# Toolbox

The toolbox key gives you access to an organized library of advanced functions. Press the toolbox key at any time while editing a calculation or expression to view a menu of functions. The advanced functions available in the toolbox menu change according to the application you are using.

When writing an expression, the first functions in the Toolbox menu include: Absolute value, n-th root and Logarithm base a. When inputting expressions for the Grapher, the menu will also include Inequalities. Within the Sequences applicaiton, the Toolbox menu will also include defined sequences.

The Toolbox menu is then divided into several thematic sub-sections:

• Calculus
• Complex numbers
• Probability
• Units and constants
• Matrices and vectors
• Lists
• Arithmetic
• Trigonometry
• Decimal numbers
• Logic
##### 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 $n$ doesn’t have to be an integer.

##### log(x,a)

Calculates the logarithm with base $a$. You must enter $a$ and $x$ in parentheses. log(x,a) gives the value of $\log_{a}(x)$.

## Calculus

##### diff(f(x),x,a)

Calculates the derivative of a function at a point. diff(f(x),x,a) gives the value of $f'(a)$. For example, to calculate the derivative of a square root at 5: diff(sqrt(x),x,5).

##### diff(f(x),x,a,n)

Calculates the nth derivative of a function at a point. diff(f(x),x,a,n) gives the value of $f^{n}(a)$. For example, to calculate the 3rd derivative of a square root at 5: diff(sqrt(x),x,5,3).

##### int(f(x),x,a,b)

Calculates the integral of a function between two bounds. int(f(x),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),x,0,5).

##### sum(f(n),n,nmin,nmax)

Calculates the sums of terms in $n$. sum(f(n),n,nmin,nmax) gives the value of $\sum_{n=n_{min}}^{n_{max}} f(n)$.

##### product(f(n),n,nmin,nmax)

Calculates the products of terms in $n$. product(f(n),n,nmin,nmax) gives the value of $\prod_{n=n_{min}}^{n_{max}} f(n)$.

## Complex numbers

##### abs(x)

Modulus 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 example, re(2+3i) returns $2$.

##### im(z)

Imaginary part of a complex number. For example, im(2+3i) returns $3$.

##### conj(z)

Conjugate of a complex number. conj(2+3i) returns the conjugate of $2+3i$, that is $2-3i$.

## Probability

### Combinatorics

##### binomial(n,k)

Number of ways to choose a subset of size $k$ elements, disregarding their order, from a set of $n$ elements. For example, $\dbinom{n}{k}$ returns $\frac{n!}{k! (n-k)!}$.

##### 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 $\frac{n!}{(n-k)!}$.

##### n!

Returns the product of the entered integer and all integers below it. For example, $5!$ returns $120$.

### Distributions

#### Normal

##### normcdf(a,µ,σ)

$P(X where X follows the normal distribution $N(\mu,\sigma)$.

##### normcdfrange(a,b,µ,σ)

$P(a where X follows the normal distribution $N(\mu,\sigma)$.

##### invnorm(a,µ,σ)

Returns $m$ where $P(X and X follows the normal distribution $N(\mu,\sigma)$.

##### normpdf(x,µ,σ)

Probability density function of $N(\mu,\sigma)$.

#### Student’s t

##### tcdf(a,k)

$P(X where X follows the t-distribution with k degress of freedom.

##### tcdfrange(a,b,k)

$P(a where X follows the t-distribution with k degress of freedom.

##### invt(a,k)

Returns $m$ where $P(X and X follows the t-distribution with k degress of freedom.

##### tpdf(x,k)

Probability density function of $t(k)$.

#### Binomial

##### binompdf(m,n,p)

$P(X=m)$ where X follows the binomial distribution $B(n,p)$.

##### binomcdf(m,n,p)

$P(X \leq m)$ where X follows the binomial distribution $B(n,p)$.

##### invbinom(a,n,p)

Returns $m$ where $P(X \leq m)=a$ and X follows the binomial distribution $B(n,p)$.

#### Poisson

##### poissonpdf(m,λ)

$P(X=m)$ where X follows the Poisson distribution with parameter λ.

##### poissoncdf(m,λ)

$P(X \leq m)$ where X follows the Poisson distribution with parameter λ.

#### Geometric

##### geompdf(m,p)

$P(X=m)$ where X follows the geometric distribution with probability p.

