Depth 1 · 37 s
Round the clock
On a clock, 9 + 5 is 2. Numbers that go round in a circle throw away the full turns and keep only the remainder - which is how you find the weekday 100 days from now.
About this video
It is 9 o'clock and a journey takes 5 hours. The clock won't show 14: an ordinary clock face goes up to 12 and then starts again, so it shows 2. On a clock, 9 + 5 = 2. This short animation shows why that isn't a mistake but a different kind of adding, called clock arithmetic (mathematicians call it modular arithmetic).
A clock is a number line bent into a circle. Five hops from 9 go 10, 11, 12, 1, 2. For 25 hours you don't need 25 hops: 24 hours are two full turns, and a full turn brings the hand back to exactly where it started. Only the 1 hour left over matters, so 25 hours after 9 o'clock the clock shows 10. The full turns are thrown away and the remainder is all that counts, just as in 25 ÷ 12 = 2 remainder 1.
A week is a circle too, of 7 days. What day is it 100 days after a Monday? 98 days are exactly 14 weeks, and 100 is 2 days more, so it is the same as 2 days after Monday: Wednesday. Whenever things come round again, only the remainder matters. This video goes with the Clock Arithmetic lesson of Number Theory Level 1 on Knaph.
This video shows depth 1 of Clock arithmetic.
Learn Clock arithmeticTranscript
- 0:00Nine o'clock, and a five-hour journey. What time do you arrive?
- 0:05Five hops round the clock. Ten, eleven, twelve, one, two. Two o'clock.
- 0:11Twenty-five hours later? Two full turns change nothing.
- 0:17Only the one hour left over counts.
- 0:20A week is a circle of seven days.
- 0:24A hundred days is fourteen weeks and two days. Monday becomes Wednesday.
- 0:31On a circle, only the remainder counts.