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.

71. i (Imaginary Unit)

The fundamental imaginary unit defined by i² = −1. It is the principal square root of negative one. Using i, the square root of any negative real number can be expressed as a real multiple of i, and the full complex number system can be constructed from real numbers and i.

72. Imaginary Axis

The vertical axis of the Argand plane, representing purely imaginary numbers. Points on the imaginary axis are of the form 0 + bi, which corresponds to bi for any real number b. The imaginary axis is perpendicular to the real axis.

73. Imaginary Part

The coefficient b in a complex number a + bi, written as Im(z) = b. It is a real number despite being called imaginary. Two complex numbers are equal only when both their real parts and imaginary parts match exactly.

74. Imaginary Unit Powers

The cyclic pattern followed by successive powers of i: i¹ = i, i² = −1, i³ = −i, i⁴ = 1, and then the pattern repeats. To evaluate iⁿ for any integer n, divide n by 4 and use the remainder to determine the result. This cycle is a common source of board exam questions.

75. Impedance (Complex)

In electrical and electronics engineering, the opposition to alternating current expressed as a complex number Z = R + jX, where R is the resistance (real part) and X is the reactance (imaginary part). The use of j instead of i is standard in electrical and electronics engineering to avoid confusion with current.

76. Inequality of Moduli

The triangle inequality applied to complex numbers, stating that |z₁ + z₂| ≤ |z₁| + |z₂| for any two complex numbers z₁ and z₂. Equality holds when both numbers have the same argument, meaning they point in the same direction.

77. Infinity in the Complex Plane

The single point at infinity added to the complex plane to form the extended complex plane. Unlike the real line which has two infinities (positive and negative), the complex plane has only one point at infinity, approached by going to infinity in any direction.

78. Inner Product (Complex)

An extension of the dot product to complex vector spaces, defined to satisfy conjugate symmetry, linearity in the first argument, and positive definiteness. For complex numbers z₁ and z₂ treated as one-dimensional complex vectors, the inner product is z₁z̄₂.

79. Integer Power of i

Any power of the imaginary unit i, evaluated using the four-cycle i¹ = i, i² = −1, i³ = −i, i⁴ = 1. For iⁿ, compute n mod 4: if the remainder is 0, the result is 1; if 1, the result is i; if 2, the result is −1; if 3, the result is −i. This cycle is a favorite topic in licensure exam problems.

80. Inverse of a Complex Number

The multiplicative inverse of a non-zero complex number z = a + bi, computed as z̄/(|z|²) = (a − bi)/(a² + b²). The inverse satisfies z · z⁻¹ = 1 and is used in division. A complex number has a multiplicative inverse if and only if it is non-zero.

81. j (Engineering Notation)

The symbol used in electrical and electronics engineering in place of i to represent the imaginary unit, since i is reserved for electrical current in circuit analysis. The notation j is standard in all EE and ECE courses, and board exam problems in these fields consistently use j rather than i.

82. Locus

The set of all complex numbers z that satisfy a given condition, plotted as a curve or region on the Argand plane. For example, the locus of all z satisfying |z| = r is a circle of radius r centered at the origin. Locus problems are common in advanced board exam questions.

83. Magnitude

Another term for the modulus or absolute value of a complex number, representing its size or distance from the origin. The magnitude of a + bi is sqrt(a² + b²). In engineering, the magnitude of a phasor gives the peak or RMS amplitude of the corresponding sinusoidal signal.

84. Modulus

The non-negative real number r = |z| = sqrt(a² + b²) representing the distance from the origin to the point z = a + bi on the Argand plane. The modulus is always non-negative and equals zero only when the complex number is zero. Also called the absolute value or magnitude.

85. Modulus-Argument Form

The same as polar form. Expressing a complex number using its modulus and argument as z = r(cos θ + i sin θ). This form emphasizes the two key properties of a complex number: its size (modulus) and direction (argument) on the Argand plane.

