How do you do division in modular arithmetic

#How do you do division in modular arithmetic how to#
#How do you do division in modular arithmetic mod#

Now for a quick example, when I was in the military, we would tell time using the 24-hour clock. The integers from 0 to 59 in our base 60 example are called Residue Classes. 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, вЂ¦).

#How do you do division in modular arithmetic mod#

We call this modulo 60 or mod 60 for short. So we will have n residue classes from the integers 0 to 59. LetвЂ™s use the timer on your microwave as an example of a base.

In general, any modulo n has n residue classes, one for each integer from 0 to n-1. However, when the context is understood, it should be OK to omit writing this every time. The вЂњ mod nвЂќ part is just notation letting us know that we are in mod n and is not actually part of the equation, per se. In mathematical terms, when a number a, leaves the same remainder as a number b, we say a and b are congruent written And we say that after one wrap around, any numbers that line up are congruent.

So the number indicating how large the circle is n, is called the modulus. Keep in mind that we are dealing with integers here and not the real numbers. This is the idea behind modular arithmetic. 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 вЂњline upвЂќ or over lap around the circle. Modular Arithmetic is a very handy and useful tool in mathematics invented by the famous Mathematician Carl Friedrich Gauss in 1801. Now that we have a brief overview about bases, we can apply the power of Modular Arithmetic to change counting bases. The circles is traditionally divided into which are also subdivided into 60 minutes of arc and further divided into 60 seconds of arc. This could prove very useful by being able to do division using more whole numbers and resulting in less fractions.Д«ase 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. In fact, 60 is the smallest number divisible by all integers from 1 to 6. So on one hand they had 1-12 and on the other they had how many 12вЂ™s, for a total of 12 x 5 = 60. 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 вЂњpointerвЂќ and the five fingers on the other as multiples of twelve. 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. Why 60? Did the children of that time have 60 fingers and/or toes? Probably not. There, children learned to count, but using 60 as a base. Now letвЂ™s go back 4,000 years ago to Sumer, a region of Mesopotamia, (modern-day Iraq). When the amount of вЂњbases reachedвЂќ 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.ДЇor example, our familiar base 10 system works as follows:ДЄ comma is added after every 3 digits for practical purposes to easily differentiate places in more complex combinations of numbers. Once the base has been reached, a second place will be added to the left indicating how many вЂњbasesвЂќ have been reached. For instructional purposes and for familiarity, we will place the units place on the far right of the sequence. This place is usually the farthest left or right place in a sequence. This is called the вЂњunitsвЂќ place or informally as the вЂњonesвЂќ place. The very first column or вЂњplaceвЂќ should be used for all the symbols preceding the base until the base itself is reached. The value of each symbol is based on the position or вЂњplaceвЂќ where the symbol is located in the sequence of symbols.ДЇor example, pick a base. вЂњPlace ValueвЂќ means that using a limited number of symbols, we can represent any number by using these symbols in a variety of combinations. How did we do this? By using a Place Value system with a base 10.

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 arenвЂ™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. 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.

#How do you do division in modular arithmetic how to#

As children, we grew up learning how to count to 10.