# Toolbox

At any time when editing a calculation or expression, you can press Toolbox. A catalogue of functions will open to help you make more specific calculations.

The Toolbox catalog is divided into several thematic sub-sections: Calculation, Complex numbers, Combinatorics, … Choose the calculation you want to perform and press OK. Complete the space between the parentheses with the arguments you need for each function.

The first three functions in the Toolbox catalogue are: Absolute value, n-th root and Logarithm to base a.

Function Description
abs(x) Calculates the absolute value of the argument you enter in parentheses. abs(-4.5) gives the value of $\mid -4.5\mid$, that is $4.5$.
root(x,n) Calculates the $n$-th root of a number. You must enter $n$ and $x$ in parentheses. root(x,n) gives the value of $\sqrt[n\,]{x}$. The value of $n$ may not be an integer.
log(x,a) Calculates the logarithm to base $a$. You must enter $a$ and $x$ in parentheses. log(x,a) gives the value of $\log_{a}(x)$.

## Calculation

Function Description
diff(f(x),a) Calculates the derivative of a function at a point. Be careful to define the function using the $x$ variable. diff(f(x),a) gives the value of $f'(a)$. For example, to calculate the derivative of a square root at 5: diff(sqrt(x),5).
int(f(x),a,b) Calculates the integral of a function between two bounds. Be careful to define the function using the $x$ variable. int(f(x),a,b) gives the value of $\int_{a}^{b} f(x) \, \mathrm{d}x$. For example, to calculate the integral of the square root between $0$ and $5$: int(sqrt(x),0,5).
sum(f(n),nmin,nmax) Calculates the sums of terms in $n$. Be careful to define the terms with the variable $n$. sum(f(n),nmin,nmax) gives the value of $\sum_{n=n_{min}}^{n_{max}} f(n)$.
product(f(n),nmin,nmax) Calculates the products of terms in $n$. Be careful to define the terms with the variable $n$. product(f(n),nmin,nmax) gives the value of $\prod_{n=n_{min}}^{n_{max}} f(n)$.

## Complex numbers

Function Description
abs(x) Absolute value of a complex number. abs(2+3i)gives the value of $\mid 2+3i\mid$.
arg(z) Argument of a complex number. arg(2+3i) gives the value of $arg(2+3i)$ in radians.
re(z) Real part of a complex number. For instance, re(2+3i) returns $2$.
im(z) Imaginary part of a complex number. For instance, im(2+3i) returns $3$.
conj(z) Conjugate of a complex number. conj(2+3i) returns the conjugate of $2+3i$, that $2-3i$.

## Combinatorics

Function Description
binomial(n,k) Number of ways to choose a subset of size $k$ elements, disregarding their order, from a set of $n$ elements. binomial(n,k) returns $\dbinom{n}{k}$, that is $\frac{n!}{k! (n-k)!}$.
permute(n,k) Number of different ordered arrangements of a $k$-element subset of an $n$-set. permute(n,k) returns $A_{n}^k$, that is $\frac{n!}{(n-k)!}$.

## Arithmetic

Function Description
gcd(p,q) Greatest Common Divisor of two integers. For instance, gcd(55,11) returns $11$.
lcm(p,q) Least Common Multiple of two integers. For instance, lcm(13,2) returns $26$.
factor(n) Integer factorization of $n$. For instance, factor(24)returns $2^3 \times 3$.
rem(p,q) Remainder of the Euclidian division of $p$ by $q$. For instance, rem(50,45) returns the remainder of the division of $50$ by $45$ that is $5$.
quo(p,q) Quotient of the Euclidian division of $p$ by $q$. For instance, quo(80,39) returns the quotient of the division of $80$ by $39$ that is $2$.

## Matrix

Function Description
inverse(M) Inverse of the matrix M. For instance, inverse([[0.25,0][0,0.25]])returns $\left[\begin{array}{cc}4 & 0 \\ 0 & 4 \end{array}\right]$.
det(M) Determinant of the matrix M. For instance, det([[1,2][3,4]]) returns $-2$.
transpose(M) Transpose of the matrix M. For instance, transpose([[1,2][3,4]]) returns $\left[\begin{array}{cc}1 & 3 \\ 2 & 4 \end{array}\right]$.
trace(M) Trace of the matrix M. For instance, trace([[1,2][3,4]]) returns $5$.
dim(M) Size of the matrix M. For instance, dim([[1,2][3,4]]) returns [2,2].

## Random and approximation

Function Description
random() Generates a random number between $0$ and $1$.
randint(a,b) Generates a random integer $a$ and $b$.
floor(x) Floor function. For instance, floor(5.8) returns $5$.
frac(x) Fractional part. For instance, frac(5.8) returns $0.8$.
ceil(x) Ceiling function. For instance, ceil(5.8) returns $6$.
round(x,n) Rounds a number to $n$ digits after the decimal point. For instance round(8.6576,2) returns $8.66$.

## Hyperbolic trigonometry

Function Description
cosh(x) Hyperbolic cosine.
sinh(x) Hyperbolic sine.
tanh(x) Hyperbolic tangent.
acosh(x) Inverse hyperbolic cosine.
asinh(x) Inverse hyperbolic sine.
atanh(x) Inverse hyperbolic tangent.

## Prediction interval

Function Description
prediction95(p,n) Prediction interval 95%. prediction95(p,n) returns $\left[ p-1.96\frac{\sqrt{p(1-p)}}{\sqrt{n}};p+1.96\frac{\sqrt{p(1-p)}}{\sqrt{n}} \right]$.
prediction(p,n) Approximation of the prediction interval. prediction(p,n) returns $\left[ p-\frac{1}{\sqrt{n}};p+\frac{1}{\sqrt{n}} \right]$.
confidence(f,n) 95% confidence interval. confidence(f,n) returns $\left[ f-\frac{1}{\sqrt{n}};f+\frac{1}{\sqrt{n}} \right]$.