Menu reference list

Number pi. The ratio of a circumference to its diameter. $\mathrm{\pi }\approx 3.1416$

Number e.$f=e^x$ is the only function such that $f\text{'}=f$ and $f\left(0\right)=1$. $e\approx 2.7183$

Imaginary unit. A number defined to satisfy $i\cdot i=-1$.

Imaginary unit. A number defined to satisfy $j\cdot j=-1$.

Infinity. Use it to calculate limits.

Addition

Subtraction

Multiplication

Division

Parentheses. Use them to control the order of the operations.

Vector. See Linear algebra section.

List. Some results or parameters are given or requested in this form.

Absolute value. Removes the sign from a number.

Norm. Length of a vector.

Fraction

Square root

Root

Power

Element of list. Use it also for a vector, a matrix, an equation, etc.

Greater-than sign

Less-than sign

Greater than or equal to

Less than or equal to

Not equal to

Absolute value. You can write | (pipe) with the keyboard, too.

Floor. Round down to the next smaller integer.

Ceiling. Round up to the next greater integer.

Round to nearest integer, and for tie-breaking round half up.

The sign for a number. It can be -1, 0 or 1.

The greater of two numbers, or a list.

The smaller of two numbers, or a list.

Generate a pseudo-random number between the two given ones (including both). Also, choose randomly from a list.

The numerator of a fraction.

The denominator of a fraction.

The quotient of the integer division of the first number (dividend) by the second (divisor).

The remainder of the integer division of the first number (dividend) by the second (divisor); also called modulus in many textbooks.

Greatest common divisor.

Least common multiple.

Prime factorization of an integer.

Tests to determine whether a number is prime. This is a predicate: a command that returns only true or false.

The degree of a polynomial

How many terms does a polynomial has.

This is one term from a polynomial. The term number is the second parameter. The terms are ordered by descending grades. Therefore, term number 1 is always the leading term.

The content of a polynomial. That is, gcd of their coefficients.

Rearrange a polynomial with multiple variables arranged around the variable in the second parameter.

Finds the roots of a polynomial or, in other words, the values of x that make it 0.

The numerator of a rational fraction

The denominator of a rational fraction

The quotient of the division of the first polynomial (dividend) by the second (divisor)

The remainder of the division of the first polynomial (dividend) by the second (divisor)

The greatest common divisor

The least common multiple

Factorization in irreducible polynomials

This tests whether a polynomial is irreducible. This is a predicate: a command that returns only true or false.

Imaginary unit

The real part of a complex number.

The imaginary part of a complex number, which is a real number.

The modulus of a complex number.

The argument of a complex number, in the range (-π,+π].

This converts a complex number from binomial form to polar form and the other way around(!). The polar form is a list formatted as {norm, argument}.

This is the conjugate of a complex number. Shift the sign of the imaginary part.

List

Mean, arithmetic mean, average.

This is used to summarize measures with different units (length, cost, weight,...) of the same object. It has no sense alone, but it is helpful for comparisons among multiple objects.

This is used for ratios and rates, as in the context of speed.

A measure of variability that is convenient for calculations

A measure of variability that has the same physical units of the data

A central measure, alternative to mean, is more robust, meaning it isn't affected by extreme data, generally known as outliers.

These values divide the data once ordered into four groups of the same size. They're used to measure variability. See the formula reference section for details.

Most frequent value in data. It can be a set if there are ties.

This is the base for the correlation coefficient. It has the same sign.

Pearson correlation coefficient. It determines whether there is a linear relationship between the paired data.

It gives the line equation that better fits the cloud of data. It finds the best $y=a+bx$.

It fits the data to a power function. It finds the best $y=ax^b$.

It fits the data to an exponential function. It finds the best $y=a\mathrm e^{bx}$.

It fits the data to a logarithmic function. It finds the best $y=a+\ln(bx)$.

$\frac1n\sum_iX_i$

$\sqrt[n]{\prod_iX_i}$

$\frac n{\displaystyle\sum_i\frac1{X_i}}$

$\frac1{n-1}\sum_i(X_i-\text{mean}(X))^2$

$\sqrt{\text{variance}(X)}$

$\left\{\begin{array}{lc}{X}_k& n=2k-1\\ \frac{X_k+X_{k+1}}{2}& n=2k\end{array}\right.$

$\begin{array}{l}\text{quartile}\left(0,X\right)=X_1\\ \text{quartile}\left(1,X\right)=\left\{\begin{array}{lc}\text{median}\left(\left\{X_1,\dots ,X_k\right\}\right)& n=2k-1\\ \text{median}\left(\left\{X_1,\dots ,X_k\right\}\right)& n=2k\end{array}\right.\\ \text{quartile}\left(2,X\right)=\text{median}\left(X\right)\\ \text{quartile}\left(3,X\right)=\left\{\begin{array}{lc}\text{median}\left(\left\{X_k,\dots ,X_n\right\}\right)& n=2k-1\\ \text{median}\left(\left\{X_{k+1},\dots ,X_n\right\}\right)& n=2k\end{array}\right.\\ \text{quartile}\left(4,X\right)=X_n\end{array}$

$\frac1{n-1}\sum_i(X_i-\text{mean}(X))(Y_i-\text{mean}(Y))$

$\frac{covariance(X,Y)}{standard\_deviation(X)\cdot standard\_deviation(Y)}$

$\frac{y-mean(Y)}{standard\_deviation(Y)}=correlation(XY)\frac{x-mean(X)}{standard\_deviation(X)}$

All intervals of the same length on the distribution's support are equally probable in the uniform distribution.

