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### 7.3 Complex Number Functions

The J (calc-conj) [conj] command computes the complex conjugate of a number. For complex number ‘a+bi’, the complex conjugate is ‘a-bi’. If the argument is a real number, this command leaves it the same. If the argument is a vector or matrix, this command replaces each element by its complex conjugate.

The G (calc-argument) [arg] command computes the “argument” or polar angle of a complex number. For a number in polar notation, this is simply the second component of the pair (r;theta)'. The result is expressed according to the current angular mode and will be in the range -180 degrees (exclusive) to +180 degrees (inclusive), or the equivalent range in radians.

The calc-imaginary command multiplies the number on the top of the stack by the imaginary number ‘i = (0,1)’. This command is not normally bound to a key in Calc, but it is available on the <IMAG> button in Keypad mode.

The f r (calc-re) [re] command replaces a complex number by its real part. This command has no effect on real numbers. (As an added convenience, re applied to a modulo form extracts the value part.)

The f i (calc-im) [im] command replaces a complex number by its imaginary part; real numbers are converted to zero. With a vector or matrix argument, these functions operate element-wise.

The v p (calc-pack) command can pack the top two numbers on the stack into a composite object such as a complex number. With a prefix argument of -1, it produces a rectangular complex number; with an argument of -2, it produces a polar complex number. (Also, see Building Vectors.)

The v u (calc-unpack`) command takes the complex number (or other composite object) on the top of the stack and unpacks it into its separate components.