Derive displays the imaginary unit (the square-root of -1) as the italicized letter i.  Note that it is distinct from the upright letter i, which can be used like any other letter in variable and function names.  The imaginary unit can be entered by clicking on the i on the math symbol toolbar, by pressing Ctrl+I, or by typing #i on the expression entry line.


A complex expression is an expression that contains the imaginary unit.  A real expression does not contain the imaginary unit.  Operations such as factoring and equation solving can yield complex expressions.  Therefore, it is important to recognize them even if you do not plan to enter any complex expressions.


A complex expression is in rectangular form if it consists of the sum of a real expression and the imaginary unit times another real expression.  Derive simplifies complex expressions to rectangular form.  For example, 

(1 + #i)^3

simplifies to

-2 + 2·i

and

(#i + 7)/(3·#i + 4)

approximates to

1.24 - 0.68·i


A point in the complex plane is determined by its magnitude (i.e. its distance from the origin) and its phase angle (i.e. the angle of the point from the x-axis).  By definition, the phase angle in radians is restricted to the interval

-π < phase angle ≤ π

If z is a point in the complex plane and n is an integer, then z^(1/n) could be any one of n distinct points, called branches, in the complex plane:  The magnitude of z^(1/n) is equal to |z|^(1/n), but the phase of z^(1/n) could be any one of n equally spaced angles.  For example, the magnitude of √9 is 3, but the phase can be either 0 radians yielding 3, or π radians yielding -3.


The principal branch of z^(1/n) is the point in the complex plane whose magnitude is |z|^(1/n) and whose phase is the phase of z divided by n.  For example, the principal branch of √9 is 3 and the principal branch of (-8)^(1/3) is 1 + √3i.


Use the Branch field of the Options > Mode Settings > Simplification command19_L5FP to change the branch Derive returns when simplifying fractional powers.  When a new branch option is selected, the command generates an expression of the form

Branch := option

where option is Principal or Real.  The branch option can also be changed by entering on the expression entry line expressions of the above form using upper and lower case exactly as shown.


Select the Principal option to force Derive to always select the principal branch.  The principal branch is the most widely accepted systematic branch choice.  Principal is the factory default branch setting.


Select the Real option to force Derive to select a real branch if one exists, even if it is not the principal branch.  For example, if this option is selected,

(-8)^(1/3)

simplifies to -2 instead of  1 + √3i.  Note that the Real branch selection may be preferable for precalculus applications.


Note that if n and m are integers, z^(m/n) is equivalent to (z^(1/n))^m.  Thus when simplifying z^(m/n) Derive first picks the appropriate branch for z^(1/n) and then raises this result to the mth power.


Other Options > Mode Settings > Simplification command19_L5FP fields

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