Introduction to Strings in Mathematica

A format convention in Mathematica is to show quotation marks around strings in input cells, but not in output cells. To show them in output cells, use FullForm.

In[18]:= "I've entered a string by using quotation marks."

Out[18]= I've entered a string by using quotation marks.

In[19]:= Head@%

Out[19]= String

In[20]:= FullForm@%%

Out[20]//FullForm= "I've entered a string by using quotation marks."

In programming languages, "String" is the name for the data type of text expressions. In other words, unlike symbols in a general expression, the symbols in a String, like a, b, c, do not refer to places in memory that store a variable, like a function or value. Symbols in a String do not refer to anything, or you could say they only refer to themselves as "literals", that is, the literal characters or tokens "a", "b", "c". An "a" is just a blank token with nothing behind it; hence the terms "literal" or "token" for a String symbol.

In[21]:= a = 5534.7; a

Out[21]= 5534.7

Trying to use a String as a variable gives an error message.

In[22]:= "a" = "Lookee here."

During evaluation of In[22]:= Set::setraw: Cannot assign to raw object a. >>

Out[22]= "Lookee here."

Consequently, Strings are not evaluated in the usual way that other Expressions are. Wellin et al. in their excellent book suggest that we think of a String as being broken up by Mathematica into a sequence of its characters and then functions are applied to the sequence.

In[8]:= aSentence = "The quick brown fox jumped over the 742 lazy white dogs.";

In[11]:= aSentenceCharacters = aSentence // Characters

Out[11]= {T,h,e, ,q,u,i,c,k, ,b,r,o,w,n, ,f,o,x, ,j,u,m,p,e,d, ,o,v,e,r, ,t,h,e, ,7,4,2, ,l,a,z,y, ,w,h,i,t,e, ,d,o,g,s,.}

Functions, like MatchQ, Cases, Position, etc., that work on non-String expressions do not work on Strings but do work on their sequence of characters.

In[14]:= Position[aSentence, "e"]

Out[14]= {}

So perhaps as Wellin et al suggest, Mathematica first breaks a String into a sequence of characters and applies some of the regular built-in functions as well as some specialized String functions to the sequence.


In[25]:= MatchQ[#, "e"] & /@ aSentenceCharacters

Out[25]= {False, False, True, False, False, False, False, False, False, False, False, False, False, False, False, False, False, False, False, False, False, False, False, False, True, False, False, False, False, True, False, False, False, False, True, False, False, False, False, False, False, False, False, False, False, False, False, False, False, True, False, False, False, False, False, False}

In[24]:= StringMatchQ[aSentenceCharacters, "e"]

Out[24]= {False, False, True, False, False, False, False, False, False, False, False, False, False, False, False, False, False, False, False, False, False, False, False, False, True, False, False, False, False, True, False, False, False, False, True, False, False, False, False, False, False, False, False, False, False, False, False, False, False, True, False, False, False, False, False, False}


In[15]:= Position[aSentenceCharacters, "e"]

Out[15]= {{3}, {25}, {30}, {35}, {50}}

In[16]:= StringPosition[aSentence, "e"]

Out[16]= {{3, 3}, {25, 25}, {30, 30}, {35, 35}, {50, 50}}

The position index is repeated in StringPosition since it shows the beginning and ending positions of the matched String.

In[20]:= StringPosition[aSentence, "lazy"]

Out[20]= {{41, 44}}

When working with Strings, you do not need to enclose a String in List brackets:

In[3]:= aString = {"abcd"}

Out[3]= {abcd}

In[4]:= aString2 = "abcd"

Out[4]= abcd

In[5]:= Head@aString2

Out[5]= String

The String is delimited by invisible quotation marks. As usual FullForm and InputForm reveal the way Mathematica sees the String.

In[6]:= FullForm@aString2

Out[6]="abcd"

In[7]:= InputForm@aString2

Out[7]="abcd"

Next we will look at the variety of String functions in Mathematica.

Mathematica Plot - Basic Plotting

Plot

For starters, be aware that Mathematica's data plotting functionality is extensive and there are a variety of Plot functions. Using Information "?" with the wildcard characters "*" shows all functions that include "Plot". I won't show them here, but you can execute the command if you wish to see them.

?*Plot*

Basic Plotting

Let's say you have a function and you want to understand its behavior. We'll do a simple one.

In[238]:= y == x^n // TraditionalForm

Out[238]//TraditionalForm= y==x^n

Let's start by generating a few sample points of the function and plotting them.  You might think in the age of computers we can dispense with plotting sample points and just let Plot do the job. But when Plot fails to give you output and you don't know why, one measure is to go ahead and plot a few sample points by hand and compare them to what Plot is doing. So let's generate the sample points the easy way with Table. We Clear the function name, define the function, give x a value of 2, and use Table to sample the domain from -5 to 5 in increments of 0.5.

In[249]:= Clear@f; f[x_, t_] := x^t;

In[250]:= samplePoints = Table[f[2, t], {t, -5, 5, .5}]

Out[250]= {0.03125, 0.0441942, 0.0625, 0.0883883, 0.125, 0.176777, 0.25, 0.353553, 0.5, \
0.707107, 1., 1.41421, 2., 2.82843, 4., 5.65685, 8., 11.3137, 16., 22.6274, \
32.}

Now we plot these values using ListPlot. Ignore for now the x-tick points, which are just the number of the data point in the List.

In[242]:= plot1 = ListPlot@samplePoints

To better visualize the function, it's often helpful to connect the sample points. We use ListLinePlot. Another way is to use ListPlot's option, Joined -> True). If you look closely, you can see that the sample points are joined by line segments; our plot is not a continuous curve. Again, ignore the abscissa values.

plot2 = ListLinePlot@samplePoints

If we wished, we could combine the plots with Show--one of the most commonly-used plot techniques.

Show[plot1, plot2]

And this is similar to what Plot does--sampling a function and plotting the connected sample points--but of course in a much more sophisticated way. For one thing, Plot samples more frequently where it thinks it might need to, such as where the value of the function changes more frequently, maybe with several tries, and automatically connects the sample points in a continuous curve. Using Plot the x-axis values are correctly labeled.

Plot[f[2, t], {t, -5, 5}]

But notice that Plot is not showing the full domain and range that we asked for. By default it tries to, in effect, show the most important part of a function by excluding what it thinks might be outliers. Two options allow you to override this behavior. PlotRange->Full tells Plot to not clip the y-axis. PlotRange->All tells Plot to include all domain points and their y-values. I usually use PlotRange->All.

Plot[f[2, t], {t, -5, 5}, PlotRange -> All]

If you wish, reveal the sample points with the Option, Mesh->All. When you don't understand what Plot is doing, revealing the Mesh is a good first step. And then as I said above, you can compare Plot's sample points to yours, which might illuminate what's going on. There are more Mesh options to allow you to control how Plot uses its sample points.

Plot[f[2, t], {t, -5, 5}, Mesh -> All, PlotRange -> All]

Next: Using Plot to Plot More than One Function