Complex numbers obey specific rules that determine how they interact under arithmetic operations. Some of these properties arise from the algebraic structure shared with real numbers, while others are unique to the complex number system. Understanding these properties provides the foundation for manipulating complex expressions, simplifying calculations, and recognizing what complex numbers can and cannot do.
Overview of Complex Number Properties
The properties of complex numbers fall into distinct categories based on their origin and nature.
Arithmetic properties govern basic operations — addition, subtraction, multiplication, and division. These properties derive from the fact that complex numbers form what mathematicians call a field, an algebraic structure shared with the real numbersR and rational numbers Q. Any rule that works for real arithmetic also works for complex arithmetic.
Conjugate, modulus, and argument properties are unique to complex numbers. The conjugatez reflects a number across the real axis. The modulus∣z∣ measures distance from the origin. The argumentarg(z) specifies direction. Each of these concepts carries its own set of rules describing how it behaves under arithmetic operations.
Algebraic closure is a property that makes C more powerful than R. Every polynomial equation has solutions in the complex numbers — a guarantee that real numbers cannot provide.
Ordering is a property that C lacks. Unlike real numbers, complex numbers cannot be arranged on a line from smallest to largest. The familiar comparisons "greater than" and "less than" do not apply.
The sections that follow examine each category in detail.
Complex Numbers as a Field
In abstract algebra, a field is a set equipped with two operations — addition and multiplication — that satisfy a specific collection of axioms. These axioms guarantee predictable arithmetic: you can add, subtract, multiply, and divide (except by zero) and always obtain a result within the same set.
The rational numbers Q form a field. So do the real numbers R. The complex numbers C extend this pattern — they too satisfy every field axiom.
Why does this matter? Because field structure ensures that algebraic techniques transfer seamlessly. Factoring, expanding, simplifying, solving equations — all the methods developed for real numbers work identically for complex numbers. No special cases, no exceptions, no hidden traps.
The field axioms divide into several groups:
• Closure: Operations keep you inside the set
• Commutativity: Order of operands does not matter
• Associativity: Grouping of operands does not matter
• Identity elements: Zero for addition, one for multiplication
• Inverse elements: Every number has a negative; every nonzero number has a reciprocal
• Distributivity: Multiplication distributes over addition
The next section lists these axioms precisely and confirms that C satisfies each one.
Arithmetic Properties (Field Axioms)
The eleven field axioms, all satisfied by C:
Closure
• z1+z2∈C for all z1,z2∈C
• z1⋅z2∈C for all z1,z2∈C
Adding or multiplying complex numbers always produces a complex number.
Commutativity
• z1+z2=z2+z1
• z1⋅z2=z2⋅z1
Order does not affect the result.
Associativity
• (z1+z2)+z3=z1+(z2+z3)
• (z1⋅z2)⋅z3=z1⋅(z2⋅z3)
Grouping does not affect the result.
Identity Elements
• Additive identity: z+0=z
• Multiplicative identity: z⋅1=z
Inverse Elements
• Additive inverse: For every z, there exists −z such that z+(−z)=0
• Multiplicative inverse: For every z=0, there exists z−1 such that z⋅z−1=1
The multiplicative inverse is z−1=∣z∣2z.
Distributivity
• z1⋅(z2+z3)=z1⋅z2+z1⋅z3
Multiplication distributes over addition.
These eleven properties guarantee that complex arithmetic behaves consistently and predictably, just like real arithmetic.
Properties of the Conjugate
The complex conjugatez of z=a+bi is z=a−bi. Conjugation interacts with arithmetic operations in predictable ways.
Conjugate of a Conjugate
z=z
Conjugating twice returns the original number.
Conjugate of a Sum
z1+z2=z1+z2
The conjugate of a sum equals the sum of conjugates.
Conjugate of a Difference
z1−z2=z1−z2
Conjugate of a Product
z1⋅z2=z1⋅z2
The conjugate of a product equals the product of conjugates.
