Very very basic guide to CalcMe
How does CalcMe work?
You can write Mathematics, and CalcMe will perform the calculations for you.
The CalcMe sheet is divided into three main areas:
Calculate
- Write what you would like to compute
- Click Calc or Enter
The default action is Calc but you can choose the most appropiate one at your convenience.
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Graph
- Write the equation or figure you would like to plot
- Click on Plot action
Define
- Store a computation in a variable
- Use it later
Arithmetic
Mathematical operations in CalcMe are represented by a symbol associated with a keyboard key.
Operation | Symbol | Keyboard |
---|---|---|
Sum | + | |
Substraction | - | |
Multiplication | * or · | |
Division | / | |
Square root | Ctrl+. | |
Root | Ctrl+, | |
Power | Ctrl+↑ |
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You can compute the quotient and remainder of a division or factorize a number into prime factors. You can also compute the greatest common divisor or the least common multiple of a set of numbers.
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Vector and Matrix constructor
Vectors
Vector are constructed with square brackets []
, and the elements are separated by commas ,
.
You can sum vectors or compute their scalar product.
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Matrices
Matrices are vectors of vectors, that is, vectors whose elements are vectors. You can create matrices with two different syntaxes
As vectors, you can sum and multiply matrices (as long as their dimensions match)
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Basic operations
As you have previously seen, you can work with vectors and matrices and perform basic arithmetic with them. However, you can do many more things: you can compute the cross product between two vectors, check if they are linearly independent; a matrix can be inverted or raised to a whole power, you can also compute its rank or its determinant.
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Element access
You can access a specific element of a vector using subindices, that start on 1. In the same way, you can retrieve an element of a matrix
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Polynomials and expressions
Polynomials are created with a number multiplied by a variable raised to a power.
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You can sum, multiply, divide and, for instance, find roots of polynomials.
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You can also create more complex expressions and operate with them.
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Differentiation
There are plenty of ways to compute the derivative of a function or expression.
Using a prime | ' |
Using the symbol | |
Using the action | |
Using the command | differentiate |
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Integration
There are also plenty of ways to compute the integral of a function or expression.
Using the symbol | |
Using the action | |
Using the command | integrate |
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Limit
It is possible to compute the limit of a function or expression. Moreover, you can also take one-sided limits.
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Taylor expansion
You can compute the Taylor series of a real function at a given point. If you are interested in the terms up to some order, you can cut the series too.
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Series
You can determine whether a series is convergent, as well as calculating the sum of a convergent series in most cases.
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Geometry
CalcMe allows us to deal with geometrical figures such points, lines, planes and conic sections. You can also create polygons and polyhedra, both in 2D and 3D. It is also possible to compute the distance between figures, the angle they form or the symmetry with respect to an object.
Points, lines and planes
It is possible to work in 2 or 3 dimensions. To create points you simply define its components.
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Given two points or a point and a vector you can construct a line.
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In a similar way, you can, for instance, construct a plane given three points.
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Figures
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Solving
Equations
It is possible to solve an equation or a system of equations exactly.
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You can also use a numerical method to solve more complicated equations.
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Inequations
It is also possible to find the solution to an inequation.
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Statistics
There are a lot of functions useful for statistical calculations such as mean, quantile, quartile, etc. You can see the complete list here.
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Probability
The most common probability distributions are also available. Analytical expressions for density and distribution functions are also available if possible.
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Combinatorics
Combinations, permutations or variations of a set of numbers can be calculated. Be careful, though: results are very large.
Counting | Symbol |
---|---|
Variations or k-permutations of n | |
Permutations | |
Combinations | |
Variations with repetition or n-tuples of m-sets | |
Permutations with repetition | |
Combinations with repetition | |
Binomial coefficient | |
Factorial You can also write ! with the keyboard. |
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Units and currencies
You can work with units, convert one into another (if it makes sense) and do basic operations with them. Units need to be introduced via the symbols available under the Units of measure tab. Find out more information such as all the units or prefixes you can use here.
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Currencies are similar to units but you cannot convert one into another. Check all the available currencies here.
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Randomness
The random function in CalcMe is adaptable to many cases of use. For example, you will see how to remove the "0" from a random selection. The normal command would be like:
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By default, this includes all of the numbers between -10 and 10. If, given the requirements of the question, the number 0 needs to be excluded from the set, you can remove it with one simple instruction (the slash /
should be used through the Logic and sets tab):
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You have to add brackets around the first list for this to work. This can of course work equally for any other number you need to exclude other than zero:
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The above would produce a random number between -10 and 10, except the number 8. You can even do this with more than one number:
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This would remove 8,-8, and 0 from the selection. As you can see, there are many more options when creating a random variable.
So far you have retrieved integer numbers but you can also work with real numbers
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Creating lists via comprehension
This section is a short guide to creating lists in CalcMe, using the commands
with
and where
. This method of defining lists is based on the common mathematical notation of "set comprehension" or "set-builder notation", for instance:
$\left\{(x,y)\in {\mathrm{\mathbb{R}}}^{2}\mid x\ge 0,y\ge 0\right\}$
We will explain the commands through the following examples.
Example 1
In the most basic level, with
simply provides a more compact form of writing long lists. You can write
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or you could greatly simplify it to the following:
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The command where
comes in handy when you would like to have additional restrictions. For example, retrieve the even numbers only:
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Alternatively, you could, of course, have done this:
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Example 2
In the first example, the size of the list was fixed. However, comprehension is specially useful when defining lists of variable size. Here, for example, is a list with random elements, of random size
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As you may have noticed, the first expression does not explicitly depend on $i$. You can think of $i$ as just a counter within a specified range. Then, you may or may not use it to define the list elements. It can also be any variable name you choose, as in the following:
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Example 3
The list comprehension notation can also be extended to more than one variable. In that case, you should specify the range for each variable used as a counter. For example, here is a list with all positive proper fractions in simplest terms, with single digit numerator and denominator:
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Another recommendation illustrated in the above example is enclosing each condition after where
in parentheses if you have more than one joined by $\wedge ,\vee $.
Example 4
Finally, note that the range for the counter variable can itself be a list, defined previously.
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Example 5
It is also possible to create matrices using this notation. For instance, creating a $4\times 3$ matrix with random coefficients is as simple as this:
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Programming
You can use some programming functions. You can see the basic ones here. For example, given a list created as described before, you can easily compute the square of the first primes.
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User functions
You can create custom functions. The
random
command is very useful but it could be a bit tedious to write every time random(-10,10)
. Instead, you can create a function that generates a random number when called:
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This eases the creation of a matrix with random coefficients. Another more elaborated example is creating a function that constructs a tri-diagonal matrix given three numbers. Therefore, every time you would like to create a tri-diagonal matrix, you just need to call this function with the upper-diagonal, diagonal and lower-diagonal terms you would like the matrix to have.
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