## STACK Documentation

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In STACK you often need to enter an answer which is an algebraic expression. You should type in your answers using the same syntax as that used in the symbolic mathematics package Maxima.

The syntax is broadly similar to the syntax used for mathematical formulae in graphical calculators; general programming languages such as Java, C and Basic; and in spreadsheet programs. You will find it useful to master it.

For example, to enter \( e^{-t}\sin(3t)\) you need to type in

```
e^(-t)*sin(3*t)
```

STACK tries quite hard to give helpful information about any syntax errors. It might also forgive some errors you make.

## Basic Notation

### Numbers

You should type in numbers without spaces, and use fractions rather than decimals where possible.
For example, \(1/4\) should be entered as `1/4`

, not as `0.25`

. Also,

- \(\pi\) is entered as either
`pi`

or`%pi`

, - \(e\), the base of the natural logarithms, is entered as either
`e`

or`%e`

, - \(i\) is entered as either
`i`

or`%i`

.- \(i\) is also sometimes entered as
`j`

if you are an engineer. If in doubt ask your teacher. - You could also use
`sqrt(-1)`

, or`(-1)^(1/2)`

, being careful with the brackets. - STACK modifies Maxima's normal input rules so that you don't get caught out with a variable
`i`

when you meant`%i`

.

- \(i\) is also sometimes entered as
- You can also use scientific notation for large numbers, e.g. \(1000\) can be entered as
`1E+3`

. Note, however, that in many situations floating point numbers are forbidden.

### Multiplication

Use a star for multiplication. Forgetting this is by far the most common source of syntax errors. For example,

- \(3x\) should be entered as
`3*x`

. - \(x(ax+1)(x-1)\) should be entered as
`x*(a*x+1)*(x-1)`

.

STACK does sometimes try to insert stars for you where there is no ambiguity, `2x`

or `(x+1)(x-1)`

.
This guessing cannot be perfect since traditional mathematical notation is sometimes ambiguous!
Compare \(f(x+1)\) and \(x(t+1)\).

### Powers

Use a caret (^) for raising something to a power: for example, \(x^2\) should be entered as `x^2`

.
You can get a caret by holding down the SHIFT key and pressing the 6 key on most keyboards.
Negative or fractional powers need brackets:

- \(x^{-2}\) should be entered as
`x^(-2)`

. - \(x^{1/3}\) should be entered as
`x^(1/3)`

. - The function
`root(x,n)`

can be used for`x^(1/n)`

. If you omit the second argument you get`root(x)=sqrt(x)`

.

### Brackets

Brackets are important to group terms in an expression. This is particularly the case in STACK since we use a one-dimensional input rather than traditional written mathematics. Try to consciously develop a sense of when you need brackets and avoid putting in too many.

For example,

\[\frac{a+b}{c+d}\]

should be entered as `(a+b)/(c+d)`

.

If you type `a+b/(c+d)`

, then STACK will think that you mean

\[a+\frac{b}{c+d}.\]

If you type `(a+b)/c+d`

, then STACK will think that you mean

\[\frac{a+b}{c}+d.\]

If you type `a+b/c+d`

, then STACK will think that you mean

\[a+\frac{b}{c}+d.\]

Think carefully about the expression `a/b/c`

. What do you think this means? There are two options

\[\frac{a}{b}\cdot\frac{1}{c} = \frac{a}{bc}\quad\mbox{or}\quad\frac{a}{\frac{b}{c}}=\frac{ac}{b}.\]

Maxima interprets this as \(\frac{a}{bc}\). If in doubt use brackets.

Note that in this context you should always use ordinary round bracket (like (a+b)), not square or curly ones (like [a+b] or {a+b}).

`{a+b}`

means a set,`[a+b]`

means a list.

### Scientific units

You may be asked to answer a question in which the answer has units. E.g. \( 9.81\mbox{m}\mbox{s}^{-2} \). To enter units you must use multiplication, so this is entered as either `9.81*m/s^2`

or `9.81*m*s^(-2)`

. Don't use a space, or another symbol such as `+`

.

### Subscripts

Use the underscore character to denote a subscript. For example, \(a_b\) should be entered as `a_b`

.