##### geomcdf(m,p)

$P(X \leq m)$ where X follows the geometric distribution with probability p.

##### geomcdfrange(m,n,p)

$P(m \leq X \leq n)$ where X follows the geometric distribution with probability p.

##### invgeom(a,p)

Returns $m$ where $P(X \leq m)=a$ and X follows the geometric distribution with probability p.

#### Hypergeometric

##### hgeompdf(m,N,K,n)

$P(X=m)$ where X follows the hypergeometric distribution with population size N, number of featured items K and sample size n.

##### hgeomcdf(m,N,K,n)

$P(X \leq m)$ where X follows the hypergeometric distribution with population size N, number of featured items K and sample size n.

##### hgeomcdfrange(m,q,N,K,n)

$P(m \leq X \leq n)$ where X follows the hypergeometric distribution with population size N, number of featured items K and sample size n.

##### invhgeom(a,N,K,n)

Returns $m$ where $P(X \leq m)=a$ and X follows the hypergeometric distribution with population size N, number of featured items K and sample size n.

### Random

##### random()

Returns a floating point number in [0,1).

##### randint(a,b)

Returns a random integer in [a,b].

##### randintnorep(a,b,n)

Returns n unique random integers in [a,b].

## Units and constants

This menu contains sub-menus for constants and each type of measurement listed below. Units and constants can be selected from the menu or typed manually.

##### a→b

This menu item is a template that allows you to perform a unit conversion.

### Length and angle

#### Length

##### Imperial
Abbreviation Unit
in Inch
ft Foot
yd Yard
mi Mile
au Astronomical unit
ly Light year
pc Parsec
##### Metric
Abbreviation Unit
pm Picometer
nm Nanometer
µm Micrometer
mm Millimeter
cm Centimeter
m Meter
km Kilometer

#### Angle

Abbreviation Unit
° Degree
°'" Degree minute second
rad Radian
gon Gradian

### Time and frequency

#### Time

Abbreviation Unit
ns Nanosecond
µs Microsecond
ms Millisecond
s Second
min Minute
h Hour
day Day
week Week
month Month
year Year

#### Frequency

Abbreviation Unit
Hz Hertz
kHz Kilohertz
MHz Megahertz
GHz Gigahertz

### Volume and area

#### Volume

##### Imperial
Abbreviation Unit
tsp Teaspoon
tbsp Tablespoon
floz Fluid ounce
cup Cup
pt Pint
qt Quart
gal Gallon
##### Metric
Abbreviation Unit
mL Milliliter
cL Centiliter
dL Deciliter
L Liter

#### Area

##### Imperial
Abbreviation Unit
acre Acre
##### Metric
Abbreviation Unit
ha Hectare

### Mass

#### Imperial

Abbreviation Unit
oz Ounce
lb Pound
shtn Short ton
lgtn Long ton

#### Metric

Abbreviation Unit
µg Microgram
mg Milligram
g Gram
kg Kilogram
t Metric ton

### Electricity

#### Current

Abbreviation Unit
µA Microampere
mA Milliampere
A Ampere

#### Voltage

Abbreviation Unit
µV Microvolt
mV Millivolt
V Volt
kV Kilovolt

#### Resistance

Abbreviation Unit
Ω Ohm
kΩ Kiloohm

#### Capacitance

Abbreviation Unit
µF Microfarad
mF Millifarad
F Farad

#### Others

Abbreviation Unit
H Henry
C Coulomb
S Siemens
T Tesla

### Force and pressure

#### Force

Abbreviation Unit
mN Millinewton
N Newton
kN Kilonewton

#### Pressure

Abbreviation Unit
Pa Pascal
hPa Hectopascal
bar Bar
atm Atmosphere

### Energy and power

#### Joule

Abbreviation Unit
mJ Millijoule
J Joule
kJ Kilojoule

#### Power

Abbreviation Unit
µW Microwatt
mW Milliwatt
W Watt
kW Kilowatt
MW Megawatt
GW Gigawatt

#### Electronvolt

Abbreviation Unit
meV Millielectronvolt
eV Electronvolt
keV Kiloelectronvolt
MeV Megaelectronvolt

### Temperature

Abbreviation Unit
K Kelvin
°F Fahrenheit
°C Celsius

### Others

Abbreviation Unit
cd Candela

#### Amount of substance

Abbreviation Unit
µmol Micromole
mmol Millimole
mol Mole

### Constants

Abbreviation Unit
c Speed of light in a vacuum
e Elementary charge
G Gravitational constant
g0 Acceleration of gravity
k Boltzmann constant
ke Coulomb constant
me Mass of an electron
mn Mass of a neutron
mp Mass of a proton
Na Avogadro constant
R Molar gas constant
ε0 Vacuum permittivity
µ0 Vacuum permeability
hplanckx Planck permeability

## Matrices and vectors

##### New matrix or vector

Create a new matrix or vector. This option creates a template. Enter your numbers using the directional keys.

