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Complex Number Explorer

How to use
  1. Click anywhere on the plane to place the point zz, or drag to move it; the point snaps to the grid. Learn more about placing the point
  2. Type exact values in the a (real part) and b (imaginary part) fields above the plane; the point moves as you type. Learn more about the input fields
  3. The Range ± buttons set the visible area from ±5 to ±10; up to ±7 the point snaps to halves, from ±8 to whole numbers. Learn more about the range buttons
  4. The right column shows z=a+biz = a + bi and three cards: Real Part in orange, Imaginary Part in navy, Modulus in blue. Learn more about the display panel
  5. The explanation panel under the cards names the state (Quadrant I to IV, Purely Real, Pure Imaginary or The Origin) and works the modulus formula for your values. Learn more about the explanation panel
  6. When both parts are nonzero a right triangle appears: orange leg aa, navy leg bb, blue hypotenuse ∣z∣|z|. Learn more about the right triangle
  7. The faint orange point mirrored across the real axis is the conjugate zˉ=a−bi\bar{z} = a - bi; the Conjugate card gives its value. Learn more about the conjugate
  8. Put the point on an axis or at the origin to see the special cases; each case and each quadrant has its own section below. Learn more about the special cases


Drag the point or type coordinates to explore

+
i
Range ±5
ReIm-5−5i-4−4i-3−3i-2−2i-1−1i11i22i33i44i55i0a = 2b = 3|z| = 3.61z̅z = 2 + 3i
z = 2 + 3i
Real Part
2
Imaginary Part
3
Modulus
3.61

Quadrant I

z = 2 + 3i has real part a = 2 and imaginary part b = 3.

Both components are positive — the point sits in the upper-right quadrant.

Its modulus |z| = √(2² + 3²) = √(4 + 9) = 3.61.

The right triangle has a horizontal leg of length 2 and a vertical leg of length 3. The hypotenuse — the distance from the origin — is 3.61.

The conjugate z̅ = 2 − 3i mirrors this point across the real axis. Same horizontal position, opposite vertical position.

Upper right is the tool’s home state — the textbook orientation of the modulus triangle. Learn more about Quadrant I · All four quadrants

Conjugate
z̅ = 2 − 3i

Click or drag on the plane, or type values above. Try the axes and the origin.


How to Use the Complex Explorer
Reading the Display Panel
Understanding the Right Triangle
Exploring the Conjugate
Special Cases: Axes and Origin
The Origin
Purely Real Numbers
Pure Imaginary Numbers
The Four Quadrants
Quadrant I: Upper Right
Quadrant II: Upper Left
Quadrant III: Lower Left
Quadrant IV: Lower Right
What is a Complex Number?
The Complex Plane (Argand Diagram)
Modulus and the Pythagorean Theorem
Complex Conjugates and Their Properties
Related Concepts and Tools




How to Use the Complex Explorer

DemoTyping a and b, setting the range
Step 0 of 5
This interactive tool visualizes complex numbers on an Argand plane (complex plane). Click anywhere on the grid to place a point, or drag to move it around. The tool responds instantly, updating all calculations as you explore.

Alternatively, type exact values in the a and b input fields above the plane. The field labeled "a" controls the real part (horizontal position), while "b" controls the imaginary part (vertical position). Press Enter or click away to confirm your input.

Use the Range ± buttons to adjust the visible area. Smaller ranges (±5) show more detail for numbers close to the origin, while larger ranges (±10) let you explore bigger values. The snap resolution adjusts automatically based on your zoom level.

Reading the Display Panel

DemoCards, panel, quadrants
Step 0 of 4
The right side of the tool shows key information about your complex number. At the top, you'll see the full representation z = a + bi with your current values.

Three cards display the components: Real Part (a) in orange matches the horizontal distance from the origin, Imaginary Part (b) in navy matches the vertical distance, and Modulus (|z|) in blue shows the straight-line distance from the origin.

Below these cards, the Explanation Panel provides context-aware descriptions. It identifies whether your point is purely real, purely imaginary, at the origin, or in one of the four quadrants. Each case includes specific calculations showing how the modulus formula applies to your values.

Understanding the Right Triangle

When your complex number has both nonzero real and imaginary parts, a right triangle appears connecting the origin to your point. This visualization reveals the geometric meaning of complex number components.

The orange horizontal leg represents the real part a, stretching along the real axis. The navy vertical leg represents the imaginary part b, extending parallel to the imaginary axis. The blue hypotenuse connecting the origin to your point represents the modulus |z|.

Labels on each side show the exact values. The small square at the corner confirms the 90° angle. This triangle demonstrates why ∣z∣=a2+b2|z| = \sqrt{a^2 + b^2} — it's simply the Pythagorean theorem applied to the complex plane.

Exploring the Conjugate

Demoz and its conjugate
Step 0 of 4
Every complex number z = a + bi has a conjugate z̄ = a − bi, shown as a faint orange point reflected across the real axis. The conjugate shares the same real part but has the opposite imaginary part.

A dashed vertical line connects z and z̄, passing through their shared position on the real axis. This reflection symmetry means that a number and its conjugate are always equidistant from the real axis — one above, one below (unless b = 0).

The Conjugate card at the bottom displays the exact value of z̄. Notice that purely real numbers equal their own conjugates (since reflecting across the real axis leaves them unchanged), while purely imaginary numbers have conjugates that are their negatives.

Special Cases: Axes and Origin

DemoAxes and origin
Step 0 of 4
Try placing your point directly on one of the axes or at the origin to see how the tool handles special cases. Each of the three cases has a dedicated section below with the tool frozen in that state.

At the origin (0, 0): The number is exactly zero. As the origin section shows, zero is the only complex number with modulus 0, and it's the unique point that is simultaneously real and pure imaginary.

On the real axis (b = 0): The number is purely real, like ordinary real numbers. The modulus simplifies to |a|, matching the familiar absolute value. The conjugate equals the number itself.

On the imaginary axis (a = 0): The number is purely imaginary. The modulus equals |b|. The conjugate is the negative of the original number, sitting on the opposite side of the origin.

The Origin

Zero has a location. On the complex plane, the number 0=0+0i0 = 0 + 0i is the point where the real and imaginary axes cross — the anchor from which every other complex number is measured.
ReIm-5−5i-4−4i-3−3i-2−2i-1−1i11i22i33i44i55i0z = 0
z = 0, frozen

The point sits exactly where the axes cross. Triangle, projections, conjugate ghost and modulus circle have all collapsed to nothing — at distance 0 from the origin there is nothing left to draw.

The origin is the only complex number with modulus zero: ∣z∣=0|z| = 0 if and only if z=0z = 0. Every other point sits at some positive distance from it, and that distance is exactly what the modulus measures. In the frozen view above, all the tool's usual decorations — the right triangle, the modulus circle, the conjugate ghost — have collapsed into the single point, because every one of them is built from a nonzero distance or height.

Algebraically, zero is the additive identity: z+0=zz + 0 = z for every complex number zz. It is also its own conjugate and its own negative, 0ˉ=−0=0\bar{0} = -0 = 0, which makes it the one fixed point shared by both reflection across the real axis and point reflection through the origin.

The origin is also the only point that belongs to both axes at once — simultaneously a purely real number and a pure imaginary one. Every classification the explanation panel can announce meets here.

Purely Real Numbers

When the imaginary part is zero, z=a+0i=az = a + 0i = a is an ordinary real number — the entire real number line lives inside the complex plane as its horizontal axis.
ReIm-5−5i-4−4i-3−3i-2−2i-1−1i11i22i33i44i55i0z = 3
z = 3, frozen

With b = 0 the point rides the orange real axis. No triangle can form — the vertical leg would have height zero — and the dashed modulus circle crosses the axis at 3 and −3, the two real numbers at distance 3 from the origin.

For a real number, the modulus formula collapses to the familiar absolute value: ∣a+0i∣=a2+02=∣a∣|a + 0i| = \sqrt{a^2 + 0^2} = |a|. Complex modulus and real absolute value agree completely on the real line — the new definition extends the old one rather than replacing it.

Conjugation does nothing here: a+0i‾=a−0i=a\overline{a + 0i} = a - 0i = a. Reflecting across the real axis leaves every point of the axis where it stands, which is why the tool shows no conjugate ghost in this state. In fact this property characterizes the real numbers: zz is real exactly when z=zˉz = \bar{z}.

