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Complex Numbers Addition /Subtraction Visualizer


Complex Addition & Subtraction

Add component-wise, see the parallelogram
ReIm-5−5i-4−4i-3−3i-2−2i-1−1i11i22i33i44i55iz₁+z₂z₁−z₂−z₂z₁z₂
Show:
z₁
3 + i
z₂
1 + 3i
z₁ + z₂ — Sum
4 + 4i
|z₁+z₂| = 5.7
z₁ − z₂ — Difference
2 − 2i
|z₁−z₂| = 2.8 (distance between z₁ and z₂)
Step-by-Step
Addition:
(3 + i) + (1 + 3i)
= (3 + 1) + (1 + 3)i
= 4 + 4i
Subtraction:
(3 + i) − (1 + 3i)
= (3 − 1) + (1 − 3)i
= 2 − 2i
Triangle Inequality
|z₁ + z₂| ≤ |z₁| + |z₂|
5.7 ≤ 3.2 + 3.2 = 6.3 ✓
Equality holds when z₁ and z₂ point in the same direction (same argument).
Key Ideas
Addition is component-wise: add the real parts, add the imaginary parts. (a+bi) + (c+di) = (a+c) + (b+d)i. No cross-terms, no i² — it is just vector addition.
The parallelogram rule: z₁+z₂ is the diagonal of the parallelogram formed by z₁ and z₂. The dashed vectors show z₂ translated to the tip of z₁ (and vice versa) — "tip to tail."
Subtraction gives the vector from z₂ to z₁. The purple dashed line between the two points has the same length and direction as z₁−z₂. The modulus |z₁−z₂| is the distance between the two points.
z₁ − z₂ = z₁ + (−z₂). Subtraction is the same as adding the negation. The ghost vector −z₂ is z₂ rotated by 180°. Adding it to z₁ via the parallelogram rule gives the same result.
Triangle inequality: |z₁+z₂| ≤ |z₁| + |z₂|. The sum can never be longer than both vectors placed end-to-end. Drag z₁ and z₂ to the same direction to see equality.

Getting Started — Drag Two Points
The Parallelogram Rule for Addition
Subtraction and the Difference Vector
Both Mode — Comparing Addition and Subtraction
Degenerate and Axis-Aligned Configurations
Reading the Step-by-Step Panel
The Triangle Inequality
How Complex Addition Works
How Complex Subtraction Works
Conjugate Pairs in Addition and Subtraction
Related Concepts and Tools




Getting Started — Drag Two Points

Two draggable points sit on the complex plane: z1z_1 (navy) and z2z_2 (orange). Grab either point and move it anywhere within the ±5\pm 5 grid to see the addition and subtraction results update instantly.

Alternatively, type precise values into the input fields on the right panel — each point has separate real and imaginary inputs accepting values from −5-5 to 55 in steps of 0.10.1. Five preset configurations are available below the plane: (3+i)(3+i) & (1+3i)(1+3i), (2+2i)(2+2i) & (−2+2i)(-2+2i), 44 & 3i3i, a conjugate pair, and (−1+3i)(-1+3i) & (2−i)(2-i). Click Random to generate two arbitrary points.

The Show toggle lets you display both operations simultaneously, or isolate addition or subtraction for a cleaner view. Start with "Both" to compare the two operations side by side, then switch to "Addition" or "Subtraction" to focus on the geometry of each one individually.

The Parallelogram Rule for Addition

Switch to Addition mode to see the parallelogram clearly. The green vector from the origin to the sum point z1+z2z_1 + z_2 forms the diagonal of the parallelogram whose sides are z1z_1 and z2z_2.

Two dashed "ghost" vectors complete the parallelogram. A dashed copy of z2z_2 (orange) is translated to the tip of z1z_1, and a dashed copy of z1z_1 (navy) is translated to the tip of z2z_2. Both ghost vectors arrive at the same destination — the sum point. This is the tip-to-tail construction: place one vector's tail at the other's tip, and the sum runs from the origin to where you end up.

The light green shaded region fills the interior of the parallelogram, making the geometric relationship immediately visible. Try the preset (3+i)(3+i) & (1+3i)(1+3i) — the parallelogram tilts toward the imaginary axis because z2z_2 has a larger imaginary component. Then try 44 & 3i3i — one side lies along the real axis and the other along the imaginary axis, producing a rectangle.

