Complex Number Subtraction Calculator
Subtract complex numbers (a + bi) − (c + di) = (a − c) + (b − d)i, calculate Euclidean distance d(z₁, z₂) = |z₁ − z₂|, verify the reverse triangle inequality theorem, and visualize tip-to-tail vector displacement on the interactive Argand plane.
z₁ = 7 + 9i (|z₁| = 11.402, θ = 52.13°)
z₂ = 3 + 4i (|z₂| = 5.000, θ = 53.13°)
Polar: 6.4031 ∠ 51.34° = 6.4031 · e^(i · 0.8961 rad)
||z₁| − |z₂|| = 6.402 ≤ |z₁ − z₂| = 6.403 ≤ |z₁| + |z₂| = 16.402
Step-by-Step Algebraic Solution
Argand Plane • Vector Subtraction
Tip-to-Tail DisplacementGeometric Rule: The difference vector (z₁ − z₂) represents the directed displacement vector originating from the tip of z₂ to the tip of z₁. Translated to the origin (0, 0), it is equivalent to vector addition with the additive inverse: z₁ + (−z₂).
Complex Number Subtraction Formula
To subtract two complex numbers, subtract their real parts and imaginary parts independently: (a + bi) − (c + di) = (a − c) + (b − d)i. In the complex Argand plane, this operation computes the directed vector displacement from z₂ to z₁, whose modulus |z₁ − z₂| equals the exact Euclidean distance between the two points.
What Is Complex Number Subtraction?
A complex number is a number that can be expressed in standard rectangular form as z = a + bi, where a and b are real numbers, and i is the imaginary unit defined by the polynomial equation:
In any complex number z = a + bi:
- Real Part: Denoted Re(z) = a.
- Imaginary Part: Denoted Im(z) = b (a real coefficient multiplying the imaginary unit i).
Because real numbers and imaginary numbers reside on mutually perpendicular (orthogonal) axes in the complex plane, they cannot be combined algebraically into a single real quantity. Subtracting one complex number from another is therefore performed component-wise by subtracting corresponding real parts and imaginary parts separately:
z₁ − z₂ = (a + bi) − (c + di) = (a − c) + (b − d)i
Rule: Real part minus real part; imaginary part minus imaginary part.
Formally, complex subtraction is defined as the addition of the additive inverse (opposite vector). For any complex number z₂ = c + di, its additive inverse is −z₂ = −c − di. Thus:
The Argand Plane & Vector Displacement
On the Argand plane (complex plane), every complex number z = a + bi corresponds uniquely to a 2D geometric vector drawn from the origin (0, 0) to the coordinate point (a, b). When we subtract z₂ from z₁, the operation has two profound geometric meanings:
1. Directed Displacement Vector
The vector (z₁ − z₂) represents the direct segment originating at the tip of z₂ and pointing directly to the tip of z₁. It answers the question: "What vector must be added to z₂ to reach z₁?"
2. Euclidean Distance Metric
The magnitude (modulus) of the difference, |z₁ − z₂|, gives the exact Euclidean distance between the points (a, b) and (c, d) in the plane, establishing the metric topology of the complex plane:
When the difference vector is translated so that its tail sits at the origin (0, 0), its tip lands at the coordinates (a − c, b − d). This is identical to using the Parallelogram Law to add z₁ to the negated vector −z₂.
Step-by-Step Worked Examples
Evaluate (7 + 9i) − (3 + 4i):
1. Distribute negative sign: 7 + 9i − 3 − 4i.
2. Real difference: 7 − 3 = 4.
3. Imaginary difference: 9i − 4i = (9 − 4)i = 5i.
4. Consolidated result: 4 + 5i.
5. Modulus: √(4² + 5²) = √(16 + 25) = √41 ≈ 6.4031.
Evaluate (−4 + 2i) − (5 − 8i):
1. Distribute negative sign: −4 + 2i − 5 − (−8i) = −4 + 2i − 5 + 8i.
2. Real difference: −4 − 5 = −9.
3. Imaginary difference: 2i + 8i = 10i.
4. Consolidated result: −9 + 10i.
5. Polar Form: r = √[(−9)² + 10²] = √181 ≈ 13.4536, θ = atan2(10, −9) ≈ 131.99°.
Let z = 5 + 3i and its conjugate z̄ = 5 − 3i. Evaluate z − z̄:
1. Expression: (5 + 3i) − (5 − 3i) = 5 + 3i − 5 + 3i.
2. Real parts cancel: 5 − 5 = 0.
3. Imaginary parts double: 3i − (−3i) = 3i + 3i = 6i.
4. Result: 0 + 6i = 6i. The difference z − z̄ is ALWAYS purely imaginary: 2 × Im(z)i.
Evaluate (9 + 4i) − (2 + 4i):
