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2.3 Basic Arithmetic Operations
As we have seen, basic arithmetic operations such as addition are performed by inserting an operation symbol between two numbers. Thus, the sum of 3 and 5 is obtained by typing 3 + 5. However, in more advanced applications it is sometimes useful to represent these operations as functions. Towards this end, Mathematica includes the following:
Plus[a, b,...] computes the sum of a, b, . . . Plus[a, b] is equivalent to a + b. Times[a, b,...] computes the product of a, b, . . . Times[a, b] is equivalent to a * b. Subtract[a, b] computes the difference of a and b. Only two arguments are permitted. Subtract [a, b] is equivalent to a b. Divide[a, b] computes the quotient of a and b. Only two arguments are permitted. Divide[a, b] is equivalent to a/b. Minus[a] produces the additive inverse (negative) of a. Minus[a] is equivalent to a. c Power[a, b] computes ab, Power[a, b, c] produces ab , etc.
Basic Concepts
EXAMPLE 30
Plus[2, 3, 4] 9 Times[2, 3, 4] 24 Power[2, 3, 4] 2 417 851 639 229 258 349 412 352
In order to see the way in which Mathematica handles functions internally, the command FullForm is quite useful.
FullForm[expression] exhibits the internal form of expression.
EXAMPLE 31
FullForm[a + b + c] Plus[a, b, c] FullForm[a b] Plus[a, Times[ 1, b]] FullForm[(a * b)^ c] Power[Times[a, b], c] FullForm may be used for any Mathematica function, not only arithmetic operators.
EXAMPLE 32
FullForm[Sin[x^3]^2]
Power[Sin[Power[x,3]],2]
In addition to the standard operational symbols discussed previously, there are a few additional commands that are useful in special situations. (Note: In order for the following to work, x and y must have numerical values.)
Increment[x] or x ++ increases the value of x by 1 but returns the old value of x. Decrement[x] or x decreases the value of x by 1 but returns the old value of x. PreIncrement[x] or ++ x increases the value of x by 1 and returns the new value of x. PreDecrement[x] or x decreases the value of x by 1 and returns the new value of x. AddTo[x,y] or x += y adds y to x and returns the new value of x. SubtractFrom[x,y] or x = y subtracts y from x and returns the new value of x. TimesBy[x,y] or x * = y multiplies x by y and returns the new value of x. DivideBy[x,y] or x /= y divides x by y and returns the new value of x.
The next two examples illustrate the various addition commands. The commands for subtraction, multiplication, and division are similar.
EXAMPLE 33
x = 3; x ++ 3 x 4
the actual value of x is 4. The old value of x is returned.
x = 3; ++ x 4 x 4
The actual value of x is 4. The new value of x is returned.
x ++ is equivalent to the sequence x x = x + 1;
++x is equivalent to the statement x=x+1
Basic Concepts
EXAMPLE 34
x = 3; y = 4; x+y 7 x 3 y 4
y remains unchanged. x remains unchanged. The sum is returned.
x = 3; y = 4; x += y 7 x 7 y 4
y remains unchanged. The new value of x is 7. The sum is returned.
x + = y is equivalent to the statement x=x+y
SOLVED PROBLEMS
2.39 How does Mathematica evaluate the expression a + bc / d
SOLUTION
FullForm[a + b * c/d] Plus[a,Times[b, c, Power[d, 1]]]
2.40 How is the function Minus[x]treated internally in Mathematica
SOLUTION
FullForm[Minus[x]] Times[ 1, x]
2.4 Strings
A string is an (ordered) sequence of characters. Strings have no numerical value and are often used as labels for tables, graphs, and other displays. In Mathematica, a string is enclosed within quotation marks. Thus "abcde" is a string of five characters. Do not confuse "abcde" with abcde, as the latter is not a string. Mathematica comes equipped with a number of string manipulation commands.
StringLength[string] returns the number of characters in string. StringJoin[string1, string2,...] or string1 <> string2 <> ... concatenates two or more strings to form a new string whose length is equal to the sum of the individual string lengths. StringReverse[string] reverses the characters in string.
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