Try steering the point along the axis with the a input field while watching the display panel: the real part card tracks the number itself, the imaginary part card stays pinned at 0, and the modulus card reports ∣a∣|a|. Compare this with the pure imaginary case, where the roles of the two cards swap exactly.

Pure Imaginary Numbers

When the real part is zero, z=0+bi=biz = 0 + bi = bi is a pure imaginary number. These numbers occupy the vertical axis — the real line's perpendicular twin, scaled by the imaginary unit ii.
ReIm-5−5i-4−4i-3−3i-2−2i-1−1i11i22i33i44i55i0z̅z = 2i
z = 2i, frozen

The point rides the navy imaginary axis at height 2. Its conjugate ghost sits directly below at −2i — reflection across the real axis sends a pure imaginary number to its own negative.

The modulus of bibi is ∣b∣|b|: the formula gives 02+b2=∣b∣\sqrt{0^2 + b^2} = |b|, the unsigned height above or below the origin. So 2i2i and −2i-2i share the same modulus, just as 33 and −3-3 do on the real axis.

Conjugation acts as negation here: bi‾=−bi\overline{bi} = -bi. That is the exact opposite of the purely real case, where conjugation changes nothing. The dashed line in the frozen view connects zz and zˉ\bar{z} straight through the origin — for pure imaginary numbers, the conjugate and the additive inverse coincide.

Squaring a pure imaginary number always produces a real number that is zero or negative: (bi)2=b2i2=−b2(bi)^2 = b^2 i^2 = -b^2. This is the defining behavior inherited from i2=−1i^2 = -1, and it is why the equation x2+1=0x^2 + 1 = 0, unsolvable on the real line, finds its two solutions ±i\pm i on this axis.

The Four Quadrants

    The complex plane divides into four quadrants based on the signs of the real and imaginary parts:

  • Quadrant I (upper right): a > 0 and b > 0
  • Quadrant II (upper left): a < 0 and b > 0
  • Quadrant III (lower left): a < 0 and b < 0
  • Quadrant IV (lower right): a > 0 and b < 0

    • Each quadrant has a dedicated section below with the tool frozen inside it. The explanation panel identifies which quadrant your point occupies and describes the sign pattern. Drag your point through different quadrants to see how the triangle orientation changes while the modulus formula remains consistent regardless of signs.

Quadrant I: Upper Right

When both the real and imaginary parts are positive, zz lands in Quadrant I — the upper-right region of the plane. This is the tool's home state: it opens at z=2+3iz = 2 + 3i, the point frozen below.
ReIm-5−5i-4−4i-3−3i-2−2i-1−1i11i22i33i44i55i0a = 2b = 3|z| = 3.61z̅z = 2 + 3i
z = 2 + 3i, frozen

The full anatomy on display: orange horizontal leg a = 2, navy vertical leg b = 3, blue hypotenuse |z| = √13 ≈ 3.61 reaching from the origin. The conjugate ghost waits below the real axis at 2 − 3i.

Quadrant I is where the right triangle appears in its textbook orientation: legs extending right and up, hypotenuse climbing away from the origin. The modulus computation reads directly off the figure: ∣z∣=22+32=13|z| = \sqrt{2^2 + 3^2} = \sqrt{13}.

Both coordinates carry positive signs, so the real part card and imaginary part card in the display panel show the point's position without any sign juggling. The conjugate zˉ=2−3i\bar{z} = 2 - 3i drops into Quadrant IV — conjugation always pairs the upper-right quadrant with the lower-right one.

Flipping the sign of the real part instead sends the point to Quadrant II; flipping both signs lands the additive inverse −z=−2−3i-z = -2 - 3i diagonally opposite in Quadrant III. All four variants share the same modulus 13\sqrt{13}, because squaring erases signs.

Quadrant II: Upper Left

A negative real part with a positive imaginary part places zz in Quadrant II, the upper-left region. The frozen example is z=−3+2iz = -3 + 2i.
ReIm-5−5i-4−4i-3−3i-2−2i-1−1i11i22i33i44i55i0a = -3b = 2|z| = 3.61z̅z = -3 + 2i
z = −3 + 2i, frozen

The triangle opens leftward: the orange leg runs to −3, the navy leg rises to 2, and the hypotenuse points up and to the left. The conjugate ghost mirrors the point into the lower-left.

