Some examples:<\/p>\n
- \n
- \\(5!! = 5 \\times 3 \\times 1\\)<\/span><\/li>\n
- \\(8!! = 8 \\times 6 \\times 4 \\times 2\\)<\/span><\/li>\n<\/ul>\n
When reading about it in a book authored by a friend of mine, the identity \\((2n)!! = 2^n n!\\)<\/span> was also presented, and I'll prove it now by induction.\nFor \\(n = 1\\)<\/span>, we have \\(2!! = 2 = 2^1 \\times 1!\\)<\/span>, which is true.\nNow, assuming the identity holds up to \\(n\\)<\/span>, we show it holds for \\(n + 1\\)<\/span>:<\/p>\n
\\[\n\\begin{align}\n(2(n + 1))!! &= (2(n + 1)) \\times (2n)!! \\\\\n&= (2(n + 1)) \\times 2^n n! \\\\\n&= (n + 1) \\times 2^{n + 1} n! \\\\\n&= 2^{n + 1} (n + 1)!\n\\end{align}\\]<\/p>\n
Done!<\/p>","summary":"Today I learned about the double factorial.","date_modified":"2025-10-20T22:34:56+02:00","tags":["combinatorics","arithmetic","induction","mathematics"],"image":"\/user\/pages\/02.blog\/04.til\/132.double-factorial\/thumbnail.webp"},{"title":"divmod for unit conversions","date_published":"2023-11-27T19:00:00+01:00","id":"https:\/\/mathspp.com\/blog\/divmod-for-unit-conversions","url":"https:\/\/mathspp.com\/blog\/divmod-for-unit-conversions","content_html":"
This article shows how to use the Python built-in
divmod<\/code> for unit conversions.<\/p>\n\nThe built-in
divmod<\/code> is the built-in that tells me that 45813 seconds is the same as 12 hours, 43 minutes, and 33 seconds.\nBut how?<\/p>\nThe built-in
divmod<\/code> gets its name from two arithmetic operations you probably know:<\/p>\n- DIVision; and<\/li>\n
- MODulo.<\/li>\n<\/ol>
So, the built-in
divmod<\/code> takes two arguments and returns two numbers.\nThe first one is the division of the arguments and the second one is the module of the arguments.\nSounds simple, right?\nThat's because it is!<\/p>\n>>> x, y = 23, 10\n>>> divmod(x, y)\n(2, 3)\n>>> (x \/\/ y, x % y)\n(2, 3)\n\n>>> x, y = 453, 19\n>>> divmod(x, y)\n(23, 16)\n>>> (x \/\/ y, x % y)\n(23, 16)<\/code><\/pre>\nSo, when can
divmod<\/code> be useful?\nLiterally when you want to divide two numbers and also know what's the remainder of that division.<\/p>\nIn practice,
divmod<\/code> shows up a lot when converting one small unit to more convenient units.\nThat's a common use case, at least.<\/p>\nExample with time units<\/a><\/h2>\n
45813 seconds is A LOT of seconds and most people just prefer to use the units of hours and minutes to talk about that many seconds!\nSo, how do we convert 45813 into its corresponding number of hours and minutes?<\/p>\n
We can use
divmod<\/code> to divide 45813 by 60, which will convert 45813 seconds to minutes, and also let us know how many seconds couldn't be converted to minutes:<\/p>\n>>> divmod(45813, 60)\n(763, 33)<\/code><\/pre>\nThen, we take the total number of minutes and repeat the same process to figure out how many hours we have.<\/p>\n
>>> divmod(763, 60)\n(12, 43)<\/code><\/pre>\nIf you put everything together, it looks like this:<\/p>\n
>>> seconds = 45813\n>>> minutes, seconds = divmod(seconds, 60)\n>>> hours, minutes = divmod(minutes, 60)\n>>> hours, minutes, seconds\n(12, 43, 33)<\/code><\/pre>\nIs this making sense?\nTry to convert 261647 seconds into days, hours, minutes, and seconds!<\/p>\n
Example with grid click<\/a><\/h2>\n
I have another example that you might enjoy.\nThere's this weird game I play sometimes called “meta tic-tac-toe”.\n(At least, that's what we called it in uni.)\nIt's 9 tic-tac-toe games embedded in a larger tic-tac-toe game.<\/p>\n
The board is shown below:<\/p>\n
![\"A](\"\/user\/pages\/02.blog\/divmod-for-unit-conversions\/_tictactoe.webp\" "\\"The")
The board of a meta tic-tac-toe game.<\/figcaption><\/figure> The rules aren't SUPER important now (although it's a pretty fun game).\nWhat matters now is that I wanted to implement this game in Python.<\/p>\n
To do that, when a user clicks on the board, I need to know what's the purple (outer) cell that is being clicked.\nI also need to know what's the orange (inner) cell that is being clicked!\nThe question is:\nhow would I compute that?<\/p>\n
Suppose the board is a 900 pixel square and everything is evenly spaced.\nIf I click at position
(423, 158)<\/code>, what are the outer and inner cells I clicked?<\/p>\n![\"A](\"\/user\/pages\/02.blog\/divmod-for-unit-conversions\/_tictactoe_play.webp\" "\\"A")
A click on the board at position (423, 158).<\/figcaption><\/figure> We can figure out the outer and inner cells with...<\/p>","summary":"This article shows how to use the Python built-in divmod for unit conversions.","date_modified":"2025-07-23T16:49:02+02:00","tags":["arithmetic","mathematics","programming","python"],"image":"\/user\/pages\/02.blog\/divmod-for-unit-conversions\/thumbnail.webp"},{"title":"Overloading arithmetic operators with dunder methods | Pydon't \ud83d\udc0d","date_published":"2023-07-31T00:00:00+02:00","id":"https:\/\/mathspp.com\/blog\/pydonts\/overloading-arithmetic-operators-with-dunder-methods","url":"https:\/\/mathspp.com\/blog\/pydonts\/overloading-arithmetic-operators-with-dunder-methods","content_html":"
