Getting Started with the Complex Numbers Calculator
The complex numbers calculator has three main sections: input panels on the left for entering your complex numbers, operation buttons in the middle for selecting calculations, and quick guide panels on the right showing formulas and explanations. The interface adjusts based on which operation you choose—some operations need two numbers while others need only one.
Start by looking at the First Number (z₁) input panel. You'll see two boxes with a plus sign and the letter i between them. The left box is for the real part (the regular number), and the right box is for the imaginary part (the coefficient of i). Enter any numbers including decimals and negatives.
Below your inputs, the calculator displays your complex number in standard form like 3+4i or −2−5i. This formatted display updates instantly as you type, helping you verify your input is correct before calculating. Watch this area to make sure the calculator interprets your numbers as you intended.
The Choose Operation section contains seven buttons for different calculations: Addition, Subtraction, Multiplication, Division, Conjugate, Modulus, and Power. Click any operation button to select it—the button turns blue when active. For Conjugate and Modulus, the second number panel disappears since these operations work on a single complex number. The result appears automatically below without needing to click Calculate.
Entering Complex Numbers in the Input Fields
Click in the first input box under First Number (z₁) to enter your real part. Type any number: positive like 5, negative like −3, decimal like 2.5, or zero like 0. The real part represents the horizontal component on the complex plane—it's just a regular number without i.
Move to the second box to enter your imaginary part. This is the coefficient of i, so if you want 4i, type 4. For −3i, type −3. You don't need to type the i—the calculator adds it automatically. The imaginary part represents the vertical component on the complex plane.
For pure real numbers like 5, enter 5 in the real box and 0 in the imaginary box. For pure imaginary numbers like 3i, enter 0 in the real box and 3 in the imaginary box. The calculator handles all combinations: 2+3i, −4−7i, 6+0i, 0+5i.
When an operation requires a second complex number, the Second Number (z₂) panel appears on the right. Enter its real and imaginary parts the same way. The formatted preview below each panel shows exactly how the calculator interprets your input: (3+4i), (0−2i), or (5+0i). If the preview shows "Enter values," you haven't typed anything yet.
Using the Four Basic Arithmetic Operations
Click the Addition (z₁ + z₂) button to add two complex numbers. Enter 3+2i in the first panel and 1+4i in the second. The calculator adds real parts together: 3+1=4. Then it adds imaginary parts: 2i+4i=6i. The result 4+6i appears immediately in the green result box below.
For Subtraction (z₁ - z₂), try (5+3i)−(2+1i). The calculator subtracts the second number's real part from the first: 5−2=3. Then imaginary parts: 3i−1i=2i. Result: 3+2i. Notice negative results work too: (1+1i)−(3+4i)=−2−3i.
Multiplication (z₁ × z₂) uses the FOIL method. Enter (2+3i) and (1+4i). The calculator multiplies: 2×1=2, 2×4i=8i, 3i×1=3i, and 3i×4i=12i2=−12 (remember i2=−1). Combining gives 2+8i+3i−12=−10+11i. Click Show Steps to see this breakdown.
Division (z₁ ÷ z₂) multiplies by the conjugate. For (1+2i)÷(3+4i), the calculator finds the conjugate of the denominator (3−4i), multiplies numerator and denominator by it, then simplifies. The result appears as a decimal like 0.44+0.08i. Division by zero (like ÷(0+0i)) shows an error: "Division by zero."
Using Single-Number Operations: Conjugate and Modulus
Select Conjugate (z₁*) to find the complex conjugate. When you click this operation, the second number panel disappears—you only need one input. Enter 3+4i in the first panel. The conjugate flips the sign of the imaginary part: 3−4i. The real part stays the same.
Try negative imaginary parts too. Enter 5−2i and the conjugate becomes 5+2i. For pure imaginary 0+7i, the conjugate is 0−7i. For pure real 8+0i, the conjugate is 8−0i=8. The conjugate is useful for division and finding the modulus squared: z×z∗=∣z∣2.
Modulus (|z₁|) calculates the distance from the origin to your point in the complex plane. Enter 3+4i. The calculator computes 32+42=9+16=25=5. This uses the Pythagorean theorem—the modulus is always a positive real number.
Try 5+12i to get modulus 13 (since 52+122=25+144=169=132). For 1+1i, the modulus is 2≈1.4142. Pure imaginary 0+3i has modulus 3. Pure real 4+0i has modulus 4. Click Show Steps to see the a2+b2 calculation broken down step-by-step.
Reading Results and Step-by-Step Solutions
After entering your numbers and selecting an operation, the result appears in the Result section at the bottom. For complex number answers, you'll see the formatted output like −10+11i, 3−2i, or 0.44+0.08i. The calculator displays large, bold text in green for successful calculations.
For modulus operations, the result is a single real number like 5.0000 or 13.0000. This appears without the +0i part since the modulus is always real. Decimal results show four decimal places for precision: 1.4142 for 2 or 0.4400 for certain divisions.
Click the green Show Steps button to reveal the step-by-step solution. A white panel expands below the result showing each calculation stage. For addition, you'll see lines like "(3+2i)+(1+4i)", then "=(3+1)+(2+4)i", then "=4+6i". For multiplication, the steps show FOIL expansion, the i2=−1 substitution, and final combination.
The Clear All red button resets everything—both input panels empty, the result disappears, and the interface returns to its initial state. Use this when starting a completely new problem. The result updates automatically as you type new numbers, so you don't need to clear between similar calculations unless you want a fresh start.
