Part 2/6:
The Origin and Development
The modern approach to modular arithmetic was formalized by Carl Friedrich Gauss in his 1801 work Disquisitiones Arithmeticae. Gauss's exploration laid the groundwork for understanding how numbers behave under division and remainders, leading to systematic applications in various fields.
Real-World Example: The 12-Hour Clock
A familiar instance of modular arithmetic is the 12-hour clock system. The hours wrap around every 12 hours, so after 12 comes 1, 2, and so forth. For example, if it’s currently 7:00, adding 8 hours results in 15:00 in a 24-hour format, but on a 12-hour clock, this corresponds to:
[
15 \equiv 3 \pmod{12}
]
meaning 15 hours is congruent to 3 hours modulo 12. Thus, the clock "wraps around," and the time reads 3:00.