Miscellaneous`RealOnly`

In high school algebra, exponents and radicals are taught early, but complex numbers are usually left to more advanced courses. Some algebra teachers have asked for a package that would allow them to avoid complex numbers. Mathematica is flexible enough to block out imaginary and complex numbers in a way that is mathematically correct.

Two ideas are implemented in the package RealOnly . Odd roots of negative numbers are defined to be negative, and calculations with unavoidable complex numbers are condensed to the symbol Nonreal. This is done by redefining the built-in functions Power and $Post.

Without loading the package, Mathematica calculates a cube root of a negative number to be complex. So no points are plotted for negative values of x and warning messages are generated.

In[1]:= Plot[x ^ (1/3), {x, -8, 8}];

Every cubic equation has three roots, counting multiplicity.

In[2]:= Solve[x^3 == -8.0]

Out[2]=

Any one of these three roots could be taken as the cube root of . Ordinarily, Mathematica chooses the one with the least positive argument (the third solution in this case).

In[3]:= (-8.0) ^ (1/3)

Out[3]=

Automatic transformations caused by the RealOnly package.

This loads the package.

In[4]:= Needs["Miscellaneous`RealOnly`"]

Power has been redefined so that an odd root of a negative number is negative.

In[5]:= (-8.0) ^ (1/3)

Out[5]=

Now the plot works for negative values of x.

In[6]:= Plot[x ^ (1/3), {x, -8, 8}];

In addition to modifying Power, the package suppresses complex numbers. This is now the solution of the cubic equation.

In[7]:= Solve[x^3 == -8.0]

Out[7]=

Very small imaginary parts are transformed to 0.

In[8]:= {23 + 0. I, Sin[ArcSin[23.]]}

Out[8]=

A number with an imaginary part that is not small is transformed to Nonreal.

In[9]:= {ArcSin[23.], Sin[23. + I]}

Out[9]=

Finally, elementary calculations involving unavoidable complex numbers are transformed to Nonreal.

In[10]:= Tan[a + 23 / (a + b I)]

Out[10]=