Subtraction and the Difference Vector

Switch to Subtraction mode to isolate the subtraction geometry. The purple vector from the origin represents z1−z2z_1 - z_2, while a purple dashed line connects the tips of z2z_2 and z1z_1 directly.

This dashed line has the same length and direction as the difference vector — it literally shows "the vector from z2z_2 to z1z_1." The modulus ∣z1−z2∣|z_1 - z_2| equals the distance between the two points on the plane.

A faint dashed vector labeled −z2-z_2 appears opposite to z2z_2 (rotated 180°). This illustrates the key identity: z1−z2=z1+(−z2)z_1 - z_2 = z_1 + (-z_2). Subtraction is addition of the negation.

Try the conjugate pair preset: z1=3+2iz_1 = 3 + 2i and z2=3−2iz_2 = 3 - 2i. The difference is 4i4i (pure imaginary) and the dashed line between the points is vertical. The sum is 66 (pure real) and lies on the real axis — conjugate pairs always produce this split.

Both Mode — Comparing Addition and Subtraction

In Both mode, the addition parallelogram (green) and subtraction vector (purple) appear together. This side-by-side view reveals how the two operations relate geometrically.

The green sum vector and the purple difference vector are actually the two diagonals of the same parallelogram. The sum is the diagonal that starts at the origin and passes through the interior. The difference connects the two input points — it is the other diagonal, translated to start at the origin.

Try dragging z2z_2 to the exact opposite of z1z_1. For example, set z1=3+2iz_1 = 3 + 2i and z2=−3−2iz_2 = -3 - 2i. The sum collapses to zero (both vectors cancel) while the difference doubles to 6+4i6 + 4i. The parallelogram degenerates to a line segment because the two vectors are antiparallel.

Now set both points in the same direction — for example z1=2+iz_1 = 2 + i and z2=4+2iz_2 = 4 + 2i. The sum is 6+3i6 + 3i (aligned), the parallelogram again degenerates to a line, and the triangle inequality reaches equality.

Degenerate and Axis-Aligned Configurations

Several input combinations produce visually distinct special cases worth capturing.

Pure real inputs: click preset 44 & 3i3i, then change z2z_2 to a real number like 22. Both vectors lie on the real axis, the parallelogram collapses to a horizontal line, and both the sum and difference are real numbers.

Pure imaginary inputs: set z1=0+3iz_1 = 0 + 3i and z2=0+2iz_2 = 0 + 2i. Both vectors are vertical, the parallelogram collapses to a vertical line, and results are purely imaginary.

One vector zero: set z2=0z_2 = 0. The sum equals z1z_1 itself, the difference also equals z1z_1, and the parallelogram disappears because one side has zero length.

Collinear vectors: when z1z_1 and z2z_2 point in the same direction (same argument), the parallelogram degenerates to a line and the triangle inequality becomes an equality. Drag both points to, say, the positive real axis to see this.

Off-screen result: when the sum exceeds the ±5\pm 5 range, the green vector is replaced by a dashed ray pointing toward the edge with an arrow and the label "z1+z2→z_1 + z_2 \to", indicating the result is beyond the visible area.

Reading the Step-by-Step Panel

The Step-by-Step section on the right breaks down each operation into three lines.

For addition, it shows: the original expression (z1)+(z2)(z_1) + (z_2), the component-wise grouping (a1+a2)+(b1+b2)i(a_1 + a_2) + (b_1 + b_2)i, and the final simplified result highlighted in green.

For subtraction, the same three-line pattern appears with minus signs: the original expression, the component-wise grouping (a1−a2)+(b1−b2)i(a_1 - a_2) + (b_1 - b_2)i, and the result in purple.

All values update dynamically as you drag or type. When "Both" mode is active, both operations appear stacked. Switching to "Addition" or "Subtraction" mode hides the irrelevant calculation, giving a cleaner view.

This panel makes the component-wise nature of complex addition explicit — the real and imaginary parts are handled independently, with no cross-terms or powers of ii involved.

The Triangle Inequality

When addition is visible, a dedicated panel displays the triangle inequality for the current configuration:

∣z1+z2∣≤∣z1∣+∣z2∣|z_1 + z_2| \leq |z_1| + |z_2|


The panel substitutes the actual modulus values and verifies the inequality with a green checkmark. This states that the length of the sum vector can never exceed the sum of the individual lengths — geometrically, the diagonal of a triangle is always shorter than the sum of the other two sides.

