Modular arithmetic: Difference between revisions

Revision as of 15:33, 16 March 2025

Revision as of 18:08, 16 March 2025

Line 5:

In [[mathematics]], '''modular arithmetic''' is a system of [[arithmetic]] for [[integer]]s, where numbers "wrap around" when reaching a certain value, called the '''modulus'''. The modern approach to modular arithmetic was developed by [[Carl Friedrich Gauss]] in his book ''[[Disquisitiones Arithmeticae]]'', published in 1801.

A familiar useexample of modular arithmetic is in the hour hand on a [[12-hour clock]],. in whichIf the day is divided into two 12-hour periods.hand Ifpoints the time isto 7:00 now, then 8 hours later it will bepoint to 3:00. Simple addition would result in {{nowrap|7 + 8 {{=}} 15}}, but 15:00 reads as 3:00 on the clock face. This is because clocksthe hour hand makes "wrapone around"rotation every 12 hours and the hour number starts againover atwhen zerothe whenhour ithand reaches 12passes12. We say that 15 is ''congruent'' to 3 modulo 12, written 15 ≡ 3 (mod 12), so that 7 + 8 ≡ 3 (mod 12). Similarly, 8:00 represents a period of 8 hours, and twice this would give 16:00, which reads as 4:00 on the clock face, written as 2 × 8 ≡ 4 (mod 12).

Similarly, if one starts at 12 and waits 8 hours, the hour hand will be at 8. If one instead waited twice as long, 16 hours, the hour hand would be on 4. This can be written as 2 × 8 ≡ 4 (mod 12). Note that after a wait of exactly 12 hours, the hour hand will always be right where it was before, so 12 acts the same as zero, thus 12 ≡ 0 (mod 12).

== Congruence ==