86. Multiple-Valued Function

A complex function that returns more than one value for a single input. The complex square root, cube root, and logarithm are all multi-valued. To work with them practically, a branch is selected to produce a single-valued principal value function.

87. Multiplication of Complex Numbers

The operation of multiplying two complex numbers by applying the distributive property and using i² = −1 to simplify. For (a + bi)(c + di), the result is (ac − bd) + (ad + bc)i. In polar form, multiplication multiplies the moduli and adds the arguments.

88. Multiplicative Identity

The complex number 1 + 0i, or simply 1, which when multiplied by any complex number leaves it unchanged. It serves the same role as the number 1 in the real number system.

89. Multiplicative Property of Moduli

The property stating that the modulus of a product equals the product of the moduli: |z₁ · z₂| = |z₁| · |z₂|. This is one of the most useful properties in polar form multiplication and extends to any finite number of factors.

90. Multiplicative Inverse

See Inverse of a Complex Number. The multiplicative inverse of z is the number 1/z, which when multiplied by z gives the identity element 1.

91. Natural Logarithm of a Complex Number

The complex-valued function ln(z) = ln(r) + iθ, where r is the modulus and θ is the argument. It is multi-valued because θ can be increased by any multiple of 2π. The principal value uses the argument in the range (−π, π].

92. Negative Imaginary Number

A complex number of the form 0 + bi where b is a negative real number, representing a point on the negative imaginary axis. In electrical engineering, a negative reactance corresponds to a capacitive element in a circuit.

93. Negative Real Number (as complex)

A real number a where a < 0, expressed as a + 0i in the complex number system. It lies on the negative real axis of the Argand plane and has argument θ = π.

94. Non-Real Complex Number

A complex number a + bi where b ≠ 0, meaning it has a non-zero imaginary part and cannot be represented on the real number line. All such numbers require the full complex plane for their representation.

95. Norm of a Complex Number

Another term for the modulus or absolute value, sometimes defined as |z|² = a² + b² (the square of the modulus) in some algebraic contexts. The norm satisfies the multiplicative property: the norm of a product equals the product of the norms.

96. nth Root of a Complex Number

Any complex number w that satisfies wⁿ = z for a given complex number z. Using De Moivre’s theorem, there are exactly n distinct nth roots, given by w_k = r^(1/n) ∠ ((θ + 2πk)/n) for k = 0, 1, 2, …, n − 1. The roots are equally spaced on a circle of radius r^(1/n).

97. nth Roots of Unity

The n complex solutions to the equation zⁿ = 1. They are equally spaced on the unit circle at angular intervals of 2π/n. The principal nth root of unity is ω = e^(2πi/n) = cos(2π/n) + i sin(2π/n), and the other roots are its successive powers.

98. Odd Function (Complex)

A complex function f(z) satisfying f(−z) = −f(z). The complex sine function is odd, while the complex cosine function is even. Decomposing functions into odd and even parts is useful in Fourier analysis and series representation.

99. One-to-One Function (Complex)

A complex function in which each element of the range corresponds to exactly one element of the domain. Also called injective. Complex analytic functions that are one-to-one are called univalent or schlicht functions and are important in conformal mapping.

100. Operations on Complex Numbers

The four basic arithmetic operations namely addition, subtraction, multiplication, and division are applied to complex numbers. All four operations produce complex number results. Rectangular form is best for addition and subtraction, while polar form simplifies multiplication, division, and exponentiation.

101. Ordered Pair Representation

The representation of a complex number a + bi as the ordered pair (a, b), treating it as a point in two-dimensional space. Under this representation, addition corresponds to vector addition and multiplication follows a specific rule that encodes the behavior of i.

102. Parallelogram Law

The geometric rule for adding two complex numbers by placing them as adjacent sides of a parallelogram, with the resulting sum being the diagonal from the origin. This is the same as vector addition and illustrates why complex numbers behave like two-dimensional vectors under addition.

