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Round brackets are used for grouping: (2 + 3) 4 means add 2 + 3 first, then multiply by 4. Never type [2 + 3]* 4. Curly brackets are used for lists: {1, 2, 3, 4}. More about lists in 3.
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Use E, not e, for the base of the natural logarithm. Since every Mathematica symbol begins with a capital letter, the base of the natural logarithm is E. This causes a bit of confusion, so be careful. Similarly, I (not i) is the imaginary unit. The symbols and from the Basic Math Input palette may be freely used if desired. Polynomials are not written in standard form. Mathematica writes polynomials with the constant term first and increasing powers from left to right. Thus, the polynomial x2 + 2 x 3 would be converted to 3 + 2 x + x2. To see the expression in a more conventional format, the command TraditionalForm may be used.
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TraditionalForm[expression] prints expression in a traditional mathematical format.
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EXAMPLE 5 Evaluate the sum of x2 + 3, 2 x + 5, and x3 + 2 and express the answer using TraditionalForm.
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(x2 + 3)+(2 x + 5)+(x3 + 2) 10 + 2 x + x2 + x3
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TraditionalForm[(x2 + 3)+(2 x + 5)+(x3 + 2)] x3 + x2 + 2 x + 10
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1.7 Compute
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81 using the Sqrt function. What happens if you do not use a capital S
Sqrt[81] 9 sqrt[81] sqrt[81]
Mathematica does not recognize the (undefined) symbol sqrt.
1.8 Use parentheses to multiply the sum of 2 and 3 by the sum of 5 and 7. What happens if you use square brackets
SOLUTION
(2 + 3)(5 + 7) 60 [2 + 3][5 + 7]
Syntax sntxb : Expression cannot begin with "[2+3][5+7]". Syntax tsntxi : "[2+3]" is incomplete; more input is needed. Syntax sntxi : Incomplete expression; more input is needed.
Click on the + to reveal the error.
1.9 Use the Sin function to compute sin( /2). What happens if you use round parentheses
SOLUTION
Sin[Pi/2] or Sin[o/2] 1 Sin(Pi/2)
Sin 2
Mathematica thinks you want to multiply the symbol Sin by and divide by 2.
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1.10 Alexis typed [4 + 1] * [6 + 2] during a Mathematica session. Why didn t she get an answer of 40
SOLUTION
Square brackets cannot be used for grouping. Round parentheses must be used.
1.11 Why didn t Ariel get an answer of 3 when she typed sqrt[9]
SOLUTION
Mathematica functions must begin with a capital letter.
1.12 Why didn t Lauren get an answer of 1 when she typed Cos(0)
SOLUTION
Square brackets, not round parentheses, must be used to contain arguments of functions.
Mathematica Gives Exact Answers
Mathematica is designed to work as a mathematician works: with 100% precision. You do not get the 10- or 12-digit numerical approximation a calculator would give, but instead get a symbolic mathematical expression.
EXAMPLE 6
12 2 3
EXAMPLE 7
1/3 + 3/5 5/7 + 2/11 463 1115
EXAMPLE 8
o+o 2
EXAMPLE 9
-1
SOLVED PROBLEMS
1.13 Simplify
SOLUTION
2 + 8 + 18 .
2 + 6 2
Sqrt[2] + Sqrt[8] + Sqrt[18]
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1.14 Compute the sum of the reciprocals of 3, 5, 7, 9, and 11.
SOLUTION
1 + 1 + 1 + 1 + 1 3 5 7 9 11 3043 3465
1/3 + 1/5 + 1/7 + 1/9 + 1/11
1.15 Compute the square root of exactly using the Sqrt function.
SOLUTION
Sqrt[Pi]
1.16 Multiply
SOLUTION
This is the only way to represent the square root of exactly.
8 by
Sqrt[8] * Sqrt[2]
1.17 Simplify
SOLUTION
3 + 12 + 27 + 48 leaving your answer in radical form.
3 + 10 3
12 +
27 +
1.5 Mathematica Basics
In this section we discuss some of the simpler concepts within Mathematica. Each will be explained in greater detail in a subsequent chapter. Symbols are defined using any sequence of alphanumeric characters (letters, digits, and certain special characters) not beginning with a digit. Once defined, a symbol retains its value until it is changed, cleared, or removed. Arithmetic operations are performed in the obvious manner using the symbols +, , , and /. Exponentiation is represented by a caret, ^, so x^y means xy. Just as in algebra, a missing symbol implies multiplication, so 2a is the same as 2 a. Be careful, however, when multiplying two symbols, since ab represents the single symbol beginning with a and ending with b. To multiply a by b you must separate the two letters with or (on the Basic Math Input palette) or a space: a b, a b, or a b .
EXAMPLE 10
a=2 b=3 c=a+b 2 3 5
Notice that the result of each calculation is displayed. This is sometimes annoying, and can be suppressed by using a semicolon (;) to the right of the instruction.
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EXAMPLE 11
a = 2; b = 3; c=a+b 5 Operations are performed in the following order: (a) exponentiation, (b) multiplication and division, (c) addition and subtraction. If the order of operations is to be modified, parentheses, ( ), must be used. Be careful not to use [ ] or { } for this purpose.
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