View source for Eng:List of Mathematical Functions Done in Scratch

{{:Eng:List of Mathematical Functions Done in Scratch/translate}}
<!--Please remember to add things in alphabetical order :) -->
This page lists ways in which '''important mathematical functions and formulas can be expressed using the Scratch [[Eng:Operators Blocks|Operators Blocks]]'''. This does not need to be limited to simple or elsewhat useful formulae.

{{Note|This article is intended for users who are skilled at mathematics.}}

==Conventional Measurement Abbreviations==
<table class="wikitable">
<tr>
<th>Abbreviation</th>
<th>Measurement</th>
</tr>
<tr>
<td>b</td>
<td>Length of a side of the base, plus the area of the base</td>
</tr>
<tr>
<td>h</td>
<td>Height</td>
</tr>
<tr>
<td>r</td>
<td>Radius. In a torus, the radius of the tube.</td>
</tr>
<tr>
<td>l</td>
<td>Slant Height</td>
</tr>
<tr>
<td>a, b, c</td>
<td>Side lengths of a triangle</td>
</tr>
<tr>
<td>A, B, C</td>
<td>Angle measures of a triangle (angle A is opposite side a)</td>
</tr>
<tr>
<td>R</td>
<td>The radius of the hole, plus the radius of the tube</td>
</tr>
<tr>
<td>m</td>
<td>Slope of a line segment</td>
</tr>
</table>
===Common Constants===
{{Note|Since [[wikipedia:pi|pi]] is an irrational number, it is up to the user to decide the number of decimal places it is rounded to. 3.14, 3.142 or 3.1416 are some conventional roundings.}}
<table class="wikitable">
<tr>
<th>Constant</th>
<th>Approximate Value</th>
</tr>
<tr>
<td>pi (''π'')</td>
<td>3.141592654</td>
</tr>
<tr>
<td>Golden Ratio (''φ'')</td>
<td>1.618033989</td>
</tr>
</table>

==Area and Surface Area==

===Cone===
<scratchblocks>
(((pi) * ((r) * (r))) + ((pi) * ((r) * (l))))
</scratchblocks>

[[File:Cone.gif|140px]]

===Cylinder===
<scratchblocks>
((((2) * (pi)) * ((r) * (r))) + ((2) * ((pi) * ((r) * (h)))))
</scratchblocks>

[[File:Cylinder.jpeg]]

===Ellipse===
<scratchblocks>
((pi) * ((a) * (b)))
</scratchblocks>

[[File:Ellipse.jpeg]]

====Circle====
Also known as a degenerate ellipse, this is the area of a circle:
<scratchblocks>
((pi) * ((r) * (r)))
</scratchblocks>

===Frustum===
<scratchblocks>
(((pi) * ((r of bottom circle) + (r of top circle))) * ( [sqrt v] of ((((r of bottom circle) - (r of top circle)) * ((r of bottom circle) - (r of top circle))) + ((h) * (h)))
</scratchblocks>

[[File:Frustum 1.gif]]

===Möbius Strip===

<scratchblocks>(((4) * (pi)) * ((r) * (w)))</scratchblocks>
w is the distance between the edges, and r is half the distance between two points on opposite sides of the strip

This formula derives from the fact that the Möbius strip is one-sided ([[wikipedia:non-orientable surface|non-orientable]]).

===Quadrilateral===

====Kite====
Here, "d1" and "d2" are the lengths of the two diagonals of the kite:

<scratchblocks>
(((d1) * (d2)) / (2))
</scratchblocks>

====Parallelogram====

<scratchblocks>
((base) * (height))
</scratchblocks>

====Trapezoid/Trapezium====
Here, "a" and "b" are the two parallel sides of the trapezoid:

<scratchblocks>((((a) + (b)) * (h)) / (2))</scratchblocks>

===Regular N-gon===
Here, "n" is the number of sides in the polygon:

<scratchblocks>
(((n) * ((side) * ((1) / ([tan v] of ((pi) / (n)))))) / (4))
</scratchblocks>

===Sphere===
<scratchblocks>((pi) * ((4) * ((r) * (r))))</scratchblocks>

[[File:Sphere.jpeg|175px]]

===Sector of a Circle===
This will find the answer in degrees:
<scratchblocks>
(((central angle) / (360)) * ((pi) * ((r) * (r))))
</scratchblocks>

===Square-based Pyramid===
<scratchblocks>
(((2) * ((b) * (l))) + ((b) * (l)))
</scratchblocks>

===Torus===
<scratchblocks>(((4) * ((pi) * (pi))) * ((R) * (r)))</scratchblocks>

[[File:Torus.jpeg|225px]]

Note that this formula is just the product of the circumferences of two circles with radii R and r.