103. Phase

The argument of a complex number expressed in the context of sinusoidal signals or phasors. In circuit analysis, the phase of a phasor indicates the timing relationship between the voltage or current and a reference sinusoid. Phase is typically measured in degrees or radians.

104. Phase Angle

The argument θ of a complex number written in polar form, specifically in the context of electrical engineering and wave analysis. A positive phase angle means the signal leads the reference, while a negative phase angle means the signal lags.

105. Phasor

A complex number used to represent a sinusoidal quantity in steady-state AC circuit analysis. A phasor encodes both the amplitude (modulus) and the phase angle (argument) of the sinusoid. Phasor arithmetic transforms differential equations into algebraic complex number equations.

106. Polar Coordinates

The coordinate system used to describe a complex number by its distance from the origin (the modulus r) and its angle from the positive real axis (the argument θ). The connection between polar and rectangular coordinates is given by a = r cos θ and b = r sin θ.

107. Polar Form

The representation of a complex number as z = r(cos θ + i sin θ) or equivalently z = r∠θ, where r is the modulus and θ is the argument. Polar form is the preferred form for multiplication, division, raising to a power, and finding roots.

108. Positive Imaginary Number

A complex number of the form 0 + bi where b is a positive real number, lying on the positive imaginary axis of the Argand plane. It has an argument of π/2 or 90°.

109. Power of a Complex Number

The result of raising a complex number to an integer or rational exponent. For integer powers, De Moivre’s theorem provides the direct formula: (r∠θ)ⁿ = rⁿ∠(nθ). For fractional exponents, the result involves roots of the complex number.

110. Principal Argument

The unique value of the argument of a complex number chosen to lie in the interval (−π, π], denoted Arg(z) with a capital A. It is the standard branch of the multi-valued argument function and is used to define the principal logarithm and principal roots.

111. Principal Root

The specific nth root of a complex number selected by using the principal argument and the positive real value of r^(1/n). When multiple roots exist, the principal root is the one corresponding to k = 0 in the nth root formula using the principal argument.

112. Principal Value

The specific value selected from the multiple values of a multi-valued complex function by applying a standard branch choice. The principal value is the most commonly used value and is determined by restricting the argument to the principal range (−π, π].

113. Product of Conjugates

The result of multiplying a complex number by its conjugate: (a + bi)(a − bi) = a² + b². This product is always a non-negative real number equal to the square of the modulus. It is the key technique used in rationalizing a complex denominator during division.

114. Proper Complex Number

A complex number with a non-zero imaginary part, meaning it is not purely real. It is a number of the form a + bi where b ≠ 0 and both a and b are real. This distinguishes it from real numbers, which are a subset of the complex numbers.

115. Pure Imaginary Number

A complex number with a real part equal to zero and a non-zero imaginary part, of the form 0 + bi = bi where b is a non-zero real number. Pure imaginary numbers lie on the imaginary axis of the Argand plane.

116. Pythagorean Relationship

The geometric basis for computing the modulus of a complex number. Since a complex number a + bi corresponds to the point (a, b), the modulus r = sqrt(a² + b²) follows directly from the Pythagorean theorem applied to the right triangle formed by a, b, and r.

117. Quadrant of a Complex Number

The region of the Argand plane in which a complex number lies, determined by the signs of its real and imaginary parts. The first quadrant has positive real and positive imaginary parts, the second has negative real and positive imaginary parts, the third has both negative, and the fourth has positive real and negative imaginary parts.

118. Quotient of Complex Numbers

The result of dividing one complex number by another. In rectangular form, multiply numerator and denominator by the conjugate of the denominator. In polar form, divide the moduli and subtract the arguments: (r₁∠θ₁)/(r₂∠θ₂) = (r₁/r₂)∠(θ₁ − θ₂).

