DPST1013 Chap.3 Complex Numbers and Polynomial Roots
Complex Numbers and Polynomial Roots
Complex numbers connect algebraic calculations with geometry in a plane. Cartesian form separates real and imaginary components, while polar form describes the same number through a modulus and an angle. Learning when to change forms is more useful than treating each formula as an isolated rule.
Addition and subtraction favour Cartesian coordinates; multiplication, division, powers and roots often become shorter in polar or exponential form. This chapter also links complex numbers to polynomial factorisation. Real coefficients impose a conjugate-pair structure on nonreal roots, allowing pairs of complex linear factors to combine into real quadratics.
Geometry provides another perspective: a modulus of a difference is a distance, so equations involving complex numbers can describe circles, lines and other sets of points. The central habits are to check quadrants, distinguish an unrestricted argument from its principal value, list every root when a power equation is solved, and verify a factorisation by expanding it.
These checks reveal mistakes that a plausible-looking calculator output can conceal.
What this chapter covers
- 01
Cartesian arithmetic and complex conjugation
- 02
Modulus, argument and the principal interval
- 03
Euler form, multiplication and De Moivre powers
- 04
Complete sets of complex roots
- 05
Distance loci in the Argand plane
- 06
Conjugate roots and real polynomial factors
Recover a complete fourth-root set
- 1Write the nonzero target as 81 exp(i·0). Its arguments are 2πk for integers k. A root therefore has modulus 81 to the power one quarter, which is three; taking a fourth root of only the real magnitude is not yet enough to find the full set.
- 2Divide the possible arguments by four. The root angles are πk/2. Choose k=0,1,2,3 to obtain four consecutive values; any later index repeats a direction because adding four to k adds a whole turn to the root angle.
- 2Evaluate the four directions: angle zero gives 3, π/2 gives 3i, π gives −3, and 3π/2 gives −3i. Each number has fourth power 81, and no two entries represent the same point. The four distinct directions are equally spaced around radius three.
Key terms
- Complex conjugate
- The number obtained by reversing the sign of the imaginary coefficient. Multiplying a number by its conjugate produces its squared modulus, which is real and permits division by a nonzero complex denominator.
- Modulus
- The nonnegative distance of a complex point from the origin. For x+iy it equals the square root of x²+y²; it multiplies across products but generally does not add across sums.
- Principal argument
- The unique argument selected in the interval −π<θ≤π for a nonzero complex number. The negative real axis uses π, and the origin has no argument because it has no direction.
- Polar form
- A representation using a positive modulus and an angle, often written r(cos θ+i sin θ). It describes the same point as Cartesian form and supports efficient multiplication and root calculations.
- Complex locus
- A set of complex points satisfying a given condition. Interpreting |z−a| as a distance often converts its equation into a familiar circle, perpendicular bisector or region in Cartesian coordinates.
- Conjugate pair
- Two complex numbers with the same real part and opposite imaginary parts. Nonreal roots of a real-coefficient polynomial occur in such pairs, and their linear factors combine into a real quadratic.
Complex Numbers and Polynomial Roots FAQ
Why can an inverse tangent give the wrong argument?
The quotient of imaginary and real parts does not retain all quadrant information. Positive ratios occur in both the first and third quadrants. Determine the signs of the original coordinates, find a compatible angle, then adjust it to the principal interval if that is requested.
Does a fourth root always mean four different numbers?
A nonzero complex target has four distinct fourth roots. The target zero is different: its only root value is zero, with multiplicity four in the polynomial equation. The distinction between distinct values and multiplicity prevents an incorrect completeness claim.
Can I add two numbers by adding their moduli and arguments?
Multiplication multiplies moduli and adds arguments; addition has no corresponding rule. Convert each summand to Cartesian coordinates, combine the real and imaginary parts, and then convert the resulting point back to polar form if the requested answer needs a modulus and argument.
When does a complex root imply its conjugate is also a root?
The polynomial must have real coefficients. Conjugating its value then leaves those coefficients unchanged, so a zero at one nonreal number implies a zero at its conjugate. With genuinely complex coefficients, that conclusion is not generally justified.
How can I check a locus equation without relying on a sketch?
Translate z into x+iy and express each modulus as a distance. For an equality of distances, square the nonnegative sides and simplify. Then test a point known geometrically to belong, such as a midpoint for a perpendicular bisector, in the original equation.
Exam move
Make a small decision table headed operation, useful representation and final check. Include addition, division, powers, roots and loci, but write a different verification for each. Division should multiply back to the original numerator; a root list should check both powers and completeness; a locus should satisfy the original distance condition.
Practise argument conversion using one number from every quadrant and one on the negative real axis. Write the reference angle and the selected principal angle separately. This makes sign errors visible before they enter a long power calculation. For root questions, first work entirely in exact exponential form. Mark the distinct angles around a circle, then convert only the values whose Cartesian forms are required.
When using conjugate roots to factor a real polynomial, build the quadratic pair before multiplying other factors, and expand the final product to check the missing coefficients as well as the visible ones.
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