugate of z = a + bi is ̅z = a – bi . It is useful because z·̅z = a 2 + b 2 = |z| 2 , a non-negative real number.
To divide, the denominator is rationalized: (a + bi)/(c + di) = /(c 2 + d 2 ) , which returns another complex in binomial form.
Polar form and practical theorems (including De Moivre)
If z = r(cos φ + i sin φ) and w = s(cos θ + i sin θ), then z·w = rs ; the magnitudes are multiplied and the angles are added.
The quotient is handled equally well: z/w = (r/s) , provided that s ≠ 0. Thus, dividing subtracts arguments and divides moduli.
For integer powers, De Moivre's theorem states that z n = r n . This is a compact and very powerful way to raise complex numbers to power.
The nth roots are given by w k = r 1/n , with k = 0,1,…,n−1. That is, there are exactly n equally spaced solutions on the circle of radius r 1/n.
Square and cube roots of i
Calculating √i is a classic. If we impose (x + yi) 2 = i and separate parts, we obtain the system x 2 – y 2 = 0 and 2xy = 1. The solution leads to ±(√2/2)(1 + i) , which are the two square roots of i.
The cube roots of i are three, located on the unit circle. One way to list them is: -i, √3/2 + (1/2)i and -√3/2 + (1/2)i . As with all roots, they form the vertices of a regular polygon inscribed in the circle of radius 1.
Multiplying and dividing by i as a rotation
Multiplying a complex number by i corresponds to rotating it 90° counterclockwise around the origin. In binomial form, (a + bi)·i = -b + ai: the coordinates are permuted with an appropriate sign change.
Dividing by i is equivalent to multiplying by its reciprocal, and since 1/i = -i , we get (a + bi)/i = (a + bi)(-i) = b – ai. Geometrically, this is a 90° clockwise rotation.
When do the “rules of the root” apply and when do they not?
The notation √· is reserved for the principal square root . On the real axis, it is only defined for values ≥ 0; on the complex plane, it is defined by choosing a principal branch. Carelessly applying rules such as √a·√b = √(ab) or √a/√b = √(a/b) outside their domain of validity leads to errors.
For example, trying to write √(-1)·√(-1) = √(1) and concluding that -1 = 1 is a trap: these equalities are only valid for real numbers a,b ≥ 0 , and in complex numbers one must take into account multivaluation and the chosen branch.
A practical guide: in ℝ, even-indexed roots of negative numbers do not exist ; in ℂ, they do exist but are multivalued and you have to fix a branch to work consistently.
Notable complex functions: i^i, i! and trigonometric functions
Using Euler's formula, i = e i(π/2 + 2kπ) , and raising i to the power of i, we obtain i i = e – (π/2 + 2kπ) for any k ∈ ℤ. The principal value corresponds to ak = 0 and is e -π/2 , a positive real number less than 1.
The “factorial” of i is defined via the gamma function: i! = Γ(1 + i) ≈ 0,4980 − 0,1549·i . Its modulus satisfies |i!| = √(π/sinh π), an elegant identity that connects real and complex analysis.
Complex trigonometric functions also present surprises: cos(i) is real and equals cosh(1) , while sin(i) is purely imaginary and equals i·sinh(1) . In general, many functions (powers, roots, logarithms, trigonometric functions) are multivalued in ℂ; it is convenient to explicitly declare the branch (Riemann surface) with which one is working.
Choosing i versus -i and the algebraic point of view
From a structural perspective, the field of complex numbers can be constructed as ℝ/(x 2 + 1) . This construction is unique up to isomorphism, but not unique in form: there are two automorphisms that fix ℝ , identity and conjugation (which sends ia -i).
This explains why there is no algebraic difference between choosing io -i as “the” imaginary unit: both roles are interchangeable through an automorphism that leaves the real intact.
Matrix interpretation and ambiguity of orientation
Another way to view complex numbers is as 2×2 real matrices . Multiplication by i can be represented by J = ,], which satisfies J 2 = -I. Also, -J satisfies the same equation , reflecting the ambivalence between counterclockwise and clockwise rotation.
More generally, there are many matrices X with X 2 = -I . Even if we consider the family of matrices ,], the condition X 2 = -I imposes z 2 + xy = -1 , which forces xy ≤ -1 and places (x,y) on branches of a hyperbola in quadrants II and IV.
Geometrically, the special orthogonal group in 2D, SO(2), has two relevant automorphisms : the identity automorphism and the automorphism that interchanges clockwise and counterclockwise rotations. Choosing what constitutes a positive rotation determines which of J or -J represents i.
Quick calculation examples
Powers mod 4 : for n = 2025, since 2025 = 4·506 + 1, we have i 2025 = i 1 = i.
Binomial product : (3 + 2i)(1 − 4i) = (3·1 − 2·4) + (3·(−4) + 2·1)i = (3 − 8) + (−12 + 2)i = −5 − 10i.
Division with conjugate : (5 − i)/(2 + i) = (5 − i)(2 − i)/(2 2 + 1 2 ) = /5 = (9 − 7i)/5 = 1,8 − 1,4i.
Polar form and De Moivre : z = √2(cos 45° + i sin 45°) = 1 + i. Then z 4 = (√2) 4 = 4(−1 + 0i) = −4.
Uses and notations in engineering and programming
In electrical engineering, to avoid confusion with the current intensity i(t), j is used for the imaginary unit . Many languages and tools follow this convention: in Python it is written 3+4j and in MATLAB/Octave there are the literals i and j (1i or 1j being recommended for robustness).
Some texts use the Greek letter iota (ι) to avoid collisions with indices. In quaternions, three “imaginary units” i, j, and k appear , with multiplication rules different from those of complex numbers; in the context of biquaternions and bivectors, another unit h may be introduced.
Helpful warnings when working with i
Whenever you manipulate expressions with complex roots or logarithms, remember that the branch choice matters . Switching branches mid-calculation can alter signs, arguments, and consequently, results.
When a real identity appears to lead to contradictions when extended to ℂ, check the domain: many familiar rules are valid only under certain conditions (e.g., a,b ≥ 0 for rules of radicals), and extending them without control can give false conclusions.
The imaginary unit is not a "trick" but a formal piece that fits precisely into the workings of algebra. From the power cycle and polar representation to i i or Γ(1+i) , all the properties fit together when rigorous definitions are adopted and the branches of multivalued functions are respected. With these ideas clear, i ceases to be "imaginary" and becomes a very real tool in your calculations.
