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Complex Numbers

The expression evaluator has built-in support for complex numbers, allowing you to write, combine, and evaluate expressions that mix real and imaginary parts without any special setup or configuration.

Imaginary Literals

An imaginary number is written using the i suffix directly after a numeric coefficient. If no coefficient is given, a value of 1 is assumed.

expr.evaluate('i');        // i
expr.evaluate('2i');       // 2i
expr.evaluate('-i');       // -i
expr.evaluate('0.5i');     // 0.5i
expr.evaluate('-3.25i');   // -3.25i
  • i on its own represents the imaginary unit (equivalent to 1i).
  • A leading - negates the imaginary coefficient as expected.
  • Decimal coefficients are fully supported.

Complex Arithmetic

Complex numbers can be freely combined with real numbers using standard arithmetic operators: addition (+), subtraction (-), multiplication (*), and division (/). The evaluator automatically applies the correct rules for complex math (e.g. i * i = -1).

Addition and Subtraction

expr.evaluate('(3 + 2i) + (1 - i)');   // "4 + i"
expr.evaluate('(5 + 3i) - (2 + i)');   // "3 + 2i"

Real parts and imaginary parts are combined independently.

Multiplication

expr.evaluate('(1 + i) * (1 - i)');    // "2"
expr.evaluate('(2 + i) * (3 + 4i)');   // "2 + 11i"

Multiplication follows the distributive law, with i * i evaluating to -1. Note that when the resulting imaginary part is 0, the result is simplified to a plain real number (as shown in the first example above).

Division

expr.evaluate('(3 + 2i) / (1 + i)');   // "2.5 - 0.5i"
expr.evaluate('10 / (2i)');            // "-5i"

Division is performed by multiplying both the numerator and denominator by the complex conjugate of the denominator, then simplifying.

Mixing Real and Complex Values

Real numbers and complex numbers can be combined directly in the same expression:

expr.evaluate('5 + 3i');     // "5 + 3i"
expr.evaluate('5 + i - 2');  // "3 + i"
expr.evaluate('2 * (1 + i)'); // "2 + 2i"

Display Formatting

Results are returned as formatted strings rather than raw numeric objects, making them easy to print or log directly.

Value Display Format
Pure imaginary unit "i"
Negative imaginary unit "-i"
Imaginary with coefficient "3 + 2i"
Real result (imaginary part is zero) "4"
Real and negative imaginary "2.5 - 0.5i"

Formatting Rules

  • If the imaginary part is 0, only the real part is shown (e.g. "2", not "2 + 0i").
  • If the real part is 0, only the imaginary part is shown (e.g. "2i", not "0 + 2i").
  • A coefficient of 1 or -1 on the imaginary part is shown as i or -i, without the numeral.
  • Negative imaginary parts are displayed with a - sign and no extra + (e.g. "2.5 - 0.5i", not "2.5 + -0.5i").
  • Positive imaginary parts following a real part are prefixed with " + " for readability (e.g. "4 + i").