In the comments of the past few weeks, and due to some of the questions raised recently, the notion of congruence has come up frequently.
In number theory (there is also geometric congruence) two whole numbers are called congruent if they result in the same remainder when divided by a third, the so-called module. Thus 7 and 19 are congruent with respect to 4, since both, when divided by 4, give a remainder of 3.
Some congruences are obvious; for example, all odd numbers are congruent with respect to 2, since they all give 1 as the remainder when divided by 2 (in this sense, what can we say about numbers ending in 1?).
More information
The congruence relation is expressed by three parallel lines and the modulus in brackets:
a ≡ b (mod m)
means that a and b are congruent with respect to m.
Congruence can also be defined as the relationship between two integers whose difference is divisible by one third. If a and b are congruent with respect to m, they give the same remainder r when divided by m, which means that:
a = pm + r
b = qm + r
where p and q are integers and therefore:
a – b = (p – q)m
then a – b is divisible by m.
Congruence is the basis of modular arithmetic, which Gauss introduced in the early 19th century with his book Disquisitiones Arithmeticae. And modular arithmetic is also called “clock arithmetic” because clocks illustrate the equivalence relationship of hours in terms of module 12 very graphically: So 7 and 19 hours are represented the same way on traditional clocks: with the big hand at 12 and that minor at 7.
Johann Carl Friedrich Gauss (1777–1855), German mathematician, astronomer and physicist, in a portrait by Christian Albrecht Jensen.
problematic clocks
One cannot talk about clock arithmetic without considering the numerous problems and mysteries (some familiar, some less so, some simple, and some less so) in which clocks play the leading role. They represent a whole section of ingenuity problems, which in turn can be broken down into three subsections: pin clocks, hourglasses, and digital clocks. Let’s look at some of the first kind:
A clock that strikes the hour takes 6 seconds to strike the 6. How long does it take for 12 to strike?
Near my house there are two clocks that strike the hours at different speeds: one strikes three times at the same time, the other twice. They are synchronized and start ringing at the same time. At what time does the slow clock strike two more chimes after the fast clock has stopped? (Based on true events, like the last one).
At 12 o’clock, the three hands of the watch – the hour hand, the minute hand and the second hand – are exactly aligned (representing the arms of the sun, as Ramón Gómez de la Serna would say). When will the three see each other again?
And as a highlight, a well-known classic, but a mandatory mention in this context. Classic and historical because the anecdote is genuine:
One afternoon Kant saw that the clock in his house had stopped. Shortly thereafter, he made his way to a friend’s house, where he noticed the time on a clock on the wall. After a long chat with his friend, Kant returned home the same way, walking as always with the firm, even gait that had not changed in twenty years. He had no idea how long it had taken him to get back, since his friend had recently moved and Kant hadn’t set the time for the trip yet. But as soon as he got home, he set the clock on time. How was it?
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