Equality holds when z1z_1 and z2z_2 have the same argument (point in the same direction). Drag both points onto the positive real axis — for example z1=3z_1 = 3 and z2=2z_2 = 2 — and watch the left side equal the right side exactly: ∣5∣=∣3∣+∣2∣=5|5| = |3| + |2| = 5.

Now rotate z2z_2 perpendicular to z1z_1 and the gap widens. The farther apart the two vectors' directions, the shorter the sum vector relative to the individual lengths.

How Complex Addition Works

Adding two complex numbers is straightforward — combine real parts and imaginary parts separately:

(a+bi)+(c+di)=(a+c)+(b+d)i(a + bi) + (c + di) = (a + c) + (b + d)i


There are no cross-terms because the real and imaginary components are independent. This is identical to adding two-dimensional vectors component-wise: if z1=(a,b)z_1 = (a, b) and z2=(c,d)z_2 = (c, d), then z1+z2=(a+c,b+d)z_1 + z_2 = (a + c, b + d).

The geometric interpretation is the parallelogram rule. Place z1z_1 and z2z_2 as arrows from the origin. The sum is the diagonal of the parallelogram they form. Equivalently, translate z2z_2 so its tail sits at the tip of z1z_1 — the arrow from the origin to the new tip of z2z_2 is z1+z2z_1 + z_2.

Complex addition is commutative (z1+z2=z2+z1z_1 + z_2 = z_2 + z_1) and associative. It also preserves the structure of the complex plane — the sum of two complex numbers is always another complex number.

How Complex Subtraction Works

Subtracting complex numbers also works component-wise:

(a+bi)−(c+di)=(a−c)+(b−d)i(a + bi) - (c + di) = (a - c) + (b - d)i


The key geometric insight is that z1−z2z_1 - z_2 equals the vector from the point z2z_2 to the point z1z_1. Its modulus ∣z1−z2∣|z_1 - z_2| is the Euclidean distance between the two points on the complex plane.

Subtraction can always be rewritten as adding the negation: z1−z2=z1+(−z2)z_1 - z_2 = z_1 + (-z_2), where −z2=−c−di-z_2 = -c - di is z2z_2 reflected through the origin (rotated by 180°180°). The visualizer shows this as the faint dashed −z2-z_2 ghost vector.

Two important special cases: when z2z_2 is the conjugate of z1z_1 (same real part, opposite imaginary part), the difference is purely imaginary (2bi2bi). When z1=z2z_1 = z_2, the difference is zero.

Conjugate Pairs in Addition and Subtraction

Click the Conjugate pair preset to load z1=3+2iz_1 = 3 + 2i and z2=3−2iz_2 = 3 - 2i. These are complex conjugates — they share the same real part but have opposite imaginary parts.

Adding conjugates always cancels the imaginary components: (3+2i)+(3−2i)=6(3 + 2i) + (3 - 2i) = 6. The sum is a pure real number lying on the real axis. In the visualizer, the green sum point sits exactly on the horizontal axis.

Subtracting conjugates always cancels the real components: (3+2i)−(3−2i)=4i(3 + 2i) - (3 - 2i) = 4i. The difference is a pure imaginary number on the vertical axis. The purple vector points straight up.

The dashed line connecting z1z_1 and z2z_2 is vertical — conjugates are always symmetric about the real axis. This pair produces one of the cleanest illustrations: real-axis sum, imaginary-axis difference, and a perfectly symmetric parallelogram.

Related Concepts and Tools

Complex addition and subtraction form the foundation of complex arithmetic. Explore these related pages for the full picture.

Complex Numbers — foundational theory covering the imaginary unit ii, rectangular form a+bia + bi, and algebraic operations.

Complex Number Explorer — a general-purpose tool for plotting and operating on complex numbers, including multiplication and division.

Polar-Rectangular Converter — convert between a+bia + bi and reiθre^{i\theta} form. Polar form simplifies multiplication and division, while rectangular form (used here) simplifies addition and subtraction.

Euler's Formula Explorer — visualize eiθ=cos⁡θ+isin⁡θe^{i\theta} = \cos\theta + i\sin\theta and see how rotation connects to the exponential function.

Powers of i Calculator — compute ini^n for any integer using the mod 4 cycle, a building block for complex arithmetic.

Variance and Expected Value — complex addition rules mirror real-number linearity: E[z1+z2]=E[z1]+E[z2]E[z_1 + z_2] = E[z_1] + E[z_2], extending to random complex variables.