Squaring wipes out signs, so the modulus ignores the quadrant: ∣−3+2i∣=(−3)2+22=13|-3 + 2i| = \sqrt{(-3)^2 + 2^2} = \sqrt{13} — exactly the distance of the frozen 2+3i2 + 3i in Quadrant I. The dashed modulus circle in the two frozen views is one and the same circle.

The conjugate −3+2i‾=−3−2i\overline{-3 + 2i} = -3 - 2i lands in Quadrant III: conjugation pairs the two left-hand quadrants with each other, just as it pairs the right-hand ones. Negating only the real part of a number is a reflection across the imaginary axis, and applied to Quadrant I it produces exactly this quadrant.

To reach Quadrant II from Quadrant I with algebra rather than dragging, multiply by ii: the product i(2+3i)=−3+2ii(2 + 3i) = -3 + 2i is precisely the frozen point above. Multiplication by ii rotates every complex number a quarter turn counterclockwise, carrying the upper-right quadrant onto the upper-left one.

Quadrant III: Lower Left

When both parts are negative, zz sits in Quadrant III, the lower-left region — diagonally opposite Quadrant I. The frozen example is z=−2−3iz = -2 - 3i.
ReIm-5−5i-4−4i-3−3i-2−2i-1−1i11i22i33i44i55i0a = -2b = -3|z| = 3.61z̅z = -2 − 3i
z = −2 − 3i, frozen

Both legs reach into negative territory: a = −2 leftward, b = −3 downward. The hypotenuse still measures √13 ≈ 3.61 — distance has no sign. The conjugate ghost now floats above the real axis.

This quadrant is home to the additive inverse of the tool's opening state: −(2+3i)=−2−3i-(2 + 3i) = -2 - 3i. Negating both parts is a point reflection straight through the origin, and it always exchanges Quadrant I with Quadrant III (and Quadrant II with Quadrant IV).

It is worth separating two reflections that are easy to conflate. The additive inverse −z-z reflects through the origin and flips both signs; the conjugate zˉ\bar{z} reflects across the real axis and flips only the imaginary sign. For the frozen point, −z=2+3i-z = 2 + 3i returns to Quadrant I while zˉ=−2+3i\bar{z} = -2 + 3i rises to Quadrant II — different destinations, both at modulus 13\sqrt{13}.

Notice how the right-triangle labels in the frozen view keep their signs — a=−2a = -2, b=−3b = -3 — while the legs themselves are unsigned lengths 2 and 3. The display panel makes the same distinction: signed coordinates in the part cards, an unsigned distance in the modulus card.

Quadrant IV: Lower Right

A positive real part with a negative imaginary part puts zz in Quadrant IV, the lower-right region. The frozen example is z=3−2iz = 3 - 2i.
ReIm-5−5i-4−4i-3−3i-2−2i-1−1i11i22i33i44i55i0a = 3b = -2|z| = 3.61z̅z = 3 − 2i
z = 3 − 2i, frozen

The triangle hangs below the real axis: orange leg to 3, navy leg down to −2. The conjugate ghost hovers above at 3 + 2i — the mirror image across the orange real axis.

Quadrant IV is the conjugate quadrant: every number here is the conjugate of a Quadrant I number, and vice versa. The frozen 3−2i3 - 2i is 3+2i‾\overline{3 + 2i}, and the tool's ghost point makes the pairing visible — the ghost of any lower-right point always hovers in the upper-right.

That pairing is what makes this quadrant matter in algebra. Whenever a polynomial with real coefficients has a complex root above the real axis, its conjugate root sits below it; the frozen pair multiplies to the real number zzˉ=∣z∣2=13z\bar{z} = |z|^2 = 13. Real answers out of complex inputs come from exactly this kind of matched pair.

Sign-wise the quadrant mirrors Quadrant II: one part positive, one negative, just distributed the other way. And the quarter-turn wheel keeps spinning here too — multiplying the frozen point by ii gives i(3−2i)=2+3ii(3 - 2i) = 2 + 3i, landing on the tool's opening state in Quadrant I.

What is a Complex Number?

A complex number extends the real number system by introducing the imaginary unit ii, defined by i2=−1i^2 = -1. Every complex number can be written as z=a+biz = a + bi, where aa and bb are real numbers.

The real part a=Re(z)a = \text{Re}(z) determines horizontal position on the complex plane. The imaginary part b=Im(z)b = \text{Im}(z) determines vertical position. Together, they specify a unique point in two-dimensional space.

