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Math

Complex Number Calculator

Add, subtract, multiply or divide two Cartesian complex numbers and inspect the result's modulus and argument.

Complex Number Calculator

Result4 + 2i

Model and interpretation

Finite Cartesian components only. Division by zero is invalid; the argument of zero is undefined. These are numerical algebra operations, not a symbolic computer algebra system.

Re(z)
4
Im(z)
2
|z|
4.47214
arg(z) (rad)
0.463648
arg(z) (°)
26.5651

Method source: OpenStax College Algebra 2e — Complex NumbersReport a correction

What this calculator answers

Complex Number Calculator applies the registered method “z = a + bi; i² = −1; |z| = hypot(a,b); arg(z) = atan2(b,a)” to the displayed Operation, Real part of A, Imaginary part of A, Real part of B, Imaginary part of B inputs. Both worked examples come from the production calculation engine, and OpenStax College Algebra 2e — Complex Numbers defines the cited method or convention; no hidden inputs or current external data are inferred.

Variables, defaults, and limits

Operation
Operation is a choice input. The displayed starter value is Add; it is an example, not a hidden assumption.
Default: Add
Accepted values: The field has no narrower HTML limit, but it must still satisfy the documented formula domain and produce a finite result. Available choices: Add, Subtract, Multiply, Divide.
Real part of A
Real part of A is a number input. The displayed starter value is 3; it is an example, not a hidden assumption.
Default: 3
Accepted values: The field has no narrower HTML limit, but it must still satisfy the documented formula domain and produce a finite result. Numeric entries must be finite and unambiguous.
Imaginary part of A
Imaginary part of A is a number input. The displayed starter value is 4; it is an example, not a hidden assumption.
Default: 4
Accepted values: The field has no narrower HTML limit, but it must still satisfy the documented formula domain and produce a finite result. Numeric entries must be finite and unambiguous.
Real part of B
Real part of B is a number input. The displayed starter value is 1; it is an example, not a hidden assumption.
Default: 1
Accepted values: The field has no narrower HTML limit, but it must still satisfy the documented formula domain and produce a finite result. Numeric entries must be finite and unambiguous.
Imaginary part of B
Imaginary part of B is a number input. The displayed starter value is -2; it is an example, not a hidden assumption.
Default: -2
Accepted values: The field has no narrower HTML limit, but it must still satisfy the documented formula domain and produce a finite result. Numeric entries must be finite and unambiguous.

Formula and calculation rule

R[complex-number-calculator]=z = a + bi; i² = −1; |z| = hypot(a,b); arg(z) = atan2(b,a); x₁=Operation; x₂=Real part of A; x₃=Imaginary part of A; x₄=Real part of B; x₅=Imaginary part of B

Complex Number Calculator evaluates the registered expression “R[complex-number-calculator]=z = a + bi; i² = −1; |z| = hypot(a,b); arg(z) = atan2(b,a); x₁=Operation; x₂=Real part of A; x₃=Imaginary part of A; x₄=Real part of B; x₅=Imaginary part of B” with the validated inputs. Full calculation precision is retained until the result is formatted for display.

Worked examples

Complex Number Calculator evaluated example 1

Use the page’s labelled starter inputs to verify Complex Number Calculator.

Inputs
  • Operation: Add
  • Real part of A: 3
  • Imaginary part of A: 4
  • Real part of B: 1
  • Imaginary part of B: -2

Evaluated result: 4 + 2i

Complex Number Calculator evaluated example 2

Change one valid input or choice and evaluate Complex Number Calculator again to check that the result responds deterministically.

Inputs
  • Operation: Divide
  • Real part of A: 3
  • Imaginary part of A: 4
  • Real part of B: 1
  • Imaginary part of B: -2

Evaluated result: -1 + 2i

Assumptions

  • Complex Number Calculator uses the visitor-entered values exactly as labelled; it does not retrieve private records or current rates.
  • The calculation is limited to the declared relationship: R[complex-number-calculator]=z = a + bi; i² = −1; |z| = hypot(a,b); arg(z) = atan2(b,a); x₁=Operation; x₂=Real part of A; x₃=Imaginary part of A; x₄=Real part of B; x₅=Imaginary part of B.
  • Intermediate values are not rounded; display rounding is applied only at the presentation boundary.
  • The deterministic calculation runs locally and does not upload calculator inputs to the application API.
  • The result is an educational arithmetic result and excludes facts that are not represented by an input.

Validation and boundaries

  • Every visible required input must be present; a missing value is never replaced with zero.
  • NaN, positive or negative infinity, unsafe overflow, and a non-finite final result are rejected.
  • Field-specific minimums, maximums, and choices apply to Operation, Real part of A, Imaginary part of A, Real part of B, Imaginary part of B.
  • Zero, negative values, and discrete counts are accepted only when the displayed field definition permits them.

Review and correction links

Common mistakes

  • Confirm the meaning and unit of Operation, Real part of A, Imaginary part of A, Real part of B, Imaginary part of B before calculating; a numerically valid value can still use the wrong convention.
  • Do not round intermediate values when checking the result, because early rounding can change the last displayed digits.
  • Changing an unstated real-world assumption does not change the calculator until the corresponding displayed input is changed.
  • Use the result as the answer to the displayed mathematical question, not to a different word problem with hidden conditions.

Frequently asked questions

What question does Complex Number Calculator answer?
Add, subtract, multiply or divide two Cartesian complex numbers and inspect the result's modulus and argument. The implemented relationship is R[complex-number-calculator]=z = a + bi; i² = −1; |z| = hypot(a,b); arg(z) = atan2(b,a); x₁=Operation; x₂=Real part of A; x₃=Imaginary part of A; x₄=Real part of B; x₅=Imaginary part of B.
Which inputs does Complex Number Calculator use?
It uses the visible fields Operation, Real part of A, Imaginary part of A, Real part of B, Imaginary part of B. No hidden value is substituted for an omitted required input.
What happens when Complex Number Calculator receives an invalid or extreme value?
The page rejects missing, ambiguous, out-of-range, or non-finite values and refuses to display a non-finite result.
Why can Complex Number Calculator differ from another result?
Different unit conventions, endpoint policies, fee or rate assumptions, formula domains, and premature rounding can produce a different answer.
Can I use Complex Number Calculator as professional advice?
No. It answers the displayed mathematical question and cannot account for omitted real-world facts.
How can I verify the Complex Number Calculator result?
Recalculate the two evaluated examples without rounding intermediate values and compare the declared formula or convention with OpenStax College Algebra 2e — Complex Numbers.

Source and review scope

OpenStax College Algebra 2e — Complex Numbers

Scope: OpenStax College Algebra 2e — Complex Numbers is used to check the formula, definition, or convention relevant to Complex Number Calculator. The citation does not supply current personal, lender, tax, medical, or market data.

Source checked:

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