# Clock arithmetic, or modular--where is the zero? **URL:** \ **Category:** Factual Questions\ **Created:** [October 29, 2003, 4:59pm UTC](https://boards.straightdope.com/t/clock-arithmetic-or-modular-where-is-the-zero/210261 "2003-10-29T16:59:32Z")\ **Posts on this page:** 8\ **Page:** 1
**Author:** ![don\_willard](https://avatars.discourse-cdn.com/v4/letter/d/a88e57/32.png) [@don\_willard](https://boards.straightdope.com/u/don_willard)\ **Post date:** [October 29, 2003, 4:59pm UTC](https://boards.straightdope.com/t/clock-arithmetic-or-modular-where-is-the-zero/210261/1 "2003-10-29T16:59:32Z")
Let’s say you are doing mod 5, then you make a circle with 1,2,3,4, and 5 on it. Then you keep going around the circle in modular and come to find out that 7 and 12 are both mod 2, etc. Or 2 = 7 mod 5. But how come you don’t put in a zero so you do 0,1,2,3,4 around the clock? Then 2 = 7 mod 5 also but 0 and 5 are mod each other. In the first way of doing the mod clock without the zero, then this time it is 5 and 10 which are mod to each other! signed Confused ---
**Author:** ![CurtC](https://avatars.discourse-cdn.com/v4/letter/c/ce73a5/32.png) [@CurtC](https://boards.straightdope.com/u/CurtC)\ **Post date:** [October 29, 2003, 5:41pm UTC](https://boards.straightdope.com/t/clock-arithmetic-or-modular-where-is-the-zero/210261/2 "2003-10-29T17:41:08Z")
So… you want clocks to start with zero? Or what are you saying? ---
**Author:** ![spingears](https://avatars.discourse-cdn.com/v4/letter/s/ebca7d/32.png) [@spingears](https://boards.straightdope.com/u/spingears)\ **Post date:** [October 29, 2003, 5:43pm UTC](https://boards.straightdope.com/t/clock-arithmetic-or-modular-where-is-the-zero/210261/3 "2003-10-29T17:43:55Z")
Zero is 5, or 12, or wherever the cycle repeats. Look at a common 12 hour clock face, were is the zero? Look at a common 24 hour clock face, were is the zero? It is to be understood that zero is identical with 12 or 24. When did the 21st century begin? Just after midnight of December 31, 2000. The 20th century ended just before midnight, same date Don’t be fooled by the popular misconception the the year 2000 was the first century of the new millenium. As far as eras, i.e. BC and AD are numbered from a 0 (zero) marker. There was/is no year zero, it is a point on the time line. Did You miss out on “Time Lines and Cycles101”? ---
**Author:** ![Thudlow\_Boink](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/thudlow_boink/32/320_2.png) [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)\ **Post date:** [October 29, 2003, 5:45pm UTC](https://boards.straightdope.com/t/clock-arithmetic-or-modular-where-is-the-zero/210261/4 "2003-10-29T17:45:56Z")
'll try to directly answer what I think your question is: You can do {0, 1, 2, 3, 4} **or** you can do {1, 2, 3, 4, 5}. It works either way, because 0 and 5 are equivalent (or congruent) mod 5. {0, 1, 2, 3, 4} is probably more standard, because it’s the set of remainders you could get when dividing by 5 (and for other, more technical, reasons). ---
**Author:** ![Hari\_Seldon](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/hari_seldon/32/5173_2.png) [@Hari\_Seldon](https://boards.straightdope.com/u/Hari_Seldon)\ **Post date:** [October 29, 2003, 6:53pm UTC](https://boards.straightdope.com/t/clock-arithmetic-or-modular-where-is-the-zero/210261/5 "2003-10-29T18:53:05Z")
Or you could use -2,-1,0,1,2, the so-called absolutely least residues (which is ambiguous when the modulus is odd. You could also use 2,4,6,8,10, if you liked. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [October 29, 2003, 8:56pm UTC](https://boards.straightdope.com/t/clock-arithmetic-or-modular-where-is-the-zero/210261/6 "2003-10-29T20:56:04Z")
I’ve always seen the digits mod n as 0, …, n - 1. You can use anything you’d like, but that’s what most people seem to use. ---
**Author:** ![spingears](https://avatars.discourse-cdn.com/v4/letter/s/ebca7d/32.png) [@spingears](https://boards.straightdope.com/u/spingears)\ **Post date:** [October 30, 2003, 2:11pm UTC](https://boards.straightdope.com/t/clock-arithmetic-or-modular-where-is-the-zero/210261/7 "2003-10-30T14:11:29Z")
No longer any excuse to be “Confused.” Susan has taken the time to provide a thoroughly interesting web site on the subject. It is at URL: [http://www.math.csusb.edu/faculty/susan/modular/modular.html](http://www.math.csusb.edu/faculty/susan/modular/modular.html) It would appear that the OP, signed “confused” was a ‘put on’ to start a thread. by asking where the zero was to be located. Susan has made a somewhat confusing subject much clearer. You could say that what goes around comes around (the circle). ---
**Author:** ![GreyWanderer](https://avatars.discourse-cdn.com/v4/letter/g/7cd45c/32.png) [@GreyWanderer](https://boards.straightdope.com/u/GreyWanderer)\ **Post date:** [October 30, 2003, 2:43pm UTC](https://boards.straightdope.com/t/clock-arithmetic-or-modular-where-is-the-zero/210261/8 "2003-10-30T14:43:19Z")
In Norway we use the 24-hour clock. It goes from 00:00 (midnight) to 23:59 (one minute before midnight).