##### transpose(M)

Transpose the matrix M. For instance, transpose([[1,2][3,4]]) returns $\left[\begin{array}{cc}1 & 3 \\ 2 & 4 \end{array}\right]$.

##### dim(M)

Size of the matrix M. For instance, dim([[1,2][3,4]]) returns [2,2].

#### Matrices

##### det(M)

Determinant of the matrix M. For instance, det([[1,2][3,4]]) returns $-2$.

##### inverse(M)

Inverse of the matrix M. For instance, inverse([[0.25,0][0,0.25]]) returns $\left[\begin{array}{cc}4 & 0 \\ 0 & 4 \end{array}\right]$.

##### identity(n)

Identity matrix of size n.

##### trace(M)

Trace of the matrix M. For instance, trace([[1,2][3,4]]) returns $5$.

##### ref(M)

Returns the scaled shape of matrix M.

##### rref(M)

Returns the scaled form of matrix M.

#### Vectors

Vectors can be row vectors or column vectors.

##### dot(U,V)

Calculates the dot product of two vectors.

##### cross(U,V)

Calculates the cross product of two vectors of size 3.

##### norm(U)

Calculates the magnitude of a vector.

## Lists

##### New list

Create a new list. This option provides opening and closing curly braces. Enter your elements using the number keys with a comma to separate each element.

##### List of f(k) for k from 1 to n

Create a new list using a function. This option creates a template. Enter your function f(k) and your upper bound.

#### Statistics

##### mean(L)

Calculates the mean of L.

##### stddev(L)

Calculates the standard deviation of L.

##### samplestddev(L)

Calculates the sample standard deviation of L.

##### med(L)

Calculates the median of L.

##### var(L)

Calculates the variance of L.

#### Operations

##### dim(L)

Returns the length of L.

##### min(L)

Returns the minimum element of L.

##### max(L)

Returns the maximum element of L.

##### sort(L)

Sorts the elements of L in ascending order.

##### sum(L)

Calculates the sum of the elements of L.

##### prod(L)

Calculates the product of the elements of L.

## Arithmetic

##### gcd(p,q)

Greatest Common Divisor of two integers. For instance, gcd(55,11) returns $11$. This function accepts more than two integers as arguments.

##### lcm(p,q)

Least Common Multiple of two integers. For instance, lcm(13,2) returns $26$. This function accepts more than two integers as arguments.

##### factor(n)

Integer factorization of $n$. For instance, factor(24) returns $2^3 \times 3$.

##### Mixed fraction

A template to input a mixed fraction.

##### 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 $2$.

## Trigonometry

### Hyperbolic

Hyperbolic sine.

##### cosh(x)

Hyperbolic cosine.

##### tanh(x)

Hyperbolic tangent.

##### arsinh(x)

Inverse hyperbolic sine.

##### arcosh(x)

Inverse hyperbolic cosine.

##### artanh(x)

Inverse hyperbolic tangent.

Cosecant

Secant

Cotangent

Arccosecant

Arcsecant

Arccotangent

## Decimal numbers

##### floor

Floor function. For instance, floor(5.8) returns $5$.

##### frac(x)

Fractional part. For instance, frac(5.8) returns $0.8$.

##### ceiling

Ceiling function. For instance, ceil(5.4) returns $6$.

##### round(x,n)

Rounds a number to $n$ digits after the decimal point. For instance round(8.6576,2) returns $8.66$.

## Logic

##### piecewise(-x,x<0,x,x≥0

A piecewise template. Input an expression followed by its domain or conditions

##### ≤

Less than or equal to

##### ≥

Greater than or equal to

Different

And

Or (inclusive)

Not

Or (exclusive)

##### nor

Not or (inclusive)

Not and