Conjugate of a Quotient
(z2z1)=z2z1
Extracting Real and Imaginary Parts
z+z=2Re(z)
z−z=2i⋅Im(z)
Product with Conjugate
z⋅z=∣z∣2
A complex number multiplied by its conjugate yields the square of its modulus — always a non-negative real number. This property is essential for division of complex numbers.
Properties of the Modulus
The modulus∣z∣ of z=a+bi is ∣z∣=a2+b2, representing the distance from the origin to z in the complex plane.
Non-Negativity
∣z∣≥0
∣z∣=0⟺z=0
The modulus is zero only for the number zero itself.
The modulus of a product equals the product of moduli.
Modulus of a Quotient
z2z1=∣z2∣∣z1∣
Modulus of a Power
∣zn∣=∣z∣n
Triangle Inequality
∣z1+z2∣≤∣z1∣+∣z2∣
The distance from the origin to z1+z2 never exceeds the sum of individual distances. Geometrically, one side of a triangle cannot exceed the sum of the other two sides.
Reverse Triangle Inequality
∣z1∣−∣z2∣≤∣z1−z2∣
The difference of moduli never exceeds the modulus of the difference.
Properties of the Argument
The argumentarg(z) is the angle from the positive real axis to the line connecting the origin to z. Arguments are determined only up to multiples of 2π, so the following properties hold modulo 2π.
Argument of a Product
arg(z1⋅z2)=arg(z1)+arg(z2)
Multiplying complex numbers adds their arguments. Geometrically, multiplication rotates by the sum of the angles.
Argument of a Quotient
arg(z2z1)=arg(z1)−arg(z2)
Dividing complex numbers subtracts arguments.
Argument of a Power
arg(zn)=n⋅arg(z)
Raising to the nth power multiplies the argument by n. This property underlies De Moivre's theorem.
Argument of a Conjugate
arg(z)=−arg(z)
Conjugation reflects across the real axis, negating the angle.
Argument of a Negative
arg(−z)=arg(z)+π
Negation rotates by 180°.
Note on Multivaluedness
Since arg(z) is defined only up to 2πk for integer k, these equalities hold in the sense that representative values from each side differ by a multiple of 2π. When using the principal argument Arg(z), results may need adjustment to stay within the chosen interval.
Algebraic Closure
A field is algebraically closed if every non-constant polynomial with coefficients in that field has at least one root in that field. The complex numbers are algebraically closed. The real numbers are not.
The Fundamental Theorem of Algebra
Every polynomial of degree n≥1 with complex coefficients has exactly n roots in C, counting multiplicity.
This theorem guarantees that polynomial equations always have solutions — no need to invent further number systems beyond C.
Why R Fails
The polynomial x2+1=0 has no real solutions. No real number squared equals −1. This limitation motivated the creation of complex numbers in the first place.
In C, the equation x2+1=0 has two solutions: x=i and x=−i.
Practical Consequence
When factoring polynomials over C, every polynomial of degree n factors completely into n linear factors:
p(z)=an(z−r1)(z−r2)⋯(z−rn)
Over R, some polynomials resist complete factorization. Over C, none do.
Despite their algebraic completeness, complex numbers lack one property that real numbers possess: order.
Not an Ordered Field
The real numbers are ordered. Given distinct a,b∈R, either a<b or b<a. This ordering respects arithmetic: if a<b, then a+c<b+c for any c, and ac<bc when c>0.
No such ordering exists for C. There is no way to define "<" on complex numbers that remains consistent with the field operations.
Why Ordering Fails
Suppose an ordering existed. Consider where i falls relative to 0.
Case 1:i>0
Then i⋅i>0, so i2>0, meaning −1>0. But this contradicts 1>0 (which must hold in any ordered field).
Case 2:i<0
Then −i>0, so (−i)(−i)>0, meaning i2>0, so −1>0. Same contradiction.
Neither case works. The number i cannot be consistently placed relative to 0, and the same argument applies to every non-real complex number.
What We Can Compare
Although z1<z2 is meaningless for complex numbers, we can compare their moduli. The statement ∣z1∣<∣z2∣ is well-defined because moduli are real numbers. We compare sizes, not positions.