This article shows you how to overload the arithmetic operators in Python with dunder methods.<\/p>\n\n
Introduction<\/a><\/h2>\n
Python lets you override the arithmetic operators like
+<\/code> for addition or*<\/code> for multiplication through dunder methods<\/a>.\nDunder methods<\/a> are special methods whose name starts and ends with a double underscore (hence, “dunder”), and some dunder methods<\/a> are specific to arithmetic operations.<\/p>\nIn this Pydon't, you will learn:<\/p>\n
- how to implement unary arithmetic operators:\n
- negation (
-p<\/code>);<\/li>\n- (
+p<\/code>);<\/li>\n- absolute value (
abs(p)<\/code>); and<\/li>\n- inverse (
~<\/code>).<\/li>\n<\/ul><\/li>\n- how to implement binary arithmetic operators:\n
- addition (
+<\/code>);<\/li>\n- subtraction (
-<\/code>);<\/li>\n- multiplication (
*<\/code>);<\/li>\n- division (
\/<\/code>);<\/li>\n- floor division (
\/\/<\/code>);<\/li>\n- modulo (
%<\/code>);<\/li>\n- (
divmod<\/code>); and<\/li>\n- exponentiation (
pow<\/code>).<\/li>\n<\/ul><\/li>\n- how to implement binary arithmetic operators for bitwise operations:\n
- left shift (
<<<\/code>);<\/li>\n- right shift (
>><\/code>);<\/li>\n- bitwise and (
&<\/code>);<\/li>\n- bitwise or (
|<\/code>);<\/li>\n- bitwise xor (
^<\/code>);<\/li>\n<\/ul><\/li>\n- what the reflected arithmetic dunder methods are;<\/li>\n
- what the augmented arithmetic dunder methods are;<\/li>\n
- what
NotImplemented<\/code> is and how it differs fromNotImplementedError<\/code>; and<\/li>\n- how Python determines which arithmetic dunder method to call.<\/li>\n<\/ul>
We will start by explaining how dunder methods work and we will give a couple of examples by implementing the unary operators.\nThen, we introduce the mechanics behind the binary arithmetic operators, basing the examples on the binary operator
+<\/code>.<\/p>\nAfter we introduce all the concepts and mechanics that Python uses to handle binary arithmetic operators, we will provide an example of a class that implements all the arithmetic dunder methods that have been mentioned above.<\/p>\n\n
\nYou can get all the Pydon'ts as a free ebook with over +400 pages and hundreds of tips<\/a>. Download the ebook “Pydon'ts – write elegant Python code” here<\/a>.<\/p>\n<\/div>\n\n
The example we will be using<\/a><\/h2>\n
The example we will be using throughout this article will be that of a
Vector<\/code>.\nAVector<\/code> will be a class for geometrical vectors, like vectors in 2D, or 3D, and it will provide operations to deal with vectors.\nFor example, by the end of this article, you will have an implementation ofVector<\/code> that lets you do things like these:<\/p>\n>>> from vector import Vector\n>>> v = Vector(3, 2)\n>>> v + Vector(4, 10)\nVector(7, 12)\n>>> 3 * v\n(9, 6)\n>>> -v\n(-3, -2)<\/code><\/pre>\nLet us go ahead and start!<\/p>\n
This is the starting vector for our class
Vector<\/code>:<\/p>\n# vector.py\nclass Vector:\n def __init__(self, *coordinates):\n self.coordinates = coordinates\n\n def __repr__(self):\n return f\"Vector{self.coordinates}\"\n\nif __name__ == \"__main__\":\n print(Vector(3, 2))<\/code><\/pre>\nRunning this code will show this output:<\/p>\n
Vector(3, 2)<\/code><\/pre>\nThis starting vector also shows two dunder methods that we are using right off the bat:<\/p>\n
- we use the dunder method
__init__<\/code><\/a> to initialise ourVector<\/code> instance; and<\/li>\n- we use the dunder method
__repr__<\/code><\/a> to provide a string representation of ourVector<\/code> objects.<\/li>\n<\/ol>This shows that dunder methods are not<\/strong> magical.\nThey look funny because of the...<\/p>","summary":"This article shows you how to overload the arithmetic operators in Python with dunder methods.","date_modified":"2025-11-14T12:33:53+01:00","tags":["arithmetic","dunder methods","programming","python"],"image":"\/user\/pages\/02.blog\/02.pydonts\/overloading-arithmetic-operators-with-dunder-methods\/thumbnail.webp"},{"title":"TIL #007 \u2013 math.nextafter","date_published":"2021-09-29T00:00:00+02:00","id":"https:\/\/mathspp.com\/blog\/til\/007","url":"https:\/\/mathspp.com\/blog\/til\/007","content_html":"
Today I learned about the
math.nextafter<\/code> method.<\/p>\n\n - we use the dunder method
- right shift (
- subtraction (
- (
- negation (
- how to implement unary arithmetic operators:\n
- \\(8!! = 8 \\times 6 \\times 4 \\times 2\\)<\/span><\/li>\n<\/ul>\n