The arguments of all Mathematica functions are enclosed in square brackets.

The names of all built-in Mathematica functions and symbols begin with capital letters.

Arithmetic is the same as on a regular calculator. Note that multiplication can be represented by a space, *, or parentheses, and a power is represented by ^. The following is the Mathematica output from simple examples

In[1]:= 2 3

Out[1]:= 6

In[2]:= 2 * 3

Out[2]:= 6

In[3]:= 2(3)

Out[3]:= 6

In[4]:= 2^3

Out[4]:= 8

Scientific notation is entered, for example, as 3.1*^5 or 3.1 10^5

By default, Mathematica produces exact results. To get an approximate numerical value, use expr //N . Also, when using a decimal point, Mathematica assumes the number is accurate only to a fixed number of decimal places. By placing a decimal after an integer, you can force Mathematica to give you approximate results. To specify a particular number of significant digits, say n, use N[expr,n] . Note, however, that unless the numbers in expr are exact or of sufficiently high precision, N[expr,n] may not be able to give results with n-digit precision. For example:

In[1] := 1 + 1/3

Out[1]:= 4/3

In[2]:= Sqrt[2]

Out[1] := 

In[3]:= 1. + 1/3

Out[3]:= 1.33333

In[4]:= Sqrt[2.]

Out[4]:= 1.41421

In[5]:= 1 + 1/3 //N

Out[5]:= 1.33333

In[6]:= Sqrt[2] //N

Out[6] := 1.41421

In[7]:= N[E, 20]

Out[7]:= 2.7182818284590452354

Some useful mathematical functions and constants:

square root of x	Sqrt[x]
exponential	Exp[x]
natural logarithm of x	Log[x]
base b logarithm of x	Log[b,x]
trigonometric functions (arguments in radians)	Sin[x], Cos[x], Tan[x]
absolute value of x	Abs[x]
closest integer to x	Round[x]
pseudo random number between 0 and 1	Random[ ]
maximum and minimum of x, y, ...	Max[x,y, ...], Min[x,y, ...]
prime factors of n	FactorInteger[n]
integer part of x	IntergerPart[x]
fractional part of x	FractionalPart[x]
k modulo n (remainder from dividing k by n)	Mod[k, n]
factorial n(n-1)(n-2)....1:	n!
binomial coefficient, =	Binomial[n,m]
Euler gamma function,	Gamma[z]
i =	I
	Infinity
	Pi
	E
: degrees to radians conversion factor	Degree

You can get descriptions of all functions directly from the Mathematica kernel by typing ?<Function> , where <Function> is the name of a function. For example:

In[8]:= ?Sqrt
   Sqrt[z] gives the square root of z

Mathematica contains an extensive list of packages of extra functions that can be loaded. Descriptions of these can be found in the Help Browser from the Add-ons & Links tab under Standard Packages. You can also find the help documentations following this [ http://www.wolfram.com/products/fields/ ] webpage. To load an add-on package, type name where name is the name of the package. For example the program listing

In[1]:= <<Miscellaneous`ChemicalElements`

In[2]:= AtomicWeight[Tungsten]

Out[2]:= 183.84

illustrates the breadth of Mathematica. This add-on provides abbreviations, atomic weights, and atomic numbers of the elements in the periodic table!

To refer to a previous result, Mathematica uses %. Use %% for the next-to-last result, and %%...% (k times) for the kth previous result. For example,

In[1]:= 5+5

Out[1]:= 10

In[2]:= %+7

Out[2]:= 17

In[3]:=  % + %%

Out[3]:= 27

Variable naming conventions

To perform repetitive operations in another way, use the Do function, which works like an iterative do loop in many programming languages. The right side of the table below gives the Mathematica commands to perform the operations described on the left side.

So far, we've used four types of bracketing: parentheses used for grouping, (whatever), square brackets used for functions, f[x], curly brackets used for lists, {a,b,c}, and double square brackets used for indexing, a[[i]].

