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.
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. Every rule that follows from the field axioms — commutativity, associativity, distributivity, identities and inverses — works for complex arithmetic just as for real 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 non-constant 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 in a way that respects addition and multiplication. The familiar comparisons "greater than" and "less than" do not apply.
The sections that follow examine each category in detail.
Five number systems. In which of them can you add, subtract, multiply and divide without ever leaving the system?
Addition and multiplication never cause trouble in any of them. Subtraction and division are the real test.
Three of the five pass every test.
number system
subtraction stays inside?
division by nonzero stays inside?
a field?
natural numbers ℕ
no: 2 − 5 = −3
—
no
integers ℤ
yes
no: 1 ÷ 2 = 1/2
no
rationals ℚ
yes
yes
yes
reals ℝ
yes
yes
yes
complex numbers ℂ
yes
yes
yes
The natural numbers fail at subtraction and the integers at division; ℚ, ℝ and ℂ never fail.
2−5=−3 is not a natural number, and 1÷2=21 is not an integer. In the last three systems every such result stays inside.
The definition
A field is a set with addition and multiplication satisfying the field axioms, so that adding, subtracting, multiplying, and dividing by anything nonzero always give a result in the set.
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 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
So Q, R and C are fields, while N and Z are not.
Being a field is what lets ordinary algebra carry over to complex numbers.
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 that rely only on the field rules work identically for complex numbers. Rules that depend on order or on real square roots, such as x2≥0 or ab=ab, do not carry over.
Whatever follows from the field rules alone holds in C; rules that need order or real square roots must be checked again.
A field need not have an order.
C satisfies every field axiom, yet it cannot be ordered in a way that respects its arithmetic; that is taken up in what complex numbers lack.
The next section lists these axioms precisely and confirms that C satisfies each one.
The rest of the page covers the field axioms, the properties of the conjugate, modulus and argument, algebraic closure, what complex numbers lack, a summary and common pitfalls.
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.
Pitfall: Dividing by the modulus instead of its square
For z=1+i, taking z−1=21−i divides by ∣z∣, and then (1+i)⋅21−i=2, not 1. Dividing by ∣z∣2=2 gives z−1=21−i.
The conjugate in the numerator is what makes the formula work. Multiplying z by z produces the real number ∣z∣2, so the product z⋅z−1 becomes ∣z∣2z⋅z, a real number divided by itself. The inverse is found without solving any equation, only by conjugating and scaling.
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.
Axiom group
Additive form (+)
Multiplicative form (·)
Closure
z₁ + z₂ ∈ ℂ
z₁ · z₂ ∈ ℂ
Commutativity
z₁ + z₂ = z₂ + z₁
z₁ · z₂ = z₂ · z₁
Associativity
(z₁ + z₂) + z₃ = z₁ + (z₂ + z₃)
(z₁ · z₂) · z₃ = z₁ · (z₂ · z₃)
Identity
z + 0 = z (additive identity 0)
z · 1 = z (multiplicative identity 1)
Inverse
−z exists; z + (−z) = 0
z⁻¹ exists for z ≠ 0; z · z⁻¹ = 1 (formula: z⁻¹ = z̄ / |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
Properties of the Modulus
∣z∣≥0
Properties of the Modulus
∣z∣=0⟺z=0
The modulus is zero only for the number zero itself.
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.
The difference of moduli never exceeds the modulus of the difference.
|z| = |z̄| = 3.6
The modulus is a distance, so it is never negative and it is unchanged by conjugation, as the two equal arrows show; it is zero only for the origin. The multiplicativity |z₁z₂| = |z₁||z₂| is the scaling half of the multiplication picture. Check the properties on a moving point on the conjugate and modulus tool.
Each property of the modulus is a statement about lengths in the plane.
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.
Raising to the nth power multiplies the argument by n. This property underlies De Moivre's theorem.
Fact: Machin's π in one product
Multiplying out gives (5+i)4(239−i)=114244(1+i), so the angles add up to that of 1+i: 4arctan51−arctan2391=4π. John Machin used this in 1706 to compute π to 100 decimal places.
The power rule turns sums of angles into products of numbers. The number 5+i has argument arctan51, so (5+i)4 has argument 4arctan51, and a further factor 239−i subtracts arctan2391. Whatever such a combination of angles adds up to can be read off the single number that the product produces.
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.
z = −4 + 3i: r = 5, argument in the second quadrant
The argument is the angle from the positive real axis to the arrow, here past 90° because the point lies in the second quadrant; a calculator’s arctan(3/−4) would report the wrong quadrant, and the picture is the check. Adding 2π to the angle changes nothing about the point. Move the point across a quadrant boundary on the polar and rectangular form converter.
The argument of a product is the sum of the arguments, which is the rotation rule of multiplication.
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:
Algebraic Closure
p(z)=an(z−r1)(z−r2)⋯(z−rn)
Over R, some polynomials resist complete factorization. Over C, none do.
Every non-constant polynomial has a root in the field?
Not always — e.g., x² + 1 = 0 has no real root
Always — guaranteed by the Fundamental Theorem of Algebra
Degree-n polynomial factors completely into n linear factors?
Not always (some factors remain irreducible quadratics)
Always: p(z) = aₙ(z − r₁)(z − r₂)⋯(z − rₙ)
Number of roots of a degree-n polynomial
0 to n
exactly n (counting multiplicity)
What Complex Numbers Lack
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.
THEOREM
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 this single case already rules out any ordering of C compatible with the field operations.
Fact: A dictionary order that fails at i
Complex numbers can be put in a line: compare real parts, and break ties by imaginary parts, as a dictionary sorts words. This order even survives adding the same number to both sides, but not multiplying: i>0 in it, yet i⋅i=−1<0.
The argument uses only two rules that every ordered field must follow: adding the same number to both sides keeps an inequality, and the product of two positive numbers is positive. Lining up the points of the plane is not the problem in itself; the problem is doing it while keeping both rules.
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.
Summary: Operation Effects on Conjugate, Modulus, and Argument
The conjugate, modulus, and argument each interact predictably with the same arithmetic operations. The table below collects how each of the three invariants behaves under sum, difference, product, quotient, power, and conjugation — exposing the parallel structure that the preceding sections established one at a time.
The same table as a grid of verdicts. Each cell shows whether the operation passes through as an exact law, gives only a bound, holds only up to a multiple of 2π, or has no simple rule. Click an operation or a column to focus on it.
Read down the columns and the three sections above line up. The conjugate passes through every operation. The modulus turns products, quotients and powers into the same operations on lengths, but a sum gives only the triangle inequality. The argument turns multiplication into addition, which is what makes De Moivre's theorem work, and has nothing simple to say about sums.
Common Pitfalls
Five mistakes come up again and again when these properties are put to work. Four are marked where they happen in the sections above; the square-root rule is the one the field section already warns about. Here each is worked through in full, with the wrong and the right working side by side.
The real rule ab=ab is carried over to negative numbers, whose square roots are imaginary. The two minus signs under the roots are multiplied away before the roots are taken.
Wrong turn
Error here
−4⋅−9=(−4)(−9)=36=6.
Right path
Take each root first, −4=2i and −9=3i, then multiply: 2i⋅3i=6i2=−6.
Why it happensThe rule holds for every pair of non-negative reals, so it feels like a general rule of arithmetic. It depends on real square roots and is not one of the field axioms.
How to checkWrite every square root of a negative number as a multiple of i before doing anything else; the factor i⋅i=−1 then appears on its own.
2Dividing by the modulus instead of its square
The inverse formula z−1=∣z∣2z is copied with ∣z∣ in the denominator instead of ∣z∣2.
Start
For z=1+i, ∣z∣=2,
Wrong turn
Error here
So z−1=21−i. Then (1+i)⋅21−i=22=2.
Right path
∣z∣2=12+12=2, so z−1=21−i=21−21i. Check: (1+i)⋅21−i=21−i2=22=1.
Why it happensThe modulus and its square look alike, and dividing z by ∣z∣ is a familiar step, the one that gives a number of length 1 in the same direction.
How to checkMultiply z by the proposed inverse. The product must be exactly 1; a result equal to ∣z∣, here 2, shows that one factor of ∣z∣ is missing.