===Triangle===
There are numerous ways to calculate the area of a triangle:

The most common formula is '''Area = (base/2)*height''' (see below) where '''base''' is any side and '''height''' is the length of a line segment perpendicular to the base, ending at the vertex opposite the base. In Scratch, this translates to:

<scratchblocks>(((base) / (2)) * (height))</scratchblocks>

If two side lengths are known (a and b) and one angle (C), the following formula may be used to find the area:
<scratchblocks>
(((a) * (b)) * (([sin v] of (C)) / (2)))
</scratchblocks>


If one side length is known (c) and two angles (A and B), the following formula may be used to find the area:
<scratchblocks>
((((c) * (c)) * (([sin v] of (A)) * ([sin v] of (B)))) / ((-2) * ([sin v] of ((A) + (B)))))
</scratchblocks>

If 3 side lengths are known (a, b, c), the following formula, known as Heron's formula, may be used to find the area:
<scratchblocks>
set [s v] to ((((a) + (b)) + (c)) / (2)) //s is called the semiperimeter; it is half the perimeter
set [area v] to ([sqrt v] of ((((s) * ((s) - (a))) * ((s) - (b))) * ((s) - (c))))
</scratchblocks>

If the triangle's vertices have integer coordinates, it is best to use [[wikipedia: Pick's theorem|Pick's theorem]] or Heron's formula (above) (using the [[Eng:#Distance between two points|#Distance between two points]] to find the lengths of the triangle's sides).

If one knows the radius of the triangle's [[wikipedia:Incircle|incircle]] and its semiperimeter (half the perimeter), we can use this formula:

<scratchblocks>((radius) * (semiperimeter))</scratchblocks>

==Volume==
===Cone===
<scratchblocks>
(((pi) * (((r) * (r)) * (h))) / (3))
</scratchblocks>

[[File:Cone.gif|140px]]

===Cylinder===
<scratchblocks>((pi) * ((r) * ((r) * (h))))</scratchblocks>

[[File:Cylinder.jpeg]]

===Frustum===

<scratchblocks>
((((pi) * (h)) / (3)) * (((r of bottom circle) * (r of bottom circle)) + (((r of bottom circle) * (r of top circle)) + ((r of top circle) * (r of top circle)))))
</scratchblocks>

[[File:Frustum 1.gif]]

===Pyramid===

<scratchblocks>
(((b) * (h)) / (3))
</scratchblocks>

[[File:Pyramid.gif|150px]]

===Regular Tetrahedron===
Here, "b" is the length of each edge:

<scratchblocks>((([sqrt v] of (2)) / (12)) * (((b) * (b)) * (b)))</scratchblocks>

[[File:Regular Tetrahedron.jpg|240px]]

===Sphere===
<scratchblocks>
(((4) / (3)) * ((pi) * ((r) * ((r) * (r)))))
</scratchblocks>

[[File:Sphere.jpeg|175px]]

===Torus===

<scratchblocks>(((2) * ((pi) * (pi))) * ((R) * ((r) * (r))))</scratchblocks>

[[File:Torus.jpeg|225px]]

Note that this formula is just the product of the area of a circle with radius r and the circumference of a circle with radius R. It is only correct when R ≥ r.