119. Radial Distance

The distance from the origin to a point on the Argand plane, equal to the modulus of the corresponding complex number. The radial distance r satisfies r ≥ 0 and is the magnitude of the complex number expressed in polar or exponential form.

120. Real and Imaginary Part Extraction

The technique of identifying the real and imaginary components of a complex expression, often used to solve complex equations by splitting them into two real equations. Re(z) = (z + z̄)/2 and Im(z) = (z − z̄)/(2i) provide formulas for extracting each part.

121. Real Axis

The horizontal axis of the Argand plane, representing the set of all real numbers embedded in the complex plane as numbers of the form a + 0i. The real axis divides the Argand plane into the upper half-plane (positive imaginary part) and the lower half-plane (negative imaginary part).

122. Real Number (as complex)

Any real number a, viewed as the complex number a + 0i with zero imaginary part. The real numbers are a subset of the complex numbers and correspond to points on the real axis of the Argand plane.

123. Real Part

The component a in the complex number a + bi, written as Re(z) = a. It is a real number representing the horizontal coordinate of the complex number on the Argand plane. Two complex numbers are equal only if both real parts and imaginary parts are equal.

124. Rectangular Form

The standard expression of a complex number as z = a + bi using Cartesian coordinates, where a is the real part and b is the imaginary part. Also called binomial or Cartesian form. This is the most natural form for addition, subtraction, and solving systems of complex equations.

125. Rectangular to Exponential Conversion

The process of writing z = a + bi as z = re^(iθ), obtained by computing r = sqrt(a² + b²) and θ = arctan(b/a) with quadrant correction. This conversion is needed when applying Euler’s formula or simplifying products and powers.

126. Roots of Polynomials

The values of z that make a polynomial expression equal to zero. Over the complex number system, every polynomial of degree n has exactly n roots counted with multiplicity, as guaranteed by the Fundamental Theorem of Algebra.

127. Rotation

The geometric operation on the complex plane performed by multiplying a complex number z by e^(iθ) = cos θ + i sin θ. This rotates z counterclockwise by the angle θ without changing its modulus. Multiplication by i rotates by 90°, and multiplication by −1 rotates by 180°.

128. Rotation by π/2

The operation of multiplying a complex number by i, which rotates its vector representation by 90° counterclockwise on the Argand plane without changing its modulus. Multiplying by −i rotates by 90° clockwise. This property explains why i² = −1 corresponds to a 180° rotation.

129. Scaling

The operation of multiplying a complex number by a positive real constant k, which changes the modulus to kr without affecting the argument. Scaling stretches or compresses the vector representation of the complex number toward or away from the origin.

130. Set of Complex Numbers

Denoted ℂ, the set of all numbers of the form a + bi where a and b are real. The set ℂ includes the real numbers ℝ as a subset (when b = 0) and the purely imaginary numbers (when a = 0). The complex numbers form a field that is algebraically closed.

131. Signum Function (Complex)

The function defined as sgn(z) = z/|z| for any non-zero complex number z. It returns the unit complex number in the same direction as z, effectively normalizing z to the unit circle. It is undefined at z = 0.

132. Similar Triangles (Complex Geometry)

A geometric relationship between triangles formed by complex numbers on the Argand plane, which can be used to establish proportionality conditions and prove geometric theorems using complex number algebra.

133. Square Root of a Complex Number

The complex number w such that w² = z. Every non-zero complex number has exactly two square roots that are negatives of each other. For z = r∠θ, the square roots are sqrt(r)∠(θ/2) and sqrt(r)∠(θ/2 + π). The square roots of a negative real number are purely imaginary.

134. Square Root of Negative One

The imaginary unit i, defined as the principal square root of −1. This definition extends the real number system to allow solutions to equations like x² = −1. The two square roots of −1 are i and −i.

135. Standard Form

The expression of a complex number as a + bi, with the real part written first followed by the imaginary part. This is the universally accepted format for presenting a complex number and is the form expected in most board exam answers unless otherwise specified.

