151 Complex Numbers Terms and Definitions | Mathematics Board Exam Review

INTRODUCTION

Complex numbers show up in the Mathematics engineering board exams more often than most reviewees expect. Whether you are taking the ECE, EE, ME, or CE licensure examination, you will encounter problems that require you to work with numbers that have both a real and an imaginary part. The challenge is not just knowing how to compute. It is knowing the language. Board exam questions are written using specific mathematical terms, and if you do not recognize the vocabulary, even a concept you understand can trip you up.

This glossary covers 151 complex numbers terms drawn from the full scope of the complex numbers topic as it appears in Philippine engineering mathematics. You will find entries on number forms, operations, graphical representations, theoretical foundations, and engineering applications. Each definition is written with the board exam in mind. That means clarity first, technical precision second, and exam relevance throughout.

Complex numbers bridge the gap between pure mathematics and real engineering problems. In circuit analysis, impedances are expressed as complex numbers. In signal processing, phase shifts are handled using polar and exponential forms. In control systems, stability is analyzed using roots plotted on the complex plane. Understanding the vocabulary of this topic does not just help you answer math problems. It helps you speak the same language as the subjects that build on top of it.

Use this glossary as your foundation review before working through practice problems. Read the terms, absorb the definitions, and pay attention to the distinctions between closely related concepts. The board exam loves to test those fine differences. Knowing that a purely imaginary number has no real part, or that the modulus is always non-negative, or that conjugate pairs always produce a real product. These small details are exactly what separates a passing score from a failing one.

The 151 Complex Numbers Terms and Definitions

1. Absolute Value of a Complex Number

The distance of a complex number from the origin on the Argand plane, computed as the square root of the sum of the squares of the real and imaginary parts. For z = a + bi, the absolute value is sqrt(a² + b²). This is the same as the modulus and is always a non-negative real number.

2. Addition of Complex Numbers

The operation of combining two complex numbers by adding their real parts together and their imaginary parts together separately. For (a + bi) + (c + di), the result is (a + c) + (b + d)i. Addition is commutative and associative for complex numbers.

3. Additive Identity

The complex number 0 + 0i, or simply 0, which when added to any complex number leaves it unchanged. It is the same additive identity as in real number arithmetic, extended to the complex number system.

4. Additive Inverse

The complex number that when added to a given complex number produces zero. For z = a + bi, the additive inverse is −a − bi, also written as −z. It is the reflection of z through the origin on the Argand plane.

5. Amplitude

Another term for the argument or angle of a complex number in polar form. It measures the counterclockwise angle from the positive real axis to the line segment connecting the origin to the point representing the complex number. Also called the phase in engineering contexts.

6. Argument

The angle θ that a complex number makes with the positive real axis on the Argand plane, measured counterclockwise. For z = a + bi, the argument is θ = arctan(b/a), adjusted for the correct quadrant. Denoted arg(z) and expressed in radians or degrees.

7. Argument Convention

The standard rule for expressing the argument of a complex number, typically choosing the principal value in the interval (−π, π] or sometimes in [0, 2π). Different problems may use different conventions, and it is important to identify which interval is being used, especially when finding roots.

8. Argand Diagram

A graphical representation of complex numbers on a two-dimensional plane where the horizontal axis represents the real part and the vertical axis represents the imaginary part. Named after Jean-Robert Argand who popularized this geometric interpretation in 1806.

9. Argand Plane

The coordinate plane used to represent complex numbers visually, with the real axis as the horizontal axis and the imaginary axis as the vertical axis. Every complex number a + bi corresponds to the point (a, b) on this plane.

10. Associative Property

The property stating that the grouping of complex numbers does not affect the result of addition or multiplication. For addition, (z₁ + z₂) + z₃ = z₁ + (z₂ + z₃). For multiplication, (z₁ · z₂) · z₃ = z₁ · (z₂ · z₃).