==Law of Sines==
[[File:Sine Rule.png]]
===Sides===
This will give an answer for side "a":

<scratchblocks>
(((b) * ( [sin v] of (A))) / ( [sin v] of (B)))
</scratchblocks>
===Angles===
This will give an answer for angle "A":

<scratchblocks>
( [asin v] of (((a) * ( [sin v] of (B))) / (b)))
</scratchblocks>

==Law of Cosines==
[[File:Cosine Rule.png]]
===Sides===
This will give an answer for side "c":
<scratchblocks>
( [sqrt v] of ((((a) * (a)) + ((b) * (b))) - ((2) * ((a) * ((b) * ( [cos v] of (C)))))))
</scratchblocks>

===Angles===
This will give an answer for angle "C":
<scratchblocks>
( [acos v] of ((((a) * (a)) + (((b) * (b)) - ((c) * (c)))) / ((2) * ((a) * (b)))))
</scratchblocks>

==Hyperbolic Functions==

There are six main [[wikipedia:Hyperbolic function|hyperbolic functions]]:

* sinh''x'' (the hyperbolic sine)
* cosh''x'' (the hyperbolic cosine)
* tanh''x'' (the hyperbolic tangent)
* coth''x'' (the hyperbolic [[wikipedia:Trigonometric functions#Reciprocal functions|cotangent]])
* sech''x'' (the hyperbolic [[wikipedia:Trigonometric functions#Reciprocal functions|secant]])
* csch''x'' (the hyperbolic [[wikipedia:Trigonometric functions#Reciprocal functions|cosecant]]), also, cosech''x''

===sinh and cosh===

Just as in normal trigonometry, the sinh and cosh functions are the fundamental units. They can be replicated with

<scratchblocks>((([e^ v] of (x)) / (2)) - ((1) / ((2) * ([e^ v] of ((0) - (x))))))</scratchblocks>

and

<scratchblocks>((([e^ v] of (x)) / (2)) + ((1) / ((2) * ([e^ v] of ((0) - (x))))))</scratchblocks>

respectively.

===sech and csch===

In normal trigonometry, the secant and cosecant are given by 1/cos''x'' and 1/sin''x'', respectively. The hyperbolic secant and cosecant can be expressed in the same way: sech''x'' = 1/cosh''x'', or:

<scratchblocks>((2) / (([e^ v] of (x)) + ([e^ v] of ((0) - (x)))))</scratchblocks>

and csch''x'' = 1/sinh''x'':

<scratchblocks>((2) / (([e^ v] of (x)) - ([e^ v] of ((0) - (x)))))</scratchblocks>

===tanh and coth===

The tangent function is expressed by sin''x''/cos''x''. Similarly, the hyperbolic tangent is expressed by sinh''x''/cosh''x''. In Scratch, that is:

<scratchblocks>((([e^ v] of (x)) - ([e^ v] of ((0) - (x)))) / (([e^ v] of (x)) + ([e^ v] of ((0) - (x)))))</scratchblocks>

Similarly, the cotangent is 1/tan''x'', so the hyperbolic cotangent is:

<scratchblocks>((([e^ v] of (x)) + ([e^ v] of ((0) - (x)))) / (([e^ v] of (x)) - ([e^ v] of ((0) - (x)))))</scratchblocks>

==Pythagorean Theorem==
[[File:Pythagorean theorem.png]]

This will give the length of the hypotenuse:
<scratchblocks>
( [sqrt v] of (((a) * (a)) + ((b) * (b))))
</scratchblocks>

And these will give the length of sides "a" and "b" (assuming the hypotenuse is "c"):
<scratchblocks>
( [sqrt v] of (((c) * (c)) + ((a) * (a))))
( [sqrt v] of (((c) * (c)) + ((b) * (b))))
</scratchblocks>
===Distance between two Points===
The distance between two points can be calculated by using the Distance Formula which is derived from the Pythagorean Theorem, The following is a [[Eng:ScratchBlocks|ScratchBlocks]] representation of the distance formula. It has been broken into three lines. The distance formula is as follows:
{{equation|<nowiki>distance = √((x2-x1)^2 + (y2-y1)^2)</nowiki>}}
<scratchblocks>
set [a v] to ( ((x2) - (x1) ) * ( (x2) - (x1) ))
set [b v] to (((y2) - (y1)) * ((y2) - (y1)))
set [distance v] to ([sqrt v] of ((a) + (b)))
</scratchblocks>

The distance formula in one line is:
<scratchblocks>
<(d) = ([sqrt v] of ((( (x2) - (x1)) * ((x2) - (x1) )) + (( (y2) - (y1)) * ((y2) - (y1) )))>
</scratchblocks>

Where d = distance, x2 = larger x value, x1 = smaller x value, y2 = larger y value, y1 = smaller y value

===Midpoint between two Points===
The coordinate of the midpoint of a line is found using the midpoint formula, which is as follows (the equation gives a ''(x,y)'' paired coordinate):
{{equation|<nowiki>midpointcoord = ((x2+x1/2), (y2+y1/2))</nowiki>}}
<scratchblocks>
set [x v] to (((x2) + (x1)) / (2))
set [y v] to (((y2) + (y1)) / (2))

</scratchblocks>

==Linear Equations==

A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. 

