Division with remainder: the basis of modulo arithmetic
It has been a very long time since I was in grade school, but back in that ancient era, our first introduction to division produced a quotient and a remainder.
Example: divide 6 into 15.
The answer would be a quotient of 2 and a remainder of 3. Soon enough, we would learn about fractions and decimals, and remainders would become a thing of the past.
But three examples of remainders linger on into what we refer to as "the real world".
Addition modulo 2: even and odd
An even number is exactly divisible by 2, which means the remainder upon division is 0.
An odd number has a remainder of 1 upon division by 2.
So all integers are split into two sets, even and odd, and they have no numbers in common. We call such sets disjoint, and if the union of disjoint sets is the entire set we were first discussing, in this case the integers Z, the disjoint sets are called a partition of the original set.
Here are the rules of modulo two addition.
even + even = even.
We will write 0+0 = 0(mod 2)
even + odd = odd + even = odd.
We will write 0+1 = 1+0 = 1(mod 2)
And the final rule, odd + odd = even.
1+1 = 0(mod 2)
This is addition, so as usual, 0 is the identity. since 1+1 = 0, 1 is its own inverse.
({0,1}, +(mod 2)) is a group of order 2.
While our first example is using 2 as our divisor, this will work for any integer n>1.
On the clock: mod 12 (hours) and mod 60 (minutes and seconds)
On the first season of the HBO series True Detective, the character played by Matthew McConaughey says "Time is a flat circle", and this is taken as a brilliant and mystical insight. But consider that, for centuries, people measured time on the face of a clock, and a clock is a flat circle.
I am writing this at 5:33 pm. What will the hour be 9 hours from now? 5+9 = 14, but because hours on a clock like the one shown are represented as numbers modulo 12, and 14(mod 12)=2.
In most modulo n systems, n is represented as 0, but there is no rule saying n can't be represent by n. For hours, the numbers used are 1 through 12 instead of 0 through 11. Here the number 12 acts as the additive identity.
Now it is 5:36. What time will it be on the minute hand 72 minutes from now? 36+72 = 108, but minutes are modulo 60, so 108(mod 60) = 48. The second hand also works on the modulo 60 system.
Yet again, 0 is the additive identity, and we can think of the inverse of k as being -k. Some examples.
-5(mod 12) = 12-5(mod 12) = 7
-43(mod 60) = 60-43(mod 60) = 17
-6(mod 12) = 12-6(mod 12) = 6
Every group that is constructed from modulo arithmetic is a finite abelian group. The word abelian comes from the last name of Niels Abel, one of two mathematicians credited with the 19th Century algebraic breakthrough that proved the general quintic equation cannot be solved just using the coefficients of the terms. (More about this tomorrow.) Abelian means that in the group a+b = b+a for all elements a and b in G.
We haven't seen a non-abelian group yet, where there are elements a and b such that a*b ≠ b*a.
Commentary
It is an honor in mathematics to have some object, measurement unit, theorem or principle named after you. Theorem names, like the Pythagorean Theorem or Green's Theorem always capitalize the person's name. But if it is a measurement unit, it is a special honor if the name is not capitalized, like a newton, a hertz or an ohm. The rule is not hard and fast. Fahrenheit and Celsius are usually capitalized.
In group theory, abelian is not capitalized, but Galois, as in Galois theory and Galois group, is capitalized. Go figure.
Tomorrow, I will take a side journey into the stories of Neils Abel and Evariste Galois, two 19th Century mathematicians who died young and independently found the proof of the insolubility of the quintic, a problem great mathematicians struggled with for centuries.
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