11. Binomial Form

An expression of a complex number written as the sum of two terms: a real part and an imaginary part. The standard form a + bi is a binomial form. Also referred to as rectangular form or Cartesian form.

12. Bound Vector

A representation of a complex number as an arrow fixed at the origin of the Argand plane and pointing to the location (a, b). The length of the arrow equals the modulus and the direction equals the argument. This interpretation connects complex numbers to vector algebra.

13. Branch Cut

A curve in the complex plane along which a multi-valued complex function is made discontinuous in order to define a single-valued branch of that function. Commonly used when defining the complex logarithm and fractional powers. The standard branch cut for the logarithm lies along the negative real axis.

14. Branch Point

A point in the complex plane around which a multi-valued function fails to return to its original value after the argument completes a full revolution. The origin is the branch point for functions like the square root and logarithm.

15. Cartesian Form

The standard representation of a complex number as z = a + bi, where a is the real part and b is the imaginary part. Named after René Descartes, this form uses rectangular coordinates and is the most common form for performing addition and subtraction.

16. Cis Notation

The shorthand notation cis θ, which stands for cos θ + i sin θ. It is used to write polar form compactly as z = r cis θ. This notation is common in Mathematics engineering board exam problems and is equivalent to r∠θ and re^(iθ).

17. Closed Form Expression

A mathematical expression for a complex quantity that can be written exactly using a finite number of standard operations and functions. Board exam problems often ask for the closed form of a complex power or root rather than a numerical approximation.

18. Closure Property

The property stating that the sum, difference, product, and quotient (excluding division by zero) of any two complex numbers is always another complex number. The set of all complex numbers is closed under these four arithmetic operations.

19. Complete Solution Set

The full set of solutions to a complex equation, including all distinct roots or values. For an nth degree polynomial, the complete solution set contains exactly n complex roots counted with multiplicity, as stated by the Fundamental Theorem of Algebra.

20. Complex Conjugate

The complex number obtained by changing the sign of the imaginary part of a given complex number. The conjugate of a + bi is a − bi, denoted with a bar over z as z̄. Multiplying a complex number by its conjugate always produces a non-negative real number equal to the square of the modulus.

21. Complex Conjugate Root Theorem

The theorem asserting that if a polynomial has real coefficients and a + bi is a root, then a − bi is necessarily also a root. This theorem is directly applicable to board exam problems asking for all roots of a real-coefficient polynomial when some roots involve i.

22. Complex Equation

An equation that involves complex numbers, either explicitly through the presence of i, or implicitly by having complex solutions. Solving complex equations often requires separating real and imaginary parts or working in polar form.

23. Complex Exponential

The expression e^(iθ) defined by Euler’s formula as cos θ + i sin θ. This form connects the exponential function to trigonometric functions in the complex number system and is central to Fourier analysis, signal processing, and differential equations.

24. Complex Field

The algebraic structure formed by the set of all complex numbers together with the operations of addition and multiplication. The complex field satisfies all field axioms and is algebraically closed, meaning every non-constant polynomial with complex coefficients has at least one complex root.

25. Complex Fraction

A fraction in which the numerator, denominator, or both contain complex numbers. To simplify a complex fraction, multiply both numerator and denominator by the conjugate of the denominator to eliminate the imaginary part from the denominator.

26. Complex Function

A function whose domain, codomain, or both consist of complex numbers. A complex function f(z) maps complex inputs to complex outputs and is the subject of complex analysis, which studies differentiability, integration, and series representation in the complex plane.

27. Complex Integer

A complex number in which both the real and imaginary parts are integers. Also called a Gaussian integer. The set of Gaussian integers forms a ring under addition and multiplication and has important applications in number theory.

28. Complex Number

A number of the form a + bi, where a and b are real numbers and i is the imaginary unit defined as the square root of −1. The real number a is the real part and the real number b is the imaginary part. Complex numbers extend the real number system to allow solutions to equations like x² + 1 = 0.

