Complex numbers and complex exponentials

Introduce the imaginary unit

The square of a real number is nonnegative, so no real number solves x² = −1. Complex numbers extend real-number arithmetic. Introduce the imaginary unit i by i² = −1, and write a complex number as z = a + ib, where a and b are real numbers. The symbol i is a number, not a unit such as metre.

The real part is Re z = a; the imaginary part is Im z = b. Both parts are real coefficients. For z = −4 + 7i, Re z = −4 and Im z = 7, rather than 7i. A real number a is also a complex number a + 0i. The term imaginary does not assign a physical direction to its coefficient.

A square means multiplication by itself. From i² = −1, i³ = i² i = −i and i⁴ = i² i² = 1. This locally supplies the rule needed in later products. Our bare-number arithmetic cases are exact mathematical choices with no physical unit or measurement claim.

Locate and combine complex numbers

Represent z = a + ib by the ordered pair (a, b) in the complex plane. The horizontal real axis gives a and the perpendicular imaginary axis gives b. Thus −4 + 7i is represented by (−4, 7). Use matching coordinate scales when interpreting distances and angles in this number plane.

Add or subtract corresponding coefficients: (a + ib) + (c + id) = (a + c) + i(b + d) and (a + ib) − (c + id) = (a − c) + i(b − d). For our choices z1 = 2 + 3i and z2 = 1 − 2i, their sum is 3 + i and their difference z1 − z2 is 1 + 5i.

Equal complex numbers have equal real coefficients and equal imaginary coefficients. The coordinate picture organizes this arithmetic. Its two axes alone do not establish two physical spatial directions; that would require a separately declared physical interpretation.

Multiply with the rule for i

Use distribution, multiplying each term in the first factor by each term in the second. Then replace i² by −1 and collect coefficients. This gives (a + ib)(c + id) = (ac − bd) + i(ad + bc). The negative bd term is essential.

For the same chosen pair, (2 + 3i)(1 − 2i) = 2 − 4i + 3i − 6i² = 8 − i. Treating i² as +1 would give an incorrect real part. The product generally differs from multiplying real parts and imaginary parts separately.

Multiplication by i sends a + ib to −b + ia, so its complex-plane coordinates become (−b, a). For instance, i(2 + 3i) = −3 + 2i. In an equally scaled complex-plane drawing this is a counterclockwise quarter-turn about the origin. This statement concerns the number representation, not a claim that an object turned.

Conjugate, measure and divide

The conjugate of z = a + ib is z* = a − ib; the star denotes conjugation here. Its imaginary coefficient changes sign, reflecting the represented point across the real axis. The modulus is |z| = √(a² + b²), the nonnegative distance from the origin in the equally scaled complex plane. The square-root symbol selects the nonnegative root. In particular, |0| = 0.

Multiplying a number by its conjugate gives z z* = (a + ib)(a − ib) = a² + b² = |z|². For our original choice z = −8 + 15i, z* = −8 − 15i and |z| = √(64 + 225) = 17. The modulus 17 is a magnitude; Re z = −8 is a signed coefficient. These answer different questions.

To divide by a nonzero complex number, multiply numerator and denominator by the denominator's conjugate. If the denominator is c + id, its resulting denominator is c² + d², which is positive whenever c and d are not both zero. This also works for a nonzero purely real or purely imaginary denominator. Division by zero remains undefined.

Introduce general unit-circle angles

Acute right-triangle ratios alone do not define the cardinal phases we need. Use an oriented angle from the positive real axis, with counterclockwise positive and clockwise negative. If the supplied angle is θ = φ rad, φ is its numerical radian value. The radian is an angle unit; angle remains a different quantity kind from an ordinary length ratio. A full turn is 2π rad, equivalent to 360°.

In a dimensionless coordinate plane, the unit circle has centre (0, 0) and radius 1. For any real numerical radian value φ, the point specified by the oriented angle has coordinates (cos φ, sin φ). This defines cosine and sine for negative angles and angles beyond an acute triangle or a full turn. Each coordinate lies between −1 and +1, and cos² φ + sin² φ = 1.

Here cos φ means cosine of the angle whose radian value is φ; sin φ has the corresponding meaning. The inputs φ and the outputs are dimensionless numbers in these formulas. A quarter-turn angle (π/2) rad therefore uses φ = π/2, rather than φ = 90. Adding 2π to φ leaves the unit-circle point unchanged. This is a mathematical angle convention, not a specified time evolution.

Use Euler's formula with its scope

In exponent notation, the base is the number being raised and the exponent is the superscript. For a positive integer n, eⁿ is a product of n factors e; e⁰ = 1. The real exponential also extends to real exponents. Its standard base e is approximately 2.718282. A complex exponent requires a mathematical extension beyond the repeated-multiplication picture.

For every real φ, Euler's established mathematical relation is e^(iφ) = cos φ + i sin φ, with φ the dimensionless numerical radian value just introduced. This lesson uses the complex exponential at purely imaginary exponents iφ. The relation is compatible with the usual exponential extension; it is not a guessed physical law or something proved by plotting a few points. Its mathematical proof is outside the prerequisites of this lesson.

