# 10 Mathematical Notations and Special Characters

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This section of Introduction
to Wolfram Notebooks
is about notations
like special characters
and notations
used in mathematics.
Much of this topic
can be illustrated
by considering various ways
of entering
this mathematical expression,
which is a definite integral.
One way of entering
this expression is with palettes.
Start by opening
the Basic Math Assistant palette
by choosing Basic Math Assistant
under the Palettes menu.
Each section of this palette
shows a grid of buttons
for pasting things
into the notebook.
There are keyboard commands
for entering all
of these things as well,
which will be described
in a moment,
but we will start
with the palette.
All of the buttons
that are needed for this input
are in the section
labeled Typesetting.
Within this section,
under the first tab
are buttons for
mathematical notations
like fractions and exponents,
and under the second tab
are buttons for entering
special symbols
and Greek letters.
Returning to the first tab,
we can start
by clicking the button
that shows the definite integral,
which enters into the notebook
a template with empty boxes,
called placeholders, for entering
parts of the integral.
The lower limit of integration
is highlighted
with a colored background,
to indicate
where input will appear.
In this example, the lower limit
of integration is 0,
so enter 0,
then press the TAB key
to move to the next placeholder,
which is the upper limit
of integration.
The upper limit of integration
in this example
is a special character,
which is the mathematical symbol
for infinity.
To enter that symbol,
click the second tab
to open the grid of buttons
for symbols and Greek letters,
and enter the upper limit
of integration,
by clicking the infinity button
in the palette.
Then press the TAB key
to move to the next placeholder
in the template,
which is the integrand.
The first part of the integrand
is an exponential,
which is under the first tab.
Pressing the button
for an exponential
enters that template.
Then return to the second tab
to enter the exponential <i>e</i>
and press the TAB key
to move to the next placeholder,
which is the exponent.
Now enter the exponent,
which is –2<i>x</i>.
Continuing to type at this point
would enter more input
into the exponent,
but to continue
with this example,
it is necessary to get out
of the exponent.
The keyboard command
to end the exponent
and move to
the enclosing expression
is CONTROL+SPACE.
So with the insertion point
in the exponent,
press CONTROL+SPACE
and then finish
entering the integrand.
You could also move
the insertion point
out of the exponent
by clicking the mouse
outside of the exponent.
Now press the TAB key
to move to the next placeholder
and enter <i>x</i>
to complete the input.
This is a live input
and can be evaluated
just like any other input
to do that calculation
and get a result.
The input is still not formatted
quite like the initial example.
There are several differences
in alignment
and in the shapes of characters.
Those differences are there
because the example is shown
in a format
called TraditionalForm
and the input
that we just entered
is shown in StandardForm.
StandardForm can be converted
to TraditionalForm
by selecting the input
and choosing
Convert To ► TraditionalForm
under the Cell menu.
TraditionalForm looks more like
traditional mathematics,
which is frequently desirable,
and the only reason
that TraditionalForm
isn't the default
is that translating traditional
mathematics into computer input
requires a somewhat
elaborate and nonuniform set
of translation rules.
For example, in StandardForm
functions are shown
with square brackets
around the arguments,
but in TraditionalForm
function arguments
are enclosed in parentheses.
This use of parentheses
is a potential ambiguity, however,
because parentheses
can also indicate
arithmetic grouping
and the computer
cannot always guess correctly
what the parentheses should mean
in every situation.
As a practical matter though,
the rules for translating
TraditionalForm input
work pretty well.
So if you prefer TraditionalForm,
it is entirely reasonable
to use it,
and if it is only being used
for display and not for input,
than there is no real problem
with it at all.
This input
and all of these notations
can also be entered
using keyboard commands.
One way to learn
about keyboard commands
is by hovering the mouse
over a corresponding button
in the palette.
For example,
hovering the mouse pointer
over the exponential <i>e</i> button
in this palette
gives a display called a tooltip
that shows the name
of the character,
which is exponential <i>e</i>,
and the keyboard command,
which is ESCAPE+ee+ESCAPE.
Typing ESCAPE+ee+ESCAPE
on the keyboard enters
the exponential constant <i>e</i>
that came up in the example.
There are also keyboard commands
for mathematical templates.
For example, under the tab
for typesetting forms,
hovering the mouse
over the button
for entering
a definite integral
shows the command
ESCAPE+dintt+ESCAPE,
which can be typed
from the keyboard
to get the definite
integral template.
The names of keyboard commands
are chosen to have some
logical way of remembering them.
For example, the keyboard command
for the integral character
is ESCAPE+int+ESCAPE
and the keyboard command
for the definite integral template
is ESCAPE+dintt+ESCAPE,
where "d-i-n-t-t" is short
for definite integral template.
To continue with the input,
enter the lower limit
of integration,
then press the TAB key
to move to the upper limit
of integration.
The keyboard command
for the infinity character
is ESCAPE+inf+ESCAPE.
Then press the TAB key
to move to the integrand.
ESCAPE+ee+ESCAPE
enters the exponential <i>e</i>
and CONTROL+6
gives a template for the exponent.