29. Complex Number Equality

The condition under which two complex numbers are considered the same: both their real parts and imaginary parts must be simultaneously equal. This gives a system of two real equations from one complex equation, a technique frequently used to solve board exam problems.

30. Complex Number System

The complete system of numbers consisting of all numbers of the form a + bi, where a and b are real. It includes the real numbers (when b = 0), the purely imaginary numbers (when a = 0), and all combinations of real and imaginary parts. The complex number system is the broadest standard number system used in engineering mathematics.

31. Complex Plane

The plane on which complex numbers are represented as points or vectors, with the real part plotted along the horizontal axis and the imaginary part plotted along the vertical axis. Also called the z-plane or the Argand plane.

32. Complex Plane Geometry

The study of geometric figures and transformations using complex numbers. Circles, lines, rotations, reflections, and inversions can all be described compactly using complex number equations. This approach unifies algebra and geometry in a powerful way.

33. Complex Polynomial

A polynomial expression whose coefficients are complex numbers. By the Fundamental Theorem of Algebra, every complex polynomial of degree n has exactly n roots counted with multiplicity in the complex number system.

34. Complex Root

A root of a polynomial equation that is a complex number with a non-zero imaginary part. For polynomials with real coefficients, complex roots always appear in conjugate pairs. If a + bi is a root, then a − bi is also a root.

35. Complex Trigonometric Functions

The extension of sine and cosine to complex arguments, defined using Euler’s formula. For a complex number z, sin(z) = (e^(iz) − e^(−iz))/(2i) and cos(z) = (e^(iz) + e^(−iz))/2. These definitions are consistent with the real-valued trigonometric functions when z is real.

36. Complex Variable

A variable that represents a complex number, typically denoted z. The study of functions of a complex variable is called complex analysis or the theory of functions of a complex variable, and it forms the mathematical foundation of many engineering tools.

37. Conjugate Pair

Two complex numbers of the form a + bi and a − bi that are conjugates of each other. Their sum is 2a, a real number. Their product is a² + b², also a real number. Conjugate pairs are always roots of polynomials with real coefficients.

38. Conjugate Product Property

The result that the product of a complex number and its conjugate equals the square of its modulus: z · z̄ = |z|². This property is the cornerstone of the division process and is used to convert a complex denominator into a real one.

39. Conjugate Quotient Property

The result that the conjugate of a quotient equals the quotient of the conjugates: (z₁/z₂)̄ = z̄₁/z̄₂ for non-zero z₂. This property, along with its analog for sums and products, simplifies the manipulation of complex expressions involving conjugates.

40. Conjugate Roots Theorem

The theorem stating that if a polynomial with real coefficients has a complex root a + bi (with b ≠ 0), then its conjugate a − bi is also a root. This theorem guarantees that complex roots come in pairs for real-coefficient polynomials.

41. Conjugate Sum Property

The result that the conjugate of a sum equals the sum of the conjugates: (z₁ + z₂)̄ = z̄₁ + z̄₂. Similarly, the conjugate of a product equals the product of the conjugates: (z₁z₂)̄ = z̄₁ · z̄₂. These properties allow conjugation to be distributed across arithmetic operations.

42. Conjugate Symmetry

The property of a function or expression where replacing z with its conjugate z̄ produces the conjugate of the original result. A function satisfying f(z̄) = f(z)̄ is said to have conjugate symmetry, which is a key property of Fourier transforms in signal processing.

43. Continuous Function

A complex function f(z) is continuous at a point z₀ if the limit of f(z) as z approaches z₀ equals f(z₀). Continuity in the complex plane requires that the function value remains close to f(z₀) regardless of the direction from which z approaches z₀.

44. Conversion from Polar to Rectangular

The process of expressing a complex number given in polar form r∠θ or r(cos θ + i sin θ) in rectangular form a + bi by computing a = r cos θ and b = r sin θ. This conversion is frequently tested in board exams involving circuit impedance and phasors.

