Compound addition is a mathematical tool that is very rarely used these days, since the problem it solves is a problem that we don't frequently deal with. Compound addition enables you to accurately add numbers of irregular
bases together, although it works for regular
Allow me to begin with a regular base, and a familiar one at that: base ten. This will demonstrate how it works. For the record, pure base ten addition can be referred to as simple addition.
Suppose you have eighteen apples. Why apples? Because apples are always used in these kinds of examples now shut up. You have eighteen apples, your friend has twelve apples and some random passing stranger has fifteen apples. For whatever odd reason, you decide to pile all the apples together, possibly mugging the random stranger in the process. Having piled them up like this, you strange, strange little man, you wonder idly to yourself exactly how many apples you might happen to have in this great big pile of yours. Instead of counting though, you decide to make use of all that edgeumacashon you received in your formative years, and add them up.
So, you need to solve 18 + 12 + 15. You begin by putting together 8 + 2 + 5 = 15. You then divide 15 by 10 and get a quotient of 1 plus a remainder of 5, so you put down the remainder and carry the quotient into the tens column, making you then add 1 + 1 + 1 + 1. The answer is thus 45 apples.
This is what you do every time you add things up manually instead of using a calculator or computer to do it for you. What's with the dividing and remainder business? Well, we just so happen to have a system of numbers that when any two digit number is divided by 10, the first digit is the quotient and the remainder is ten. Makes things quite simple, wouldn't you say? That's why this is simple addition.
The thing is, this works because each column of digits is ten times bigger than the column to the right of it, forever in both directions. You can apply this same principle to bases other than ten, which I'll leave up to you as it's such a simple manner. Just replace "divide by ten" with "divide by <whatever the base number is>". The more interesting thing is when you want to add numbers that aren't regular like this... that have weird relationships between each of the columns.
What number systems am I talking about? You're sitting there right now thinking that only madmen would come up with such a system, and how could it possibly be useful? But then I'll go and point you to the imperial measurement system. Twelve inches make a foot, so you might think we can work in base 12, but then you need to remember that three feet make a yard. Oops.
So we've got some string. Two pieces of string. One piece of string is one yard, two feet, and seven inches long. The other piece of string is three yards, one foot, and eleven inches long. Lay the two end to end, and how long are the two pieces of string combined? If I told you to solve this problem yesterday, you'd probably use some multiplication to convert yards to feet, then feet to inches, add them up, then do some division at the end to separate it back out into yards, feet, and inches. But now, you can use compound addition and still tear your hair out because it's a bloody annoying process and we should all switch to metric but we Americans are too bloody stubborn aren't we? ARENT WE!!?!?!
Sorry, got a little carried away there. Let's do this.
1 yard, 2 feet, 7 inches
+ 3yards, 1 foot, 11inches
7 + 11 = 18. 18 divided by 12 is 1 remainder 6, so put down 6, carry the 1.
1(carried) + 2 + 1 = 4. 4 divided by 3 is 1 remainder 1, so put down 1, carry the 1.
1(carried) + 1 + 3 = 5.
The answer is 5 yards, 1 foot, 6 inches.
Don't believe me? Well, let's convert things out like you would have done it. 1 yard, 2 feet, 7 inches = 5 feet, 7 inches = 67 inches. 3 yards, 1 foot, 11 inches = 10 feet, 11 inches = 131 inches. 67 + 131 = 198. 198 / 12 = 16.5. 12*.5 = 6, so 16 feet, six inches. 16 / 3 = 5.3333..., 1/3 of 3 is 1, so we get 5 yards, 1 foot, 6 inches.
Which is more work? Personally, I did the compound addition in my head and on paper. On the other hand, dividing 198 by 12 was a little much for me, so I resorted to a calculator. Your mileage may vary.
Other uses for compound addition: ounces, pounds; cups, pints, gallons; pennies, shillings, pounds; units, dozens; seconds, minutes, hours, days, weeks, months, years; firkins, kilderkins, barrels, hogsheads, puncheons, butt. There are others, and most of them aren't needed. Generally, time measurements aren't going to change, so compound addition remains useful there, but elsewhere, metric is so much nicer.
Reference: Asimov on Numbers by Isaac Asimov, copyrights 1959, 1960, 1962, 1963, 1964, 1965, 1977
Asimov, in turn, referenced Pike's Arithmatic, also known as A New and Complete System of Arithmatic Composed for the Use of the Citizens of the United States, by Nicolas Pike, A.M. which was published in 1785, and then reprinted (Second Edition, Enlarged) in 1797. This book, Asimov reports, is over 500 pages long, "crammed full of small print and with no relief whatsoever in the way of illustrations or diagrams." It was apparently an elementary school textbook.