Creating Repeated Sequences With The Modulus (MOD) Operator
Posted August 10, 2011 at 9:49 AM by Ben Nadel
This almost doesn't warrant its own post; but, it seems to be one of those small factoids that I can't keep in my head without writing it down. And so, a quick post to demonstrate creating repeated sequences with the modulus operator (MOD or %). To be fair, the modulus operator already creates repeated sequences; but, those sequences end with a zero. This is perfect for when you only care about one number in a particular sequence; but, when you want to deal with a sequence as a set of numbers, typically, you want your sequence to start at 1 and proceed to N.
In order to end your sequence on "N", you have to add some math to your modulus calculation. But, before we look at that, let's take a look at a simple modulus loop:
- <!---
- Loop to 15 using a modulus of 3 to get a result
- at each index.
- --->
- <cfloop
- index="i"
- from="1"
- to="15"
- step="1">
-
- #i# % 3 = #(i % 3)#<br />
-
- </cfloop>
Here, we are looping from 1 to 15 and outputting the mod-3 for each index. When we run this code, we get the following output:
1 % 3 = 1
2 % 3 = 2
3 % 3 = 0
4 % 3 = 1
5 % 3 = 2
6 % 3 = 0
7 % 3 = 1
8 % 3 = 2
9 % 3 = 0
10 % 3 = 1
11 % 3 = 2
12 % 3 = 0
13 % 3 = 1
14 % 3 = 2
15 % 3 = 0
As you can see, this creates the repeated sequence, {1,2,0}. Like I said, standard modulus sequences end with zero. This is great if you want to determine which index you are at (1 == first, 0 == last); but, if you wanted to get the index within the repeated sequence, your logic gets a bit more tricky.
To create a repeated sequence that ends in N rather than zero, we have to add some more math to the mix:
- <!---
- Now, loop to 15 using a modulus of 3. However, this time,
- rather than repeating 1,2,0, we want to repeat 1,2,3. To do
- this, we have to use a slightly different formula:
-
- ((n - 1) % 3) + 1
- --->
- <cfloop
- index="i"
- from="1"
- to="15"
- step="1">
-
- ((#i# - 1) % 3) + 1 = #(((i - 1) % 3) + 1)#<br />
-
- </cfloop>
This time, we are subtracting one from the index and then adding one to the result. When we run this code, we get the following output:
((1 - 1) % 3) + 1 = 1
((2 - 1) % 3) + 1 = 2
((3 - 1) % 3) + 1 = 3
((4 - 1) % 3) + 1 = 1
((5 - 1) % 3) + 1 = 2
((6 - 1) % 3) + 1 = 3
((7 - 1) % 3) + 1 = 1
((8 - 1) % 3) + 1 = 2
((9 - 1) % 3) + 1 = 3
((10 - 1) % 3) + 1 = 1
((11 - 1) % 3) + 1 = 2
((12 - 1) % 3) + 1 = 3
((13 - 1) % 3) + 1 = 1
((14 - 1) % 3) + 1 = 2
((15 - 1) % 3) + 1 = 3
As you can see, this time, we created the repeated sequence, {1,2,3}. With the extra math, we were able to end on N (3 in our case) rather than zero.
And, of course, we could abstract this concept out into its own function:
- <cffunction
- name="modSequence"
- access="public"
- returntype="numeric"
- output="false"
- hint="I return the modulus result of the given value into the given sequence such that the resultant sequence goes from 1 to N (rather than zero).">
-
- <!--- Define arguments. --->
- <cfargument
- name="index"
- type="numeric"
- required="true"
- hint="I am the index that we will fit into the given sequence."
- />
-
- <cfargument
- name="sequence"
- type="numeric"
- required="true"
- hint="I am sequence (max value) into which we will be fitting the given index."
- />
-
- <cfreturn (((arguments.index - 1) % arguments.sequence) + 1) />
- </cffunction>
-
-
- <!--- loop to 15, repeating 1,2,3. --->
- <cfloop
- index="i"
- from="1"
- to="15"
- step="1">
-
- modSequence( #i#, 3 ) = #modSequence( i, 3 )#<br />
-
- </cfloop>
Here, we're simply encapsulating the details behind a function invocation. When we run the above code we get the following page output:
modSequence( 1, 3 ) = 1
modSequence( 2, 3 ) = 2
modSequence( 3, 3 ) = 3
modSequence( 4, 3 ) = 1
modSequence( 5, 3 ) = 2
modSequence( 6, 3 ) = 3
modSequence( 7, 3 ) = 1
modSequence( 8, 3 ) = 2
modSequence( 9, 3 ) = 3
modSequence( 10, 3 ) = 1
modSequence( 11, 3 ) = 2
modSequence( 12, 3 ) = 3
modSequence( 13, 3 ) = 1
modSequence( 14, 3 ) = 2
modSequence( 15, 3 ) = 3
The modulus operator is pretty awesome, no doubt. Hopefully, I'll now be able to remember how to use it to create a more "standard" repeated sequence. And if I can't, at least I know where to look.
Reader Comments
When I first started programming, I thought the MOD operator was the coolest, most awesome and interesting thing. I still love it to this day. Thanks for writing about it. You get extra brownie points, as well, for adding extra math. :-)
@Anna,
Yeah, the mod operator is totally awesome :) It's one of my favorite operators (which may very well be the geekiest thing I say all day).
@Ben,
haha...you can be geeky, that's ok. I am geeky all day, every day, probably, but it's gotten so bad with me, that I don't even really notice it anymore, and don't recognize it when I am.
@Ben,
If you cfloop from -6 to 6, instead of from 1 to 15, you'll see that % is actually a remainder operator, not a true modulo. In a true modulo, -5 % 3 would 1, not -2.
It can make a difference if you're trying to generate a random number within a range. Say you want to generate a random number in the range of 0 to 59, to represent the number of seconds to wait before doing something. Ordinarily, % 60 does that quite nicely. But if the first operand accidentally goes negative somehow, you instead get a number in the range of -59 to 0.
Just something to watch out for.
This might be a bit more visual. Switching to code for its monofont.
- Remainder versus true modulo:
- i i \ 3 i % 3 true i mod 3
- -6 -2 0 0
- -5 -1 -2 1
- -4 -1 -1 2
- -3 -1 0 0
- -2 0 -2 1
- -1 0 -1 2
- 0 0 0 0
- 1 0 1 1
- 2 0 2 2
- 3 1 0 0
- 4 1 1 1
- 5 1 2 2
- 6 2 0 0
You can take integer quotient times divisor and add in the remainder to get the dividend, on either side of 0.
You can't do that with true modulo.
Aw, shucks. I wish you would allow pre in addition to code.
@Ben, I forgot to mention that I like the other operators, too. It's just that by the time I got to the mod operator, I had used the others so much, I had just grown kind of tired of them. They were kind of like old news at that point. But I still like them. Just think the mod is super-cool. (and can be helpful for a lot of solutions to different problems you can't really solve easily in other ways).
We use MOD to place things in row, three columns wide. Thus, if there are eight items, the output ends up looking like:
1 2 3
4 5 6
7 8
is that what prompted this post, Ben?
@WebManWalking,
It's funny you mention that - I almost never think about numbers in the negative. I know that sounds odd; but, I can't even remember the last time that I really had to work with negative numbers. ... I am sure that is a ridiculous thought - I am sure I use them all the time :)
@Randall,
That's the kind of situation where I really started to use MOD for the first time on a regular basis. When I was using Table tags to lay things out, especially, I remember I used to use the MOD operator to determine where I should add the closing / opening TR tags.