Here is an example of a linear equation: 2x+4=14, for which in standard form would look like this:
<scratchblocks>
((((a) * (x)) + (b)) = (c))
</scratchblocks>

In scratch, we could solve for x using the following script:
<scratchblocks>
(((c) - (b)) / (a))
</scratchblocks>

Inputting the values of a, b and c scratch could solve the equation with ease.

==Quadratic Formula==
[[File:Quadratic.png]]

These will give the two possible values of "x":
<scratchblocks>
((((b) * (-1)) + ( [sqrt v] of (((b) * (b)) - ((4) * ((a) * (c)))))) / ((2) * (a)))
((((b) * (-1)) - ( [sqrt v] of (((b) * (b)) - ((4) * ((a) * (c)))))) / ((2) * (a)))
</scratchblocks>
<br/>
{{Caution|If b<sup>2</sup>-4ac is negative, you will get a script error. Here's how you can avoid the error... }}


In this case these will give you the two possible values of "x":
<scratchblocks>

(join (((b) * (-1)) / ( (2) * (a))) (join (join [+] (([sqrt v] of ((((b) * (b)) - ((4) * ((a) * (c)))) * (-1))) / ((2) * (a)))) [i]))

(join (((b) * (-1)) / ( (2) * (a))) (join (join [-] (([sqrt v] of ((((b) * (b)) - ((4) * ((a) * (c)))) * (-1))) / ((2) * (a)))) [i]))

</scratchblocks>


{{warning|}} a≠0. If you plug in a=0, you will receive an invalid solution. As a rule in math, the "a" coefficient in Quadratic equations should not have a value of 0.

==Cubic Formula==
{{note|This formula may deal with complex numbers that cannot be handled by Scratch. If it were to be worked out properly, the complex numbers would cancel out, but unfortunately here a lot of problems will be unsolvable because Scratch does not support complex roots.}}

This will give the answer to the equation:

[[File:Cubic function.png]]

The analogous formula for this is:

[[File:Cubic function 2.gif|600px]]

In Scratch this translates to this (scroll the equation to see more):

<scratchblocks>((([10 ^ v] of (((1) / (3)) * ([log v] of ((((((0) - (((b) * (b)) * (b))) / ((27) * (((a) * (a)) * (a)))) + (((b) * (c)) / ((6) * ((a) * (a))))) - ((d) / ((2) * (a)))) + ([sqrt v] of (((((((0) - (((b) * (b)) * (b))) / ((27) * (((a) * (a)) * (a)))) + (((b) * (c)) / ((6) * ((a) * (a))))) - ((d) / ((2) * (a)))) * (((((0) - (((b) * (b)) * (b))) / ((27) * (((a) * (a)) * (a)))) + (((b) * (c)) / ((6) * ((a) * (a))))) - ((d) / ((2) * (a))))) + ((((c) / ((3) * (a))) - (((b) * (b)) / ((9) * ((a) * (a))))) * ((((c) / ((3) * (a))) - (((b) * (b)) / ((9) * ((a) * (a))))) * (((c) / ((3) * (a))) - (((b) * (b)) / ((9) * ((a) * (a))))))))))))) + ([10 ^ v] of (((1) / (3)) * ([log v] of ((((((0) - (((b) * (b)) * (b))) / ((27) * (((a) * (a)) * (a)))) + (((b) * (c)) / ((6) * ((a) * (a))))) - ((d) / ((2) * (a)))) - ([sqrt v] of (((((((0) - (((b) * (b)) * (b))) / ((27) * (((a) * (a)) * (a)))) + (((b) * (c)) / ((6) * ((a) * (a))))) - ((d) / ((2) * (a)))) * (((((0) - (((b) * (b)) * (b))) / ((27) * (((a) * (a)) * (a)))) + (((b) * (c)) / ((6) * ((a) * (a))))) - ((d) / ((2) * (a))))) + ((((c) / ((3) * (a))) - (((b) * (b)) / ((9) * ((a) * (a))))) * ((((c) / ((3) * (a))) - (((b) * (b)) / ((9) * ((a) * (a))))) * (((c) / ((3) * (a))) - (((b) * (b)) / ((9) * ((a) * (a)))))))))))))) - ((b) / ((3) * (a)))</scratchblocks>

There are ways to work around Scratch's incapability of calculating complex roots with the aid of the Trigonometric Form of Solutions.