136. Subset Relationships

The hierarchical relationship among number systems: natural numbers ⊂ integers ⊂ rational numbers ⊂ real numbers ⊂ complex numbers. Every real number is a complex number with zero imaginary part, but not every complex number is real.

137. Subtraction of Complex Numbers

The operation of finding the difference between two complex numbers by subtracting the real parts and imaginary parts separately. For (a + bi) − (c + di), the result is (a − c) + (b − d)i. Geometrically, subtraction produces the vector from the second number to the first.

138. Sum of a Complex Number and Its Conjugate

For z = a + bi, the sum z + z̄ = (a + bi) + (a − bi) = 2a. This sum is always a real number equal to twice the real part of z. This property is used to extract the real part from a complex expression.

139. Symmetric Point

The point on the Argand plane that is the reflection of a complex number z across a given axis or point. The reflection across the real axis gives the conjugate z̄. The reflection across the imaginary axis gives −z̄. The reflection through the origin gives −z.

140. Trigonometric Form

Another name for polar form, expressing a complex number as z = r(cos θ + i sin θ). Sometimes abbreviated as z = r cis θ, where cis stands for cosine plus i sine. This form directly connects complex numbers to trigonometry and is the foundation for De Moivre’s theorem.

141. Triangle Inequality

The inequality stating that |z₁ + z₂| ≤ |z₁| + |z₂| for any two complex numbers, reflecting the geometric fact that the length of one side of a triangle cannot exceed the sum of the other two sides. It applies to any number of terms: |z₁ + z₂ + … + zₙ| ≤ |z₁| + |z₂| + … + |zₙ|.

142. Trigonometric Identities (Complex)

Standard trigonometric identities extended to complex arguments using Euler’s formula. For example, cos(α + β) = cos α cos β − sin α sin β and sin(α + β) = sin α cos β + cos α sin β can be derived efficiently using the exponential form of complex numbers.

143. Unit Circle

The circle of radius 1 centered at the origin of the Argand plane, described by |z| = 1. All complex numbers on the unit circle have the form e^(iθ) = cos θ + i sin θ. The nth roots of unity are equally spaced points on the unit circle.

144. Unit Complex Number

A complex number with modulus equal to 1, lying on the unit circle. Every unit complex number can be written as e^(iθ) = cos θ + i sin θ for some angle θ. Unit complex numbers represent pure rotations in the complex plane.

145. Unit Imaginary Number

The imaginary unit i itself, which is the unit complex number at angle 90° on the unit circle. Its properties include i² = −1, i³ = −i, and i⁴ = 1. It is the base element from which all imaginary and complex numbers are constructed.

146. Vector Addition (Complex)

The interpretation of complex number addition as vector addition on the Argand plane. Adding a + bi and c + di corresponds to placing the vectors (a, b) and (c, d) tip-to-tail and finding the resultant (a + c, b + d). This is the parallelogram rule applied to complex numbers.

147. Vector Representation

The representation of a complex number a + bi as the directed line segment or arrow from the origin to the point (a, b) on the Argand plane. The length of the arrow is the modulus and its direction angle is the argument. This connects complex number theory to vector analysis.

148. Vertical Component

The imaginary part of a complex number, corresponding to its projection onto the imaginary axis of the Argand plane. For z = a + bi, the vertical component is b. In phasor analysis, the vertical component represents the out-of-phase or quadrature component of a signal.

149. z-Notation

The conventional use of the letter z to represent a complex variable or a general complex number, paralleling the use of x for real variables. The complex plane is often called the z-plane in this notation. Complex functions are typically written as f(z) or w = f(z).

150. z-Plane

The complex plane expressed in terms of the complex variable z = x + iy, commonly used in complex analysis, Laplace transforms, and control systems. The z-plane is the domain on which complex functions f(z) are analyzed, and its geometry  including poles, zeros, and contours which determines the behavior of engineering systems.

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.