Complex numbers arose historically to solve equations like x2+1=0x^2 + 1 = 0, which has no real solutions. Today they appear throughout mathematics, physics, and engineering — from signal processing and quantum mechanics to electrical circuits and fluid dynamics.

The Complex Plane (Argand Diagram)

The complex plane represents complex numbers geometrically. The horizontal axis displays real values, while the vertical axis displays imaginary values. A complex number z=a+biz = a + bi corresponds to the point (a,b)(a, b).

This representation, also called an Argand diagram, transforms algebraic operations into geometric ones. Addition becomes vector addition. Multiplication involves rotation and scaling. The conjugate becomes reflection across the real axis.

The plane extends infinitely in all directions. The Range control in this tool lets you focus on different portions — zoom in for detail near the origin, or zoom out to see larger values. Grid lines help you read coordinates at a glance.

Modulus and the Pythagorean Theorem

The modulus (or absolute value) of a complex number measures its distance from the origin:

∣z∣=∣a+bi∣=a2+b2|z| = |a + bi| = \sqrt{a^2 + b^2}


This formula follows directly from the Pythagorean theorem. The point (a,b)(a, b) forms a right triangle with legs of length ∣a∣|a| and ∣b∣|b|. The hypotenuse — the straight-line distance to the origin — equals a2+b2\sqrt{a^2 + b^2}.

Key properties of modulus include: ∣z∣≥0|z| \geq 0 always, with equality only when z=0z = 0. For real numbers, modulus matches absolute value. The modulus of a product equals the product of moduli: ∣z1z2∣=∣z1∣∣z2∣|z_1 z_2| = |z_1| |z_2|.

Complex Conjugates and Their Properties

The complex conjugate of z=a+biz = a + bi is zˉ=a−bi\bar{z} = a - bi. Geometrically, conjugation reflects a point across the real axis — the real part stays the same while the imaginary part changes sign.

Conjugates have important algebraic properties. The product z⋅zˉ=a2+b2=∣z∣2z \cdot \bar{z} = a^2 + b^2 = |z|^2 always yields a non-negative real number. This makes conjugates essential for division: to compute wz\frac{w}{z}, multiply numerator and denominator by zˉ\bar{z} to get a real denominator.

Other properties include: z1+z2‾=z1ˉ+z2ˉ\overline{z_1 + z_2} = \bar{z_1} + \bar{z_2} and z1⋅z2‾=z1ˉ⋅z2ˉ\overline{z_1 \cdot z_2} = \bar{z_1} \cdot \bar{z_2}. The conjugate of a conjugate returns the original: zˉ‾=z\overline{\bar{z}} = z.

Related Concepts and Tools

The explorer shows the full picture of a single complex number at once. Each tool below takes one of those properties and goes deeper.

Complex Conjugate Explorer — the ghost point zˉ\bar{z} you see reflected below the real axis here gets its own dedicated tool. Drag zz and watch the conjugate mirror it in real time, with the identity z⋅zˉ=∣z∣2z \cdot \bar{z} = |z|^2 verified step by step.

Polar & Rectangular Converter — the modulus and right triangle visible in this explorer are the core of the polar representation reiθre^{i\theta}. This converter makes the switch between the two forms explicit, with live conversion formulas and an angle arc.

Addition & Subtraction Visualizer — move from one complex number to two. This tool places z1z_1 and z2z_2 on the same plane and shows the parallelogram rule for addition and the difference vector for subtraction.

Multiplication Visualizer — see what happens when two complex numbers are multiplied: moduli scale and arguments rotate. The single modulus circle you see in the explorer becomes a geometric transformation.

Distance & Midpoint Tool — extends the modulus concept to two points. The distance ∣z1−z2∣|z_1 - z_2| between any two complex numbers is computed and visualized directly on the plane.

De Moivre's Theorem Visualizer — takes the modulus and argument shown here and raises them to a power, rnr^n and nθn\theta.

Division Visualizer — the inverse of the rotation-and-scaling this explorer displays for a single point.

Euler's Formula Explorer — explains why the right triangle in this explorer can be written as reiθre^{i\theta} at all.

Powers of i Calculator — what happens when the point in this explorer is ii itself and you keep multiplying: a four-step cycle around the unit circle.