Using the Quick Guide Panels
The right side of the calculator contains two helpful guide panels that stay visible as you scroll. The top panel, Complex Numbers Guide, displays general information about complex numbers: the standard form z=a+bi, what real and imaginary parts mean, example numbers, and the key property i2=−1.
This general guide is always visible regardless of which operation you select. Read it first if you're new to complex numbers—it explains the notation and basic concepts you need to understand your calculations. The guide includes bullet points and examples formatted with proper mathematical notation.
Below it, the Operation Guide panel changes based on your selected operation. When you click Addition, it shows the addition formula (a+bi)+(c+di)=(a+c)+(b+d)i, explains how it works, and provides an example. Switch to Multiplication and the panel updates to show FOIL method, the i2=−1 rule, and a multiplication example.
Each operation guide includes the relevant formula, step-by-step explanation of the method, important properties or rules, and at least one worked example. For Division, you'll see the conjugate multiplication technique. For Modulus, you'll see the distance formula ∣z∣=a2+b2. These guides help you understand what the calculator is doing behind the scenes without needing to leave the page.
What Are Complex Numbers
Complex numbers extend the real number system by introducing i, the imaginary unit, defined as i=−1 or equivalently i2=−1. This allows us to take square roots of negative numbers and solve equations like x2+1=0, which has no real solutions but has complex solutions x=i and x=−i.
A complex number has the form z=a+bi where a is the real part and b is the imaginary part (both a and b are real numbers). For example, in 3+4i, the real part is 3 and the imaginary part is 4. Don't be confused by the name "imaginary"—these numbers are just as valid as real numbers, just more abstract.
Pure real numbers like 5 can be written as 5+0i (imaginary part is zero). Pure imaginary numbers like 3i can be written as 0+3i (real part is zero). Complex numbers with both parts nonzero, like 2+3i, represent points in the two-dimensional complex plane where the horizontal axis is real and vertical axis is imaginary.
For comprehensive coverage of complex number theory, notation, and the complex plane visualization, see complex number fundamentals and imaginary unit properties.
How Complex Number Operations Work
Addition and subtraction treat complex numbers like binomials. Add real parts together and imaginary parts together separately: (3+4i)+(1+2i)=(3+1)+(4i+2i)=4+6i. Subtraction works the same: (5+3i)−(2+1i)=(5−2)+(3i−1i)=3+2i. Think of it as combining like terms.
Multiplication uses the distributive property (FOIL for binomials). Multiply (2+3i)(1+4i) by expanding: 2(1)+2(4i)+3i(1)+3i(4i)=2+8i+3i+12i2. Then substitute i2=−1: 2+11i+12(−1)=2+11i−12=−10+11i. The i2=−1 substitution is crucial.
Division requires the conjugate trick. To divide (a+bi)÷(c+di), multiply both numerator and denominator by the conjugate (c−di): c+dia+bi×c−dic−di. The denominator becomes (c+di)(c−di)=c2+d2, a real number, making the division straightforward.
The conjugatez∗=a−bi flips the imaginary sign. The modulus∣z∣=a2+b2 measures distance from the origin. These concepts connect: z×z∗=∣z∣2. For detailed operation rules and properties, see complex arithmetic theory and complex number algebra.
Understanding Conjugate and Modulus
The conjugate of a complex number z=a+bi is written zˉ or z∗ and equals a−bi. To find it, keep the real part the same and flip the sign of the imaginary part. The conjugate of 3+4i is 3−4i. The conjugate of 2−5i is 2+5i. For pure real 7+0i, the conjugate is 7−0i=7.
Conjugates are essential for division: multiplying by a conjugate eliminates the imaginary part in the denominator. They also satisfy z×zˉ=a2+b2, a real number. For 3+4i: (3+4i)(3−4i)=9−16i2=9+16=25. This equals ∣z∣2.
The modulus∣z∣=a2+b2 represents the distance from the origin (0,0) to the point (a,b) in the complex plane. For 3+4i, plot the point (3,4) and measure its distance from origin: 32+42=25=5. This is the Pythagorean theorem.
The modulus is always non-negative (zero or positive). For 0+0i, the modulus is 0. For all other complex numbers, it's positive. The modulus has useful properties: ∣z1×z2∣=∣z1∣×∣z2∣ and ∣z1÷z2∣=∣z1∣÷∣z2∣. For deeper understanding, see complex conjugate properties and modulus applications.
Related Calculators and Mathematical Tools
Quadratic Equation Solver - Solve equations like x2+1=0 that have complex solutions. When the discriminant is negative, the roots are complex numbers. Use the complex calculator to verify and work with these solutions.
Polynomial Calculator - Factor and solve polynomials with complex roots. Every polynomial can be factored into linear factors over the complex numbers, following the Fundamental Theorem of Algebra.
Trigonometry Calculator - Complex numbers connect deeply with trigonometry through Euler's formula: eiθ=cosθ+isinθ. This links exponential, trigonometric, and complex functions.
Matrix Calculator - Matrices can have complex number entries. Use complex arithmetic for eigenvalues, eigenvectors, and matrix operations in quantum mechanics and signal processing.
Vector Calculator - Complex numbers can be viewed as two-dimensional vectors. Addition works like vector addition, and the modulus is the vector magnitude.
For theoretical background, explore complex number theory, complex plane visualization, polar form of complex numbers, De Moivre's theorem, Euler's formula, and applications in engineering and physics.