The normal distribution is determined by its mean $\mu$ and standard deviation $\sigma$. It is widely used in natural science, among other fields.

The exponential distribution is the probability distribution that describes the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate.

The $\chi^2$ distribution with $k$ degrees of freedom is the distribution of a sum of the squares of $k$ independent standard normal random variables.

Student's $t$-distribution arises when estimating the mean of a normally distributed population in situations where the sample size is small, and population standard deviation is unknown.

The Bernoulli distribution is the probability distribution of a random variable which takes the value $1$ with probability $p$ and the value 0 with probability $q=1-p$.

The binomial distribution with parameters $n$ and $p$ is the discrete probability distribution of the number of successes in a sequence of $n$ independent experiments, each asking a yes-no question, i.e. each ruled by the same Bernoulli distribution.

The geometric distribution with parameter $p$ is the discrete probability distribution of the number of failures before the first success. A Bernoulli variable rules each try with the parameter $p$.

The Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant rate and independently of the time since the last event.

The $F$-distribution is a continuous probability distribution that frequently arises as to the null distribution of a test statistic, most notably in the analysis of variance (ANOVA), e.g., $F$-test.

It retrieves a random number following a given distribution.

Distribution function of a random variable at a given point. In some cases, the analytical expression is given. $distribution(X,x)=F_X(x)=P(X\leq x)=\int_{-\infty}^x{f_X(t)\;}dt$, cumulative distribution function (CDF)

Density function of a random variable at a given point. In some cases, the analytical expression is given. $density(X,x)=f_X(x)=\frac d{dx}F_X(x)$, probability density function (PDF).

Quantile function of a random variable for a given probability. $quantile(X,p)=x⟺distribution(X,x)=p$, inverse function of CDF.

Histograms are often used to represent continuous data visually.

As opposed to histograms, bar charts are helpful for displaying discrete data, as well as categorical data.

Pie charts are another standard tool for categorical data when we want to show the proportion that each category occupies out of the whole.

Boxplots are very useful for a concise representation of a data set. In one boxplot, we see the median, the interquartile range (IQR), as well as outliers.

Number pi This is useful when working with radians. $\mathrm\pi\approx3.1416$

Angle degree

Direct

Sine, related to the side opposite the angle

Cosine, related to the side adjacent to the angle

Tangent, sin/cos

Reciprocal

Cosecant, 1/sin

Secant, 1/cos

Cotangent, 1/tan

Inverse

This is one of the many angles whose sine is the given number. It's the one in [-π/2,π/2].

This is one of the many angles whose cosine is the given number. It's the one in [0,π].

This is one of the many angles whose tangent is the given number. It's the one in (-π/2,π/2).

Number e This is the basis of the Napierian logarithm. $e\approx2.7183$

Logarithm with base

Exponential, e powered to the given number.

Natural or Napierian logarithm

Logarithm

$\sinh(x)=\frac{e^x-e^{-x}}2$

$\cosh(x)=\frac{e^x+e^{-x}}2$

$\tanh(x)=\frac{\sinh(x)}{\cosh(x)}$

n

nano

0.000 000 001

µ

micro

0.000 001

m

mili

0.001

c

centi

0.01

d

deci

0.1

da

deca

10

h

hecto

100

k

kilo

1000

M

mega

1 000 000

G

giga

1 000 000 000

Meter

Gram

Second

Ampere

Kelvin

Mol

Candela

Angle degree

Angle minute

Angle second

Radian

Steradiant

Hour

Minute

Second

Liter

Newton

Hertz

Pascal

Watt

Joule

Coulomb

Volt

Ohm

Farad

Siemens

Weber

Bar

Henry

Tesla

Lux

Lumen

Gray

Becquerel

Sievert

Katal

Atmosphere

Molar

Dalton

Electronvolt

Pond

Yard

Foot

Inch

Mile

Nautical mile

Gallon

Ounce

Pound

Fluid ounce

Pint

Percent

Permil

Dollar

Euro

Pound

Franc

Krone

Bitcoin

Ruble

Rupee

Won

Yen

if statement. If the condition is held, then performs the action inside the block.

else statement. It should be preceded by an if. If the condition specified in the if statement is not held, then runs the commands inside the else's block.

It should be preceded by an if. If the condition specified in the previous if statement is not held and the current condition is satisfied, then performs the action inside the block.

for statement. Write quickly a loop that needs to be run a specific number of times.

while statement repeats the code block until the condition is not satisfied. Be sure that the condition does not hold in some cases. Otherwise, the code will run infinitely.

repeat statement. While the condition does not hold, duplicate the code block. Again, be sure that the condition is satisfied sometimes.

This block is extremely useful when one defines its own functions. It allows performing different actions inside one block and defining local variables.

return statement. Returns a value in a user function.

Allows defining local variables: variables defined just in one code block.