Complex numbers appear everywhere from signal processing labs to geometry sketches: they let you combine a real and imaginary part to represent rotation, oscillation, and two-dimensional points with a compact notation. Seeing concrete examples helps bridge abstract definitions and practical use.
There are 21 Examples of Complex Numbers, ranging from -0.50+0.87i to i. For each entry you’ll find below the value listed as a+bi, its Modulus (r), and a short Category / use note to indicate how that number is typically applied — you’ll find below.
How do I read the Modulus (r) and Category / use for each complex number?
The Modulus (r) is the magnitude, computed as sqrt(a^2 + b^2), and tells you the distance from the origin when plotting a+bi on the plane; the Category / use label clarifies whether a number is, for example, purely real, purely imaginary, on the unit circle, or commonly used in specific applications like phasors or eigenvalues.
Can these examples help me with calculations or plotting?
Yes — using the a+bi form you can add and subtract components directly, while the Modulus (r) (and implied angle) lets you convert to polar form for multiplication, division, and plotting; the Category / use notes give quick context for which operations or interpretations matter most for each example.
Examples of Complex Numbers
| Name | a+bi | Modulus (r) | Category / use |
|---|---|---|---|
| i | 0+1i | 1.00 | Fundamental unit / algebra |
| -i | 0-1i | 1.00 | Fundamental unit / algebra |
| 0 | 0+0i | 0.00 | Zero element / algebra |
| 1 | 1+0i | 1.00 | Real unit / normalization |
| -1 | -1+0i | 1.00 | Real negative / Euler identity |
| 1+i | 1.00+1.00i | 1.41 | Diagonal point / geometry |
| 1-i | 1.00-1.00i | 1.41 | Conjugate example / geometry |
| 2+3i | 2.00+3.00i | 3.61 | Arbitrary example / visualization |
| 3+4i | 3.00+4.00i | 5.00 | Gaussian integer / Pythagorean |
| 3-4i | 3.00-4.00i | 5.00 | Conjugate / Gaussian integer |
| 0+5i | 0.00+5.00i | 5.00 | Pure imaginary / vertical axis |
| 0.50+0.87i | 0.50+0.87i | 1.00 | e^{iπ/3} / root of unity |
| -0.50+0.87i | -0.50+0.87i | 1.00 | Primitive cube root / algebra |
| 0.71+0.71i | 0.71+0.71i | 1.00 | e^{iπ/4} / rotation |
| 0.54+0.84i | 0.54+0.84i | 1.00 | e^{i} / complex exponential |
| 0.43-0.25i | 0.43-0.25i | 0.50 | Damped phasor / applied signals |
| 1.73+1.00i | 1.73+1.00i | 2.00 | Scaled phasor / polar form |
| 1+2i | 1.00+2.00i | 2.24 | Gaussian integer / algebraic example |
| 0.31+0.95i | 0.31+0.95i | 1.00 | Primitive 5th root / symmetry |
| -2.00+1.50i | -2.00+1.50i | 2.50 | Illustrative negative real / applied |
| 4.00+3.00i | 4.00+3.00i | 5.00 | Large Gaussian integer / geometry |
Images and Descriptions
i
i is the imaginary unit whose square is -1. Its modulus is 1 and argument 90°. It’s the basic building block of complex numbers, essential in algebra, signal theory, and rotating-vector interpretations.
-i
-i is the negative imaginary unit with modulus 1 and argument -90°. It pairs with i as a conjugate and appears in solutions to differential equations and in phasor rotations pointing downward on the complex plane.
0
Zero is the only complex number with modulus zero and undefined argument. It’s the additive identity in the complex plane, often used as a reference point and origin for geometric interpretations and analytic formulas.
1
1 sits on the real axis with modulus 1 and argument 0. It’s frequently used as a multiplicative identity, a trivial complex number, and as a phase reference for rotations in engineering and physics.
-1
-1 is the real number at 180° on the unit circle. Famous from Euler’s formula, e^{iπ}=-1, it links exponentials, trigonometry, and complex analysis and often appears in symmetry and wave problems.
1+i
1+i lies on the 45° line in the first quadrant with modulus √2≈1.41. It’s a simple nontrivial complex number used to illustrate addition, multiplication by i (rotation), and Gaussian integer arithmetic.
1-i
1−i is the complex conjugate of 1+i, mirrored across the real axis, modulus √2. Conjugate pairs like this are important for obtaining real results from complex expressions and in solving polynomial equations.
2+3i
2+3i is a typical off-axis complex number with modulus √13≈3.61 and argument about 56.3°. It’s useful for plotting, testing algebraic operations, and modeling two-dimensional quantities like impedance or waves with phase.
3+4i
3+4i is a Gaussian integer with modulus 5, illustrating integer-coordinate points in the complex plane. Its tidy 3–4–5 relationship makes it a standard example in geometry, number theory, and lattice visualizations.
3-4i
3−4i is the conjugate of 3+4i, same modulus 5 and argument negative. Conjugates are central to taking magnitudes, rationalizing denominators, and studying symmetry in complex-valued signals and equations.
0+5i
5i is a pure imaginary number on the positive imaginary axis with modulus 5. Pure imaginaries model 90° phase-shifted components in AC circuits and appear in transforms where real and imaginary parts separate cleanly.
0.50+0.87i
This is e^{iπ/3}, a sixth-root-of-unity point at 60° on the unit circle with modulus 1. It’s common in symmetric polygon geometry, phasor analysis, and as a simple complex rotation by one-sixth turn.
-0.50+0.87i
This is a primitive cube root of unity (e^{2πi/3}), located at 120° with modulus 1. Such roots solve z^3=1, appear in discrete Fourier transforms, cyclotomic polynomials, and symmetric factorization problems.
0.71+0.71i
0.71+0.71i approximates e^{iπ/4}, a 45° rotation on the unit circle. It’s an elementary phasor used to illustrate equal real and imaginary parts, quarter-turn symmetries, and simple digital-signal rotations.
0.54+0.84i
0.54+0.84i is e^{i} (cos1 + i sin1) with modulus 1. It’s a transcendental point on the unit circle showing how real exponent inputs create rotations; important in Fourier, quantum phases, and complex dynamics.
0.43-0.25i
This number equals 0.5·e^{-iπ/6}, a damped phasor with modulus 0.50 and angle −30°. Engineers use such scaled complex numbers for decaying oscillations, transient analysis, and representing magnitude-plus-phase of signals.
1.73+1.00i
1.73+1.00i is 2·e^{iπ/6} (modulus 2, angle 30°). It demonstrates converting between polar and rectangular forms and models amplitude-and-phase quantities like currents, voltages, or rotating vectors in mechanics.
1+2i
1+2i is a simple Gaussian integer with modulus √5≈2.24 and nontrivial argument. It’s often used to demonstrate integer factorization in the Gaussian integers and to build intuition about magnitude and angle changes under arithmetic.
0.31+0.95i
Approximately e^{2πi/5}, this primitive 5th root of unity lies on the unit circle near 72°. It’s central to pentagonal symmetry, discrete Fourier components, and algebraic structures of cyclic polynomials.
-2.00+1.50i
−2+1.50i has modulus 2.50 and an argument in the second quadrant. It serves as a nontrivial example with negative real part, appearing in stability analysis, control theory poles, and complex-plane visual checks.
4.00+3.00i
4+3i is another Gaussian integer with modulus 5, showing that different integer pairs can share a modulus. It’s useful in lattice geometry, number-theoretic examples, and visualizing integer-packed points in the complex plane.
