on to the primes

 What subtraction is to addition, division is to multiplication, kind of the reverse.  Also like subtraction it is non commutative. 8x7 = 7x8, but 8/7 does not equal 7/8. If I have 80 sheep and I want to know how many herds of 20 I can make, I can divide them 20 into 80 and get 4.  But now it is a two step problem, for multiplication I just have to multiply but for division I have to first multiply and then divide. 1x20, no.  2x20, no.  3x20, no.  4x20, ah there we are.

And in the majority of cases you do not get an integer, you get an arrested development.  Divide 4 into 3 and you get 3/4 which is really just a description of the operation, but there it is, you can't go any further.  You now have fractions, and really there is no way to do proper arithmetic without them, you have to admit them into the kingdom, and now you have the rational numbers, which includes all the integers, and the negatives, and that odd fellow zero, and all those abominable fractions. 

It is my opinion, as a substitute teacher for grades k to 8, that at this point (third, fourth) grade, arithmetic becomes hard for the kids.  For all the other operations you can use those little cubes, line them up, toss them together, take some away, no problem.  But for division you have to use the pie on the blackboard, half a pie, a third of a pie, etc.  I always called them pizza pies to make them more fun for the kids, but they were not fooled wrong.

And then there is that awful long division.  To multiply 2,345,678 by 876,543 the main problem is keeping your lines straight and remembering what number to carry.  To do long division you have to do all that multiplication, and then you have to subtract big numbers with all the carrying and borrowing that that entails, and it's just a lot of fucking work.

And then there are the operations on fractions, before you can add or subtract them you have to multiply their denominators so that they are the same, and in that process you have to remember to multiply the numerators also. For multiplication it's not that bad you just multiply through, but division is excruciating.  You have to remember to turn one of them, I think the second one upside down and then multiply.  "Mr Schadt, why do I turn the second one upside down?"  Mr Schadt smiles indulgently and begins blah, blah, blah, and ends up sputtering, "Because I told you to, that's why."

And here is another thing.  Those fractions, those arrested developments, if you are dividing by two you get an integer every other time, if you divide by three you get an integer every third time, four, every fourth time, five, every fifth time.  In between are the fractions.  And there are certain numbers that unless you divide them by themselves always give you a fraction.  At first they are plentiful, 2,3,5,7,11,13, but by say 5,000 they get sparser 5,003, 5,009, 5,011, 5,021, and the farther you go the sparser they get, but they never completely fade away because there is an infinite number of them.  You would think there would be a pattern for them in something logical like a number system, but there is not.