### More examples

- \(2^{a+b}\) should be entered as
`2^(a+b)`

- \(2 \cos 3x\) should be entered as
`2*cos(3*x)`

- \(e^{ax}\sin(bx)\) should be entered as
`exp(a*x)*sin(b*x)`

- \( (ax^2 + b x + c)^{-1}\) should be entered as
`(a*x^2 + b*x + c)^(-1)`

.

## Functions

**Standard functions**: Functions, such as \(\sin\), \(\cos\), \(\tan\), \(\exp\), \(\log\) and so on can be entered using their usual names. However, the argument must always be enclosed in brackets: \(\sin x\) should be entered as`sin(x)`

, \(\ln 3\) should be entered as`ln(3)`

and so on.**Modulus function**: The modulus function, sometimes called the absolute value of*x*, is written as |*x*| in traditional notation. This must be entered as`abs(x)`

.

### Trigonometrical functions

Things to remember:

- STACK uses radians for the angles not degrees!
- The function \(1/\sin(x)\) should be typed in as
`csc(x)`

rather than`cosec(x)`

.

You can type`cosec(x)`

or just call it`1/sin(x)`

if you prefer. - \(\sin^2x\) must be entered as
`sin(x)^2`

(which is what it really means, after all). Similarly for \(\tan^2(x)\), \(\sinh^2(x)\) and so on. - Recall that \(\sin^{-1}(x)\) traditionally means the number \(t\) such that \(\sin(t) = x\),
which is completely different from the number \(\sin(x)^{-1} = 1/\sin(x)\).
This traditional notation is really rather unfortunate and is not used by the CAS; instead,
\(\sin^{-1}(x)\) should be entered as
`asin(x)`

. Similarly, \(\tan^{-1}(x)\) should be entered as`atan(x)`

and so on.

### Exponentials and Logarithms

- To enter the exponential function type
`exp(x)`

. Typing`e^x`

should work in STACK, but gets you into bad habits when using a CAS later! - Type
`ln(x)`

or`log(x)`

to enter the*natural logarithm*of \(x\) with base \(e\approx 2.71\cdots\). Note that both of these start with a lower case l for logarithm, not a capital I (`i`

). - The logarithm of \(x\) to base \(10\) is entered as
`lg(x)`

. - The logarithm of \(x\) to base \(a\) is entered as
`lg(x,a)`

, sometimes written \(\log_{a}(x)\).

## Matrices

You may be given a grid of boxes to fill in. If not, the teacher may provide a hint as to the correct syntax. Otherwise you will need to use Maxima's notation for entering the matrix.

The matrix:

\[ \left( \begin{array}{ccc} 1 & 2 & 3 \\ 4 & 5 & 6 \end{array} \right)\]

must be entered as `matrix([1,2,3],[4,5,6])`

.

Each row is entered as a list, and these should be the same length. The function matrix is used to indicate this is a matrix and not a "list of lists".

### Equations and Inequalities

Equations can be entered using the equals sign. For example, to enter the equation \(y=x^2-2x+1\) type `y=x^2-2*x+1`

.

Inequalities can be entered using the greater than and less than signs on the keyboard.
Notice that there are four possibilities for you to choose from: `<`

or `>`

or `<=`

or `>=`

.
Note there is no space between these symbols, and the equality must come second when it is used, i.e. you cannot use `=<`

.

Sometimes you will need to connect inequalities together as `x>1 and x<=5`

. You must use the logical connectives `and`

and `or`

. "Chained inequalities" such as \(1<x<5\) are not permitted as input syntax. You should enter this as `1<x and x<5`

.

### Reasoning line by line

Sometimes you will be asked to reason line by line. This is called reasoning by equivalence.

## Other notes

**Greek letters**can be entered using their English names. For example, enter \(\alpha+\beta\) as`alpha+beta`

, and \(2\pi\) as`2*pi`

.**Sets**: To enter a set such as \(\{1,2,3\}\) in Maxima you could use the function`set(1,2,3)`

, or use curly brackets and type`{1,2,3}`

.**Lists**: can be entered using square brackets. For example, to enter the list*1,2,2,3*type`[1,2,2,3]`

.- Note that you do
**not**need a semicolon at the end, unlike when you are using a CAS directly.

You can also learn about the right syntax by doing tests in practice mode and asking for the solutions; as well as displaying the right answers in ordinary mathematical notation, STACK will tell you how they could be entered. However, there are often several possible ways, and STACK will not always suggest the easiest one.

If you have difficulties in entering your answer into STACK you should contact your teacher.

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