CNO

Math & Statistics

Complex Number Operations Calculator

Perform Cartesian-form operations on two complex numbers. Calculate real and imaginary parts of the sum, difference, and product; both magnitudes; the distance between inputs; and the distance from the sum to an entered comparison point.

First number real partFirst number imaginary partSecond number real partSecond number imaginary partEntered real comparisonEntered imaginary comparison

Sum real part-

Sum imaginary part-

First minus second real part-

First minus second imaginary part-

Product real part-

Product imaginary part-

First-number magnitude-

Second-number magnitude-

Distance between complex points-

Sum-to-comparison distance-

Decision view

Complex-plane operation map

Complex-plane operation mapInput, sum, product, and comparison points share real and imaginary axes so component arithmetic and geometric distance can be inspected together.

Exact scenario comparisonSecond number real part changes while all other entered assumptions remain constant.

Second number real part	Sum real part	Sum imaginary part	First minus second real part	First minus second imaginary part	Product real part	Product imaginary part	First-number magnitude	Second-number magnitude	Distance between complex points	Sum-to-comparison distance

How to use Complex Number Operations Calculator

Enter real and imaginary components for both complex numbers.
Read addition and subtraction component by component.
Use the expanded product to audit the i-squared sign change.
Inspect magnitudes and points on the complex plane.

Calculator guide

Understanding Complex Number Operations Calculator

Complex arithmetic combines real and imaginary components algebraically and geometrically. This calculator shows addition, subtraction, multiplication, magnitudes, and distances while placing the inputs, sum, product, and comparison on the complex plane.

Components add directly Real combines with real and imaginary with imaginary.

Multiplication mixes Cross terms affect the imaginary component.

i squared changes sign The bd term subtracts from the real product.

Geometry verifies Every result is a point on the complex plane.

Calculation method

How the calculation works

Perform component-wise addition and subtraction and distributive multiplication, then compute magnitudes and geometric distances in the complex plane. In the Complex Number Operations Calculator, the live scenario varies second number real part and tracks product real part while the remaining results preserve the reconciliation path. Add and subtract matching components. Multiply distributively and use i squared equals -1, which changes the real product term to ac - bd and the imaginary term to ad + bc.

Detailed calculation process

Combine Cartesian components and verify them on the complex plane

The defaults use z1 = 3 + 4i and z2 = 2 - i, with comparison point 5 + 3i.

General formula: (a+bi)+(c+di) = (a+c)+(b+d)i; (a+bi)(c+di) = (ac-bd)+(ad+bc)i; |a+bi| = sqrt(a^2+b^2) Addition is component-wise. Multiplication distributes four terms and converts bdi^2 to -bd because i^2 = -1. Magnitude is the point's distance from the origin.

What each symbol means

a, b Real and imaginary components of the first complex number.

c, d Real and imaginary components of the second complex number.

i Imaginary unit satisfying i^2 = -1.

Re(z) Real coordinate of complex number z.

Im(z) Imaginary coordinate of complex number z.

|z| Complex magnitude, measured in the component unit.

Worked substitution with the default inputs

1. Add the two numbers: (3+4i) + (2-i) = (3+2) + (4-1)i = 5+3i The default sum exactly matches the entered comparison point.

2. Subtract the second number: (3+4i) - (2-i) = (3-2) + (4-(-1))i = 1+5i Subtracting a negative imaginary component increases the imaginary difference.

3. Expand the product: (3+4i)(2-i) = 6 - 3i + 8i - 4i^2 All real-imaginary cross terms remain visible before simplification.

4. Apply i squared equals negative one: 6 - 3i + 8i + 4 = 10 + 5i The i-squared term contributes +4 to the real component.

5. Calculate magnitudes and distances: |z1| = sqrt(3^2+4^2) = 5; |z2| = sqrt(2^2+(-1)^2) = 2.236068; |z1-z2| = sqrt(26) = 5.099020; sum gap = 0 Complex distance is ordinary Euclidean distance on the real-imaginary plane.

The defaults produce sum 5+3i, difference 1+5i, product 10+5i, magnitudes 5 and 2.236, input distance 5.099, and zero distance from the sum to the comparison.

Complex-plane map

Track inputs, sum, product, and comparison as points

Coordinate arrows make the algebraic operations spatially inspectable.

First number Point (a,b).

Second number Point (c,d).

Sum Vector addition endpoint.

Product Mixed-component result point.

Worked situations

Practical examples

The sum of 3+4i and 2-i is 5+3i.
The product simplifies to 10+5i after applying i squared equals -1.
The first number has magnitude 5 by the 3-4-5 triangle.

Better inputs

Useful tips

Keep the sign of every imaginary component explicit.
Plot results to distinguish component arithmetic from magnitude.
Use polar form when repeated multiplication, powers, or phase are central.

Before relying on the result

Limitations and common mistakes

The calculator covers Cartesian addition, subtraction, multiplication, magnitude, and distance only.
Division, conjugates, arguments, roots, and branch choices are outside this page.
Component units must be compatible when the complex numbers represent physical quantities.

Reference

Key terms

Complex plane: Coordinate plane with real values horizontally and imaginary values vertically.
Imaginary unit: Number i defined by i squared equals -1.
Magnitude: Euclidean distance from a complex point to the origin.

Important note

Calculated directly from the entered values using the displayed formula and rounding settings.

Frequently asked questions

Why is the real product ac minus bd?

The term bdi squared equals -bd because i squared is -1.

Is magnitude the same as the real part?

No. Magnitude uses both coordinates: sqrt(a squared plus b squared).

Why does the sum plot like vector addition?

Complex addition adds the horizontal and vertical components just like 2D vectors.

Can I use this for complex division?

No. Division and conjugate-based formulas are outside this calculator's current scope.