45. Conversion from Rectangular to Polar

The process of expressing a complex number a + bi in polar form by computing the modulus r = sqrt(a² + b²) and the argument θ = arctan(b/a) with appropriate quadrant adjustment. This is needed when applying De Moivre’s theorem or multiplying complex numbers efficiently.

46. Cubic Roots of Unity

The three complex numbers that satisfy the equation z³ = 1. One root is the real number 1, and the other two are complex: (−1 + i√3)/2 and (−1 − i√3)/2. These roots are equally spaced at 120° intervals on the unit circle.

47. De Moivre’s Theorem

The theorem stating that for any real number n and complex number in polar form, (r∠θ)ⁿ = rⁿ∠(nθ), or equivalently (cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ). It is the primary tool for computing powers and roots of complex numbers in polar form.

48. Difference of Complex Numbers

The result of subtracting one complex number from another by subtracting the real parts and the imaginary parts separately. For (a + bi) − (c + di), the result is (a − c) + (b − d)i.

49. Direction Angle

The angle that the vector representation of a complex number makes with the positive real axis, measured counterclockwise. It is the same as the argument of the complex number and determines the direction of the corresponding vector on the Argand plane.

50. Distance Between Two Complex Numbers

The modulus of the difference between two complex numbers: d(z₁, z₂) = |z₁ − z₂| = sqrt((a₁ − a₂)² + (b₁ − b₂)²). This formula is the complex plane analog of the distance formula in coordinate geometry and appears in locus problems.

51. Distributive Property

The property that multiplication distributes over addition for complex numbers. For any complex numbers z₁, z₂, and z₃, z₁(z₂ + z₃) = z₁z₂ + z₁z₃. This property is essential for expanding products of complex expressions.

52. Division of Complex Numbers

The operation of dividing one complex number by another, performed by multiplying both numerator and denominator by the conjugate of the denominator. For (a + bi)/(c + di), multiply top and bottom by (c − di) to obtain a real denominator and separate real and imaginary parts.

53. Equality of Complex Numbers

Two complex numbers a + bi and c + di are equal if and only if a = c and b = d. This means both the real parts and the imaginary parts must be equal simultaneously. Equality is the basis for solving complex equations by separating components.

54. Euler’s Formula

The identity e^(iθ) = cos θ + i sin θ, which establishes a deep connection between the complex exponential function and the trigonometric functions. It is one of the most important formulas in mathematics and engineering, providing the foundation for the exponential form of complex numbers.

55. Euler’s Identity

The special case of Euler’s formula when θ = π, giving e^(iπ) + 1 = 0. This equation relates five fundamental mathematical constants: e, i, π, 1, and 0, and is widely regarded as one of the most elegant results in all of mathematics.

56. Even Function (Complex)

A complex function f(z) is even if f(−z) = f(z) for all z. The real part of an analytic function satisfying this condition is an even function of z, while the imaginary part behaves accordingly. Even complex functions appear frequently in Fourier series with symmetric intervals.

57. Exponential Form

The representation of a complex number using Euler’s formula as z = re^(iθ), where r is the modulus and θ is the argument. This form is equivalent to polar form and is particularly compact and powerful for multiplication, division, and computing powers.

58. Extended Complex Plane

The complex plane augmented by a single point at infinity, also called the Riemann sphere or the one-point compactification of the complex plane. Adding the point at infinity allows functions like 1/z to be defined at z = 0 by mapping it to infinity.

59. Fixed Point

A complex number z₀ that satisfies f(z₀) = z₀ for a given function f. Fixed points play an important role in the analysis of iteration, fractals, and dynamical systems in the complex plane.

60. Fourth Roots of Unity

The four complex solutions to z⁴ = 1, which are 1, i, −1, and −i. These are equally spaced at 90° intervals on the unit circle. The imaginary unit i is one of the four fourth roots of unity.

61. Free Vector

A representation of a complex number as an arrow that can be positioned anywhere in the plane, as opposed to a bound vector which must start at the origin. The direction and length of the free vector correspond to the argument and modulus of the complex number.