On common keyboards
the caret character,
which is used in computer code
for exponents,
is on the key for the number 6,
so a good way to remember
CONTROL+6 for an exponent
is that it is the CONTROL key
and the key
with the caret character on it.
Then enter the exponent
and press the TAB key
to move to the last placeholder
and complete the input.
This input could have been
entered in many different ways.
For example,
to enter the exponential
rather than starting
with the template,
we can start with
the exponential <i>e</i> character,
select that character
and then click the button
for the template.
This had the effect of inserting
the exponential <i>e</i> character
in place of the filled
black square
that is shown on the button.
Several of the buttons
in this palette
have placeholders
that are filled black squares
rather than empty black squares.
The filled black squares
are selection placeholders,
which work just like
the empty square placeholders
except that if something
in the notebook is selected
when one of those buttons
with a filled black square
selection placeholder is clicked,
then the selected expression
gets inserted
in place of
the selection placeholder.
For example,
the definite integral example
could have been entered
by starting
with the inner expression.
The button that was used
earlier for entering
the definite integral template
does not have
a filled black square
selection placeholder,
but there is a similar button
in the Basic Commands section
of the palette
that does have
a selection placeholder.
To enter the input,
select the expression
and click the button,
which inserts
the selected expression
in place
of the selection placeholder.
This template is also different
in that it has
descriptive placeholders
that display as shaded boxes
with reminders about the purpose
of each part of the input.
These descriptive placeholders
work just like
the empty box placeholders
and can be filled in just as
before to complete the input.
All of these notations
and special characters
can be used in Text cells
and in graphics
and almost anywhere else
in the notebook.
For example, here is a Text cell
with some mathematical notations
within the text.
This Text cell can be entered
using much the same process
as was used for entering input.
Start with a Text cell
and use either the palette
or keyboard commands
to enter mathematical notations.
An important aspect
of this Text cell
is that the formatted mathematics
is actually in a separate cell
called an Inline cell.
Clicking within
the formatted mathematics
changes the background color
to highlight the Inline cell.
To leave the Inline cell
and continue entering text,
you can either
click beyond the Inline cell
or press CONTROL+0,
which is the keyboard command
to end the Inline cell.
In that example, the Inline cell
was created automatically
as soon as some notation
other than text was entered,
but in general it is necessary
to first create
a new Inline cell,
which you can do
by choosing Start Inline Cell
from the Typesetting menu
under the Insert menu,
and right below that
is End Inline Cell,
which is the command
that was used
to end an Inline cell.
The keyboard command
for Start Inline Cell
is CONTROL+9
and the keyboard command
for End Inline Cell
is CONTROL+0.
So to enter another Inline cell,
press CONTORL+9
to create the cell,
then enter the fraction,
press CONTROL+0
to end the inline cell
and return to
the enclosing text cell
and complete the input.
These notations
can also be used in graphics.
For example, this input
uses the Epilog option
to add a formatted label
to a plot.
The label is entered here
just as it would be
in any other input.
Keyboard commands
for special characters
are referred to as aliases.
Many common characters
have aliases
and often several aliases.
For example, the tooltip
for the Greek letter alpha
shows that the long name
of the character
is the capitalized word Alpha,
which can be entered
by typing backslash,
square bracket,
followed by the long name,
and as soon as the closing
square bracket is entered,
the display turns
into the character.
Characters can also be entered
using tech or SGML names.
For example,
typing ESCAPE+BACKSLASH+alpha,
which is the tech name
for this character,
and pressing the ESCAPE key
a second time
gives the same alpha character.
The SGML name,
which comes up
in HTML documents,
can be entered as
ESCAPE, ampersand,
then the SGML name
of the character,
which is also Alpha,
followed by a semicolon
and then pressing
the ESCAPE key a second time
to get the character.
So if you already know
character names from tech
or from SGML,
you can use those names
to enter characters
in Wolfram Notebooks.
There are many thousands
of special characters.
At the bottom of each grid
of characters
in the basic math
assistant palette
is a button that opens
the Special Characters palette,
which is the same palette
that is opened
by choosing Special Character
under the Insert menu.
If you see a character
like this one
and would like to know
how to enter it,
you can select the character
and choose Find Selected Function
from the Help menu
to bring up
the documentation page
for that character,
which for this character
shows the long name
of the character,
which is DoubleLongRightArrow
and the Unicode number
and the alias
and some other information.
Not all characters have aliases
and some less common characters
do not even have long names,
but it is essentially
always possible
to enter a character as Unicode.
For example, backslash, colon,
followed by the Unicode number
gives the double long
right arrow character.
That's the end of the examples
for this section.
You can find more information
in the Wolfram Documentation
in this tutorial on
two-dimensional expression input.
This section covered
the basic process
for entering mathematics
and special characters,
but there are many details
that can also be controlled
involving things like the size
and alignment of characters
and ways to customize
the formatting
and customize the handling
of various inputs.
If your application requires
that sort of control,
you can find more information
in the section
on math typesetting
options and tweaking
and in links from this guide page
on defining custom notation.