The table gives e^0 = 1, e^(iπ/2) = i, e^(iπ) = −1, e^(i3π/2) = −i and e^(i2π) = 1. Setting φ = π gives Euler's identity e^(iπ) + 1 = 0. The modulus is always |e^(iφ)| = √(cos² φ + sin² φ) = 1. These are exact identities; the circle illustrates their representation without proving the exponential relation.

Recover a stated physical quantity

Cosine displacement and a positive amplitude appear in elementary models of oscillation. Our original hypothetical case supplies only the scalar relation q(φ) = A cos φ, with A = 0.025 m and any supplied real φ. The displacement q is measured from a fixed reference position along one line, with right positive. The positive amplitude A is the maximum value of |q|. These stipulated values are not measurements.

Define an auxiliary complex representation W(φ) = A e^(iφ) = A cos φ + i A sin φ. Recover the model's physical scalar using q = Re W. Both Re W and Im W have metre units, and W/A = e^(iφ) is dimensionless because the positive reference amplitude has matching length units. Here Im W is a second coefficient used in the representation; it does not describe a second physical displacement axis.

For θ = (π/2) rad, φ = π/2, W = 0 + i(0.025 m) and q = 0 m, although |W| = A = 0.025 m. At θ = π rad, q = −0.025 m. The auxiliary circle therefore has fixed modulus while the scalar's absolute value can vary. No relation between φ and time, frequency or force is supplied. The representation cannot by itself specify an object's trajectory, speed or physical cause.

Worked examples

Try it yourself

1. For the exact dimensionless number z = 6 − 11i, give Re z, Im z and z*. Does Im z include i?

   Show the answer and reasoning

   Re z = 6, Im z = −11 and z* = 6 + 11i. Im z does not include i.

   In a + ib, a and b are real coefficients. Here b = −11; conjugation changes b to +11.

2. For exact dimensionless choices u = −3 + 2i and v = 5 − 7i, calculate u + v and u − v.

   Show the answer and reasoning

   u + v = 2 − 5i; u − v = −8 + 9i.

   Addition gives (−3 + 5) + i(2 − 7). Subtraction gives (−3 − 5) + i[2 − (−7)]. Subtracting a negative coefficient increases the imaginary coefficient.

3. Calculate i(−2 + 7i). Give its complex-plane coordinates before and after multiplication, without assigning a physical motion.

   Show the answer and reasoning

   i(−2 + 7i) = −7 − 2i. The coordinates change from (−2, 7) to (−7, −2).

   Distribution gives −2i + 7i² = −7 − 2i. The coordinate rule (a, b) → (−b, a) describes the mathematical quarter-turn when both axes use the same scale.

4. In exact dimensionless arithmetic, calculate 1/i using a conjugate. Is 1/0 defined, and what is |0|?

   Show the answer and reasoning

   1/i = −i; 1/0 is undefined; |0| = 0.

   The nonzero denominator i has conjugate −i. Multiplying numerator and denominator by −i gives (−i)/(−i²) = −i. A nonzero purely imaginary denominator is allowed; zero is not. The modulus is nonnegative and need not be strictly positive.

5. A supplied angle is θ = −90°. Use 1° = (π/180) rad to find its numerical radian value φ, then calculate e^(iφ). What happens if 2π is added to φ?

   Show the answer and reasoning

   θ = (−π/2) rad, φ = −π/2 and e^(iφ) = −i. Adding 2π gives φ = 3π/2 and the same complex number −i.

   A clockwise quarter-turn from the positive real axis reaches (0, −1) on the unit circle. Thus cos φ = 0 and sin φ = −1. The exponent uses the numerical radian value, and a full turn returns to the same point.

6. In the supplied model A = 0.025 m, W = A e^(iφ) and q = Re W, compare φ = 0 and φ = π/2. Give q and |W| at each. Do these two supplied phases determine how fast an object moves?

   Show the answer and reasoning

   At φ = 0, q = +0.025 m and |W| = 0.025 m. At φ = π/2, q = 0 m and |W| = 0.025 m. They do not determine a speed.

   The real coefficient changes from A to zero while the auxiliary modulus remains A. There are no times or phase–time rule in the stipulated model, so the phase comparison is not a speed calculation or evidence for circular object motion.

What to carry forward

• For z = a + ib, Re z = a and Im z = b are real coefficients; i² = −1.
• Complex arithmetic combines coefficients and uses i² = −1; division requires a nonzero denominator.
• Conjugation reverses the imaginary coefficient, and the modulus √(a² + b²) is nonnegative.
• For θ = φ rad, φ is the numerical radian input; unit-circle coordinates define cosine and sine for any real φ.
• Euler's formula e^(iφ) = cos φ + i sin φ is an established mathematical relation; a diagram illustrates it without proving it.
• A physical interpretation must be declared: in the selected scalar model q = Re W, the imaginary coefficient and auxiliary circle supply no second physical axis or time evolution.