1. Real difference: 9 − 2 = 7.
2. Imaginary difference: 4i − 4i = 0i.
3. Result: 7 + 0i = 7 (lies entirely on the real axis Re).
Algebraic Field Properties of Complex Subtraction
While complex addition satisfies commutativity and associativity, complex subtraction behaves very differently. Understanding these algebraic differences is vital for mastering complex algebra:
| Property | Algebraic Identity | Mathematical Interpretation |
|---|---|---|
| Non-Commutative | z₁ − z₂ = −(z₂ − z₁) | Swapping the operands negates the difference vector (rotates it by 180°). |
| Non-Associative | (z₁ − z₂) − z₃ ≠ z₁ − (z₂ − z₃) | Grouping order changes the result because the second term acquires a positive sign. |
| Self-Subtraction (Nilpotence) | z − z = 0 | Subtracting any complex number from itself yields the additive zero identity 0 + 0i. |
| Conjugation Distributivity | conj(z₁ − z₂) = conj(z₁) − conj(z₂) | The conjugate of a difference equals the difference of the conjugates. |
| Conjugate Subtraction | z − z̄ = 2i · Im(z) | Subtracting a conjugate always eliminates the real part, leaving a pure imaginary number. |
The Reverse Triangle Inequality Theorem
While the standard Triangle Inequality governs complex addition (|z₁ + z₂| ≤ |z₁| + |z₂|), complex subtraction is governed by the Reverse Triangle Inequality:
||z₁| − |z₂|| ≤ |z₁ − z₂| ≤ |z₁| + |z₂|
This double inequality provides fundamental bounds in complex analysis:
- Lower Bound: The distance between z₁ and z₂ can never be less than the absolute difference between their individual distances from the origin: |d₁ − d₂|. Equality occurs if and only if z₁ and z₂ lie on the same ray from the origin (same argument angle θ).
- Upper Bound: The distance between z₁ and z₂ can never exceed the sum of their individual distances from the origin: |z₁| + |z₂|. Equality occurs if and only if z₁ and z₂ point in exactly opposite directions (Δθ = 180° or π radians).
Subtraction: Rectangular Form vs. Polar Form
Students often ask whether complex numbers can be subtracted directly in polar form r ∠ θ or exponential Euler form r e^(iθ):
Why Direct Polar Subtraction Does Not Work
Unlike complex multiplication and division—where polar coordinates multiply moduli and subtract angles: (r₁ ∠ &theta₁) / (r₂ ∠ &theta₂) = (r₁/r₂) ∠ (&theta₁ − &theta₂)—subtraction has no simple angle formula.
r₁ ∠ &theta₁ − r₂ ∠ &theta₂ ≠ (r₁ − r₂) ∠ (&theta₁ − &theta₂) [FATAL ERROR]
To subtract numbers given in polar or trigonometric form, you must follow the three-step standard protocol:
- Convert both numbers to rectangular coordinates: a = r₁ cos(&theta₁), b = r₁ sin(&theta₁); c = r₂ cos(&theta₂), d = r₂ sin(&theta₂)
- Subtract the rectangular components: x_diff = a − c, y_diff = b − d
- Convert the difference back to polar coordinates: r_diff = √(x_diff² + y_diff²), θ_diff = atan2(y_diff, x_diff)
Common Pitfalls & Sign Distribution Errors
Mistake: Writing (5 + 4i) − (2 − 3i) = (5 − 2) + (4 − 3)i = 3 + 1i.
Correct: The negative sign applies to both terms: −(2 − 3i) = −2 + 3i. The true result is (5 − 2) + (4 + 3)i = 3 + 7i. Always keep the parentheses intact until you distribute the negative sign explicitly.
Mistake: Believing that z₁ − z₂ = z₂ − z₁.
Correct: Subtraction reverses direction in the complex plane. (7 + 2i) − (3 + 5i) = 4 − 3i, whereas (3 + 5i) − (7 + 2i) = −4 + 3i. They are exact opposites: z₂ − z₁ = −(z₁ − z₂).
Mistake: Combining real and imaginary terms like 4 + 5i = 9i or 7 − 2i = 5i.
Correct: Real numbers and imaginary numbers are linearly independent vectors. They can never be combined into a single term without violating the axioms of the complex field.
Applications in Electrical Engineering & Physics
Complex number subtraction is an indispensable mathematical operation across modern STEM disciplines:
AC Electrical Circuits (Kirchhoff's Voltage Law)
In alternating current (AC) circuit analysis, voltages and currents are modeled as complex phasors. The voltage drop across an impedance between two nodes A and B is computed as the complex subtraction of their node phasors: V_AB = V_A − V_B.
Audio Signal Processing & Noise Cancellation
Active noise cancellation (ANC) systems compute the complex Fourier spectrum of incoming ambient noise and subtract it from the desired audio signal phasor: S_clean(ω) = S_mic(ω) − S_noise(ω), extinguishing background acoustic interference.
Quantum Mechanics & Wave Interference
Quantum states are complex probability amplitudes. Destructive quantum interference occurs when two state amplitudes differ in phase by π radians, creating a subtraction of state vectors that cancels the probability of transition.
Robotics & Relative Vector Navigation
In planar kinematics, autonomous mobile robots represent 2D positions as complex numbers. The relative vector and distance from robot position z_robot to target destination z_target is simply Δz = z_target − z_robot.
Frequently Asked Questions
How do you subtract two complex numbers?
Is complex number subtraction commutative?
What is the geometric meaning of subtracting complex numbers?
What happens when you subtract a complex number from its complex conjugate?
What is the Reverse Triangle Inequality for complex subtraction?
Can you subtract complex numbers directly in polar form?
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