62. Fundamental Theorem of Algebra

The theorem stating that every non-constant polynomial of degree n with complex coefficients has exactly n complex roots, counted with multiplicity. This theorem guarantees that the complex number system is algebraically complete and closed under root-finding.

63. Gaussian Integer

A complex number of the form a + bi where both a and b are integers. Gaussian integers form a ring that generalizes the behavior of ordinary integers to the complex plane, and they have applications in number theory and cryptography.

64. Geometric Interpretation of Multiplication

When two complex numbers are multiplied, the resulting modulus is the product of their moduli and the resulting argument is the sum of their arguments. Geometrically, multiplication by a complex number r∠θ rotates a vector by θ and scales it by r.

65. Geometric Series (Complex)

An infinite series of the form 1 + z + z² + z³ + … where z is a complex number. When |z| < 1, the series converges to 1/(1 − z). Complex geometric series appear in signal processing, z-transforms, and Fourier analysis.

66. Half-Plane

A region of the complex plane consisting of all complex numbers whose real parts (or imaginary parts) satisfy a given inequality. The right half-plane refers to all z with Re(z) > 0, which is important in stability analysis and Laplace transforms.

67. Harmonic Conjugate

A real-valued function v(x, y) that is related to another real-valued function u(x, y) through the Cauchy-Riemann equations. If u and v satisfy these equations, then f(z) = u + iv is an analytic function. The functions u and v are called harmonic conjugates of each other.

68. Harmonic Function

A real-valued function that satisfies Laplace’s equation, meaning its second partial derivatives sum to zero. The real and imaginary parts of an analytic complex function are both harmonic functions. Harmonic functions appear in heat transfer, fluid flow, and potential theory.

69. Horizontal Component

The real part of a complex number, corresponding to its projection onto the real axis of the Argand plane. For z = a + bi, the horizontal component is a. In engineering applications involving phasors, the horizontal component represents the in-phase component.

70. Hyperbolic Functions (Complex)

The complex analogs of the real hyperbolic sine and cosine, defined using the complex exponential. The complex hyperbolic cosine is cosh(z) = (e^z + e^(−z))/2 and the hyperbolic sine is sinh(z) = (e^z − e^(−z))/2. These are related to the circular trigonometric functions through the imaginary unit.

151. Zero Complex Number

The complex number 0 + 0i, located at the origin of the Argand plane. It is the additive identity of the complex number system. Its modulus is 0 and its argument is undefined since it has no direction.

Conclusion

For the Mathematics engineering board exam, complex numbers demand both conceptual fluency and operational skill. The most tested areas are operations in rectangular and polar form, powers and roots using De Moivre’s theorem, the behavior of i and its integer powers, and conversions between the different forms of representing a complex number. If you are an ECE or EE candidate, you also need to be completely comfortable with the phasor interpretation and the use of j notation in circuit problems. These topics appear not just in the mathematics section but in your major subjects as well.

Pay particular attention to the terms that involve conditions and properties rather than just definitions. The conjugate root theorem, the triangle inequality, the modulus-argument relationship, and the Fundamental Theorem of Algebra are all sources of problem types that the PRC examinations have historically used. Knowing what a term means is only half the work. You need to know what it implies, what conditions it requires, and how to apply it when the problem does not explicitly name the technique you should use.

Finally, build fluency with the graphical interpretation. Many complex number problems become significantly easier when you can visualize what is happening on the Argand plane. Distance between two complex numbers, locus conditions, and rotation problems all have clean geometric meanings that rectangular form computation alone does not reveal. Strong reviewees combine algebraic skill with geometric intuition, and that combination is what allows them to work quickly and accurately under exam conditions.

For practice problems on all these topics, head over to our Advanced Mathematics Problems and Solutions section here on PinoyBix. Hundreds of solved exam-type questions, complete with step-by-step solutions, organized by topic so you can drill exactly what you need to work on.