3Forgetting that i squared is −1
In z⋅z=(a+bi)(a−bi), the difference-of-squares pattern gives a2−(bi)2, and (bi)2 is then taken to be b2.
Wrong turn
Error here
(2+3i)(2−3i)=4−9=−5.
Right path
(2+3i)(2−3i)=4−(3i)2=4−9i2=4+9=13. Since ∣2+3i∣=4+9=13, this is ∣2+3i∣2=13, as the identity requires.
Why it happensThe real pattern (x+y)(x−y)=x2−y2 is applied with y=3 instead of y=3i.
How to checkz⋅z is never negative. A negative result such as −5 shows the slip at once; the answer must equal a2+b2.
4Adding the moduli of a sum
The modulus of a sum is replaced by the sum of the moduli, as if the triangle inequality were an equation. It only says that ∣z1+z2∣ is at most ∣z1∣+∣z2∣.
Wrong turn
Error here
∣3+4i∣=∣3∣+∣4i∣=3+4=7.
Right path
∣3+4i∣=32+42=25=5. The sum of the moduli, 7, is only an upper bound.
Why it happensThe conjugate passes through sums and the modulus passes through products, so it is tempting to expect the modulus to pass through sums as well.
How to checkPicture the two numbers as arrows. Unless they point in the same direction, the arrow of the sum is shorter than the two arrows laid end to end. Here 3 points right and 4i points up, so the result must be less than 7.
5Leaving the sum outside the principal range
The rule arg(z1z2)=arg(z1)+arg(z2) is applied to principal arguments, and the sum is reported as the principal argument of the product even when it falls outside the interval (−π,π].
Wrong turn
Error here
Arg(−1)=π and Arg(i)=2π, so Arg((−1)⋅i)=23π.
Right path
The sum 23π is a valid argument of −i, but it lies outside (−π,π]. Subtracting 2π gives Arg(−i)=23π−2π=−2π.
Why it happensThe rule holds for arg, which allows any multiple of 2π, while Arg must land in one fixed interval.
How to checkCompute the product, here −i, and locate it. It lies on the negative imaginary axis, where the principal argument is −2π.
Each slip applies a rule outside the place where it holds. ab=ab fails for two negative numbers, the inverse divides by ∣z∣2, zzˉ=a2+b2 because i2=−1, the modulus of a sum is at most the sum of the moduli, and a sum of principal arguments may need 2π subtracted. Multiplying back, or locating the product on the plane, catches each slip.
Properties of Complex Numbers FAQ
How does the argument behave under multiplication and division?
+
Arguments add when you multiply and subtract when you divide, while the moduli multiply and divide. This is the whole reason polar form makes those operations easy. The sum may leave the principal range, so a full turn is often added or removed afterwards to bring it back.Read more →
Does conjugation distribute over addition and multiplication?
+
Yes, over both, and over division too. The conjugate of a sum is the sum of the conjugates, and the same holds for products. This is why conjugating an entire polynomial equation with real coefficients leaves the coefficients alone while turning each root into its own conjugate.Read more →
How do complex numbers prove Machin's formula for π?
+
By multiplying, since arguments add. The number 5+i has argument arctan51 and 239−i has argument −arctan2391, and the product works out to (5+i)4(239−i)=114244(1+i). A positive multiple of 1+i has argument 4π, so 4arctan51−arctan2391=4π. John Machin used this in 1706 to compute π to 100 decimal places.Read more →
Can complex numbers be sorted in dictionary order?
+
Yes as a list, but not in a way that respects multiplication. Compare real parts first and break ties with imaginary parts, the way a dictionary sorts words. That order even survives adding the same number to both sides. It fails for products: i>0 in it, since 0+1i beats 0+0i, yet i⋅i=−1<0, so the product of two positive numbers comes out negative.Read more →
Is the argument of a sum the sum of the arguments?
+
No, that rule belongs to products. For 1 and i the arguments are 0 and 2π; they add to 2π, the argument of the product i, while the sum 1+i has argument 4π. The argument of a sum has no simple rule, because it depends on the lengths of the two numbers as well as their angles. Compute the sum first, then read off its argument.Read more →