==Tetration==
[[wikipedia: Tetration|Tetration]] is iterated exponentiation. It is written as <sup>''b''</sup>''a'', ''a''^^''b'', or ''a''[4]''b'', where ''a'' is the base and ''b'' is the height. Tetration is computed by the following procedure: <sup>''b''</sup>''a''=''a''<sup>''a''<sup>⋰<sup>''a''</sup></sup></sup>, where there are ''b'' ''a'''s So <sup>4</sup>2 is 2<sup>2<sup>2<sup>2</sup></sup></sup>, which is 65536. It can be written in Scratch as:

<scratchblocks>repeat ((b) - (1))
repeat (a)
set [a^^b v] to ((a^^b) * (a))
end
end</scratchblocks>

Here, "a^^b" is your answer. This method only works for whole heights, however, but there is no known algorithm for calculating the value for non-integral heights.

==xth root==
The [[wikipedia: nth root|''x''th root]] of a number ''n'' is number that must be multiplied ''x'' times to get ''n''. For example, the cube root (3rd root) of 2 (written ∛2) is about 1.2599, because 1.2599*1.2599*1.2599≈2.

A universal formula for finding the '''''x'''''th root of a number can be found using logarithms.  To find the '''''x''''' root of '''''n''''', use the following formula:
<scratchblocks>
([e ^ v] of (([ln v] of (n)) / (x)))
</scratchblocks>

This is based on a method of using exponents and logarithms, which Scratch supports, to find powers, which Scratch does not directly support (see [[Eng:Solving Exponents#Logarithmic Method|Solving Exponents#Logarithmic Method]]).

===Fourth root, Eighth root, etc.===

The fourth root of a number is the number that, when squared twice, gives the original number. The fourth root of 81, for example, is 3. Therefore, it can be replicated in Scratch with this script:

<scratchblocks>([sqrt v] of ([sqrt v] of (a)))//a is the input number</scratchblocks>

Similarly, the eighth root of a number is the number, when squared three times, gives the original number. The eighth root of 256, for example, is 2. It can be replicated with this script:

<scratchblocks>([sqrt v] of ([sqrt v] of ([sqrt v] of (a))))</scratchblocks>

In general, to get the 2<sup>''n''</sup>th root of a number, nest ''n'' [[Eng:() of () (Operators block)|() of () (Operators block)]] [[Eng:blocks|blocks]] together, and then enter the desired value, or insert a [[Eng:variable|variable]].

==Physics==

This section is for physical constants and equations.

===Important Physical Constants===


<table class="wikitable">
<tr>
<th>Constant</th>
<th>Symbol</th>
<th>Approximate value</th>
</tr>
<tr>
<td>Speed of light</td>
<td>''c''</td>
<td>299792458 (exact value)</td>
</tr>
<tr>
<td>Universal gravitational constant</td>
<td>''G''</td>
<td>6.67*10<sup>−11</sup></td>
</tr>
<tr>
<td>Fine-structure constant</td>
<td>''α''</td>
<td>0.007297</td>
</tr>
<tr>
<td>Planck's constant</td>
<td>''h''</td>
<td>6.626*10<sup>−34</sup></td>
</tr>
</table>

==Pick's Theorem==
[[wikipedia:Pick's theorem|Pick's theorem]] says that the area of a polygon with integer coordinates can be determined by:

{{equation|<nowiki>A=½a+b-1</nowiki>}}

In the equation, a is the number of integer coordinates on the perimeter of the shape, and b is the number of coordinates inside the figure. In Scratch this translates to:

<scratchblocks>((((a) / (2)) + (b)) - (1))</scratchblocks>

[[Category:Math Tutorials]] [[Category:Lists|Mathematical Functions Done in Scratch]]