As children, we grew up learning how to count to 10. Why 10? Well this could be easily justified using the fact that we as humans have 10 fingers and any whole number up to 10 could be easily represented by a quick show of fingers. But what happened when we, as children with this new found power of counting objects up to 10, encountered a number greater than 10? Did we take off our shoes and start counting with our toes? That might have solved the issue for numbers greater than 10 but less than 20 (assuming you aren’t polydactylic) but in all reality, we needed a way to transcend the idea of representing objects with our fingers and/or toes and represent *any* number, no matter how large.

How did we do this? By using a Place Value system with a base 10. “Place Value” means that using a limited number of symbols, we can represent any number by using these symbols in a variety of combinations. The value of each symbol is based on the position or “place” where the symbol is located in the sequence of symbols.

For example, pick a base. The very first column or “place” should be used for all the symbols preceding the base until the base itself is reached. This is called the “units” place or informally as the “ones” place. This place is usually the farthest left or right place in a sequence. For instructional purposes and for familiarity, we will place the units place on the far right of the sequence.

Once the base has been reached, a second place will be added to the left indicating how many “bases” have been reached. When the amount of “bases reached” has reached the base amount, then a new place is added, again to the left, indicating how many bases of bases have been reached and so on and so forth.

For example, our familiar base 10 system works as follows:

_____ . . . _____ _____ _____ _____

A comma is added after every 3 digits for practical purposes to easily differentiate places in more complex combinations of numbers.

Now let’s go back 4,000 years ago to Sumer, a region of Mesopotamia, (modern-day Iraq). There, children learned to count, but using 60 as a base. Why 60? Did the children of that time have 60 fingers and/or toes? Probably not. The reason for using this number as a base has not been explicitly recorded but there are two convincing hypothesis on why a base 60 number system developed.

One idea, is that instead of using their whole finger to represent a single number, the Babylonians actually counted the 12 knuckles of the four fingers on one hand, using the thumb as a “pointer” and the five fingers on the other as multiples of twelve. So on one hand they had 1-12 and on the other they had how many 12’s, for a total of 12 x 5 = 60.

The other idea is that the number 60 has many divisors, 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60. In fact, 60 is the smallest number divisible by all integers from 1 to 6. This could prove very useful by being able to do division using more whole numbers and resulting in less fractions.

Base 60 is still used in many aspects of our lives today such as the 60 seconds in a minute and the 60 minutes in an hour. The circles is traditionally divided into which are also subdivided into 60 minutes of arc and further divided into 60 seconds of arc.

**Modular Arithmetic**

Now that we have a brief overview about bases, we can apply the power of Modular Arithmetic to change counting bases. Modular Arithmetic is a very handy and useful tool in mathematics invented by the famous Mathematician Carl Friedrich Gauss in 1801. We know that the number line is infinitely long but if we were to wrap this infinitely long number line around a circle of a given circumference *n*, we would notice that numbers would “line up” or over lap around the circle. This is the idea behind modular arithmetic. Keep in mind that we are dealing with integers here and not the real numbers.

So the number indicating how large the circle is *n*, is called the *modulus*. And we say that after one wrap around, any numbers that line up are *congruent*. In mathematical terms, when a number a, leaves the same remainder as a number *b*, we say *a* and *b* are ** congruent** written

a ≡ b mod n

The “*mod n*” part is just notation letting us know that we are in mod *n* and is not actually part of the equation, per se. However, when the context is understood, it should be OK to omit writing this every time.

In general, any modulo *n* has *n* residue classes, one for each integer from 0 to *n-1*.

Let’s use the timer on your microwave as an example of a base. So we will have *n *residue classes from the integers 0 to 59.

0, 1, 2, 3, … 56, 57, 58 59

We call this **modulo 60** or **mod 60** for short. When we add 1 to 59, we return to 0. This is true for any modulus, even our own familiar base 10 (when we add 1 to 9, we return to 0) or even every day objects like traffic lights (Red, Green, Yellow, Red, …). The integers from 0 to 59 in our base 60 example are called *Residue Classes*.

Now for a quick example, when I was in the military, we would tell time using the 24-hour clock. This is different than the usual 12 hour clock where all 24 hours are represented twice and distinguished using A.M. or P.M.

So when I would get asked what time I would be ready to get picked on Friday for the weekend, I would reply 1600. Of course this did not make sense to most people because the face of a clock only has the numbers 1-12 listed on it. How could I explain correctly to them what time to pick me up so as to maximize our time together on the sunny beaches of San Diego? Using modular arithmetic of course!

Numbers are said to be *congruent* if their *difference* is divisible by the modulus. Or stated more succinctly, *a* is congruent to *b * if *a-b * is divisible by *n* shown algebraically

a ≡ b mod n if a-b / kn for some k

This basically means that the difference must be divisible by the base.

In our example, let’s show that 1600 is congruent to 4:00. For lingo purposes, just think of the colon as “hundred hours” to be in step with 1600. 1600 – 400 is 1200, a multiple of 12. Written

1600 ≡ 400 mod 1200

1600-400 /1200

“So 4:00 P.M. civilian. Don’t be late.”

Another cool example of things you can do with modular arithmetic is calculate the last digit or remainder of a huge number like . Try doing that by hand! Here is how we would do it mod 10.

19^{19} ≡ 9^{19}; (because 19 is congruent to 9 mod 10)

(9^{2})^{9}*9 ≡ (81)^{9}*9 ;

(1)^{9}*9 ≡ 9; (because 81 is congruent to 1 mod 10)

References:

Use of base 60 using hands

https://www.youtube.com/watch?v=_XBJBG3vKS8&list=PLMpRfUAf1yQGhrBCe6moyrW49mJlNgf0n

Base 60 as a base

http://www.storyofmathematics.com/sumerian.html

Sub-divisions of angles into minutes and seconds

http://en.wikipedia.org/wiki/Minute_of_arc

Modular Arithmetic

http://en.wikipedia.org/wiki/Modular_arithmetic

https://brilliant.org/discussions/thread/modular-arithmetic-2/