When doing calculations, often you'll go through multiple steps. You can do each step on a separate line or combine them into one using semicolons. Mathematica won't print output from any command with a semi-colon after it. For example,

In[1]:= a={1,2,3,4,5}; b=2*a; b[[3]]

Out[1]:= 6

Mathematica can do symbolic computations using the same operators as in numerical calculations. For example:

In[1]:= 3 x^2 + 9 x - 5 x^2

Out[1]:= 9 x - 2 x^2

The function Expand multiplies out products and powers. The function Factor does the reverse and factors a long expression. The function Simplify attempts to find the simplest form of an expression by using standard algebraic transformations. The function FullSimplify tries to find the simplest form of an expression by using a larger range of algebraic transformations than included in the Simplify function.

In[1]:= Expand[(x + y) (x + y) (x + y)]

Out[1]:= x3 + 3 x2 y + 3 x y2 + y3

In[2]:= Factor[%]

Out[2]:= (x + y)^3

In[3]:= Simplify[1/(4 (-1+x)) - 1/(4 (1+x)) - 1/(2 (1+x^2))]
Out[3]:= 
               

Some more useful algebraic functions:

Mathematica does a variety of the plethora of calculations we know and love as calculus. Some useful calculus functions: (note that many of these can be entered using the palettes in the palette window.)

D[f,x]	The partial derivative,
D[f, x1, x2, ...]	Multiple derivative
Integrate[f, x]	The indefinite integral
Integrate[f, x, y, ...]	The multiple integral
Integrate[f, {x, xmin, xmax}]	The definite integral
Integrate[f, {x, xmin, xmax}, {y, ymin, ymax}, ...]	The multiple integral
Sum[f, {i, imin, imax}]	The sum
Sum[f, {i, imin, imax, di}]	The sum with i increasing in steps of di
Sum[f, {i, imin, imax}, {j, jmin, jmax}]	The nested sum
Product[f, {i, imin, imax}]	The product
Solve[lhs==rhs, x]	Solution to an equation for x
Solve[{lhs1==rhs1, lhs2==rhs2, ...}, {x, y, ...}]	Solve a set of simultaneous equations for x, y, ...
Series[f, {x, x0, order}]	A power series expansion of f about the point x = x0
Limit[f, x -> x0]	The limit limx->x0f
Eliminate[{lhs1==rhs1, lhs2==rhs2, ...}, {x, ...}]	Eliminate x, ... in a set of simultaneous equations
Reduce[{lhs1==rhs1, lhs2==rhs2, ...}, {x, y, ...}]	Give a set of simplified equations, including all possible solutions
Normal[series]	Truncate a power series to give an ordinary expression

Mathematica contains the following add-on statistical packages. Refer to the Mathematica Help for detailed information about these. These packages are nice if you want to know something about a common distribution such as the normal or binomial distribution (especially expected values), but fall far behind programs like SAS or Stata for data manipulation functions and any functions requiring advanced statistical theory.

Matrices in Mathematica are entered as lists of lists. For example, entering {{a,b},{c,d}} results in the matrix Matrix multiplication is represented by ., and matrix division is, of course, not defined. * represents scalar or element wise multiplication and + and - operators represent matrix addition and subtraction.

Some useful functions for matrices are:

Some useful graphics functions:

A few examples of the above functions:

Plot[x^2, {x, 0, 10}]
Plot3D[Exp[x^2+y^2], {x,-2,2}, {y,-2,2}]
ParametricPlot3D[{Cos[t] (3+Cos[u]), Sin[t] (3+Cos[u]), Sin[u]}, {t, 0, 2Pi, {u, 0, 2Pi}]

For attractive examples of some more complicated uses of these graphics functions, have a look at the Graphics Gallery in the Help Browser by choosing the Demos tab and looking under Graphics Gallery.

Functions begin with a CAPITAL letter.

Arguments of functions use SQUARE brackets.

Statements end with a semi-colon ONLY if you don't want output.

You press SHIFT-return to execute input

Once you've got these down, you can just refer to the Mathematica Help for the appropriate function name and arguments. Get friendly with the Help! It is definitely worth your time to spend a few moments browsing though it, especially through all the add-on packages. Every time you do, you'll learn about additional Mathematica functions.

Contact information for Wolfram Research and Mathematica: