How to Add, Subtract, Multiply, and Divide Natural Numbers

History

The name given by mathematicians to the numbers we count with is the natural numbers. For a long long time in history these were simply the numbers. They are probably the only numbers we really have a strong intuitive feel for. But this is okay, because out of these familiar numbers all the other numbers that mathematicians talk about were created.

The details of how arithmetic was invented are not known, but it is clear that people have not always done it. The rudiments of addition and subtraction probably go back even to before people had the kind of idea that we have nowadays of number. In early times the idea of number was much more concrete than it is nowadays. Instead of giving a name to how many there was of something, they used something smaller and cheaper to model it like a pile of pebbles or sticks or tally marks.

Archaeologist have found a number of clay tokens that they think were used by the ancient Babylonians for keeping track of how many or various things that they had. Some of these are nearly 10,000 years old. At around 8000 BCE these people started domesticating plants and animals and started being able to produce more that they could eat at once, so they felt the need to keep track of how much they had of various things. It is not known if they even had words for numbers at this time, but that needn't have stopped them. Clay was plentiful and easy to use, so they formed it into various shapes to represent the things that they had. One small ball of clay might represent a certain measure of grain and different shaped piece of clay might represent a lamb.

Using these sorts of methods of accounting one might easily imagine that the idea of adding and subtracting in the physical sense of adding a certain pile of tokens to another piles of pebbles or tokens or taking away some tokens as a way of modeling the similar action with the bigger more valuable items that they represented. Multiplication and Division are more complicated, so they were probably invented much later. Even in early civilizations they didn't view them quite the way we view them nowadays. In ancient Egypt, for example, the basic operations were addition, subtraction, and doubling, and the various problems that we would solve by multiplying were probably viewed as separate individual problems to be solved by the use of these operations.

Throughout history various notations for numbers have been used. The notation system we finally settled on came originally from India. Roman Numerals were used in Europe as late as the early Renaissance, but the dark ages were really only the dark ages for Europe. Even before the fall of Rome the Indians were making good use of this superior system of number notation that we use today and call the Hindu-Arabic system. The reason Arabic is part of the name is that it was introduced to Europe by the Arabs, the people who really were having a good time in the 'dark' ages.

One of the most important of these Arabs for elementary mathematics is Al-Khwarizmi.

He wrote an arithmetic book where he explained the Indian number notation system and the methods for doing arithmetic with it. This book became so popular that systematic methods for doing things got named after him and became called algorithms.

Another important person for arithmetic is a man living in Italy in the 13th century named Leonardo of Pisa, known also as Fibonacci

He is probably the European most responsible for the fact that we use Hindu-Arabic numerals and compute the way the Indians and Arabs did instead of on a counting board. His father was a diplomat and he traveled widely with his father and was educated in North Africa and learned about the Hindu-Arabic number notation and computation methods and became convinced that they were greatly superior to the systems used in Europe. When he returned to Italy he wrote a book called Liber Abacci explaining these numbers and how to do arithmetic with them and the book was very influential.

The advantages of this system for addition and subtraction are not a lot, but it is much better for multiplication and the big advantage was that you got a written record of your work so that it could be later checked. From the 13th c. onward the commercial world was getting increasingly complicated, so such things were increasingly important. Because of this Leonardo's book influenced very much the merchants of the time as well as the mathematicians. So a number of people were able to make their living teaching this new arithmetic and writing books about it and new innovations developed so that eventually the methods that you were taught in school evolved.

My Philosophy about Paper and Pencil Arithmetic

In recent years there has been a certain amount of controversy about the teaching of the standard algorithms for paper and pencil arithmetic, in view of the fact that nowadays the far more efficient way to do any of these operations is to use calculators or computers. Viewpoints seem to range from wanting paper and pencil arithmetic banned to wanting calculators banned. I think my view is fairly moderate on the matter. I think that with the existence of calculators and computers the importance of doing paper and pencil arithmetic for practical reasons is indeed considerably less, but I think that some experience with it is a good thing for a number of reasons. I think you learn some things about how numbers work by learning the algorithms, particularly if you learn why they work. I think also that you learn something important simply by learning how to follow an algorithm in the first place. It doesn't really matter whether it is the standard ones that you learn, and the standard ones probably aren't even the most efficient, but they have become to some extent part of our culture, so why not? Others will be familiar with them if for some reason you need to justify a calculation to someone when there is no calculator around. Like in Fibonacci's time, the existence of a written record can still in some situations be an advantage that paper and pencil arithmetic has. And even if they become totally outdated, at least they can be learned for historical interest preferably along with some other methods that have been used in the past as well.

As for the understanding of numbers aspect to it, this can be somewhat of a weak argument, because too often people learn the basic operations without the slightest idea of the reasons for them. This is indeed, I believe, a big weakness in the Hindu-Arabic notation and computational techniques that its advantages of conciseness and efficiency also allow us to lose sight of the meanings. For this reason I think it is also good to learn some other less efficient, but more concrete earlier methods, so I have included here some links to articles that I have written about some. Since these are methods that have actually been used by people, as well as learning something about numbers you can also learn something about how mathematics got to where it is today. There are certainly plenty others that could be explored as well, and making up your own, as many advocate is a good idea too. Ultimately my view is that it is a lot more important to learn what the arithmetic operations mean than how to do them, and mainly you are doing some computations by more primitive methods than a calculator to get some hands on experience, and how you get that experience doesn't really matter that much as long as you get enough of it and get it in a way that aids understanding rather than promoting rigidness.

Addition

The earliest addition was probably physical addition. You have a flock of sheep and a pile of pebbles to keep track of them with. Your brother dies and you get his sheep and to keep track of how many sheep you have now, you physically add his pile of pebbles to yours. Later when number notation was invented adding wasn't much changed. To find out out how the ancient Egyptians would have done addition see Egyptian Addition. To find out how a medieval merchant would have done addition with a counting board see How to Add and Subtract with a Counting Board.

Now, if you look at the examples in those articles you can see very easily that the Hindu-Arabic numbers that we use today represented quite a good short hand for writing numbers over earlier notations. In Hindu-Arabic Numerals instead of repetition of letters we use distinct symbols for the numbers from 1 to 9 (see Egyptian Addition and How to Add and Subtract with a Counting Board) and the position of these symbols takes the place of the letter to tell whether for example a 3 represents three units or three tens or three hundreds or three thousands, etc.. It is more like the counting board than it is like Roman Numerals in this way, except that the order is left to right instead of up to down and there are no spaces. To use this shorthand system for adding there is the disadvantage that without the physical counters instead of pushing counters together you need to memorize the answers to the addition problems involving the numbers between 1 and 9. But on the other hand once you have memorized them, it should be faster just to write down the memorized answer than to push the counters around. If you don't already know these addition facts one way to get familiar with them is to do any that you forget on the counting board. So lets say you forget momentarily what 8+7 is. Just represent 8 and 7 on the two sides of the counting board like this.

The 8 is represented on the left side and the 7 on the right side. Push them together like this.

Then regroup and get this,

which is 1 ten and 1 five, so represents 15, the 1 in the tens place representing 1 ten and the 5 in the ones place representing 5 ones. Our system has no fives place, so the five has to be represented in singles, but our system has other advantages to compensate for this.

I also think it is perfectly okay to use your fingers as counters, and count on your fingers if you forget one of your addition facts. If you do it often enough, you will probably get tired of it and memorize them.

Once you have the additions of numbers from 1 to 9 memorized, here is how to do the addition example I did in the How to Add and Subtract with a Counting Board article with Hindu-Arabic Numerals .

111

2737

3874

6611

7+4=11, write down the one on the right and carry the one on the left, 3+7+1=11, write down the one the right, carry the one on the left, 7+8+1=16, write down the six, carry the one, 2+3+1=6.

But what is this carrying thing about? To see that, we have to remember what our notation means. The additions being done here represent adding various numbers of ones, tens, hundreds, and thousands. When we add 7 and 4, we are adding 7 individual things to 4 individual things to get 11 individual things, like sticks, or sheep, or loaves of bread, and we get 11 of those same kind of things. But in our number notation system we are not allowed to have more than 9 in the ones column, so we turn the 11 ones into 1 ten and 1 one. You might think of this as tying up ten of the sticks into a bundle of 10 and having one left over, or cashing in 10 pennies for a dime. The addition of the 3 and the 7 represent adding 3 groups of ten to 7 groups of 10, but we also have the 1 group of ten that we got from 11, so all together we get 11 groups of 10. But the same thing is wrong with that, we have too many, so we can bundle up 10 of them as a group of 100. The addition of the 7 and 8 represents adding 7 hundreds and 8 hundreds, but we also have the one we bundled up from the tens, so altogether that makes 16 of them. Again too many, so we bundle up 10 of them to make 1 thousand, that we carry over to the thousands places and add it to the 2 and the 3 thousands in the thousands place to get 6 thousands.

What is going on here is really not too different from what we did on the counting board, except that there were a couple of shortcuts taken. To make it look more like what we did on the counting board, we could write it like this.

2 7 3 7

3 8 7 4

5 15 10 11

The answer line here is simply the results of the additions of the thousands, hundreds, tens, and ones, just like when we pushed all of the counters together on the counting board. Then to follow what we did on the counting board we would regroup by first taking ten out of the one's column and adding 1 to the tens column to get

5 15 11 1.

Then we would see that there are too many in the tens column too, so we could turn ten of them into one hundred and promote it to the hundreds column and this would give us

5 16 1 1,

Then finally the sixteen hundreds are too many, so ten of them need to be removed and one has to be added to the thousands' column to give the proper answer of

6611.

For writing, though, unlike with the counters this involves a lot of inefficient rewriting, so what we do is combine the adding and the regrouping together by carrying. When we add the 7 and the 4 instead of writing 11 and having to regroup later, we just regroup as we go, by immediately splitting up the eleven into 10+1 and putting the 1 in the ones column and the writing the 1 ten above the tens column to be adding to the other tens, and the same with the 10 in the tens column and the 15 in the hundreds column.

Even with this method adding, particularly adding a lot of numbers can get tedious, so inventive people over the years have come up with a number of inventions to help with it. Particularly for adding, there have been a great variety of them. Before electricity they were mechanical. Now we have electronic computers and calculators that we can use for far faster and reliable ways to add numbers. Computers and calculators allow you to add really by the most primitive of methods, without any groupings by 5's or 10's, but just by adding 1's, like the old piles of pebbles methods, so it makes all of the cleverness of later methods unnecessary. We can do this with computers and calculators simply because they work so fast.

As to the method of addition with a calculator, there is not really much to teach. Just key in the first number, then press the + button, then key in the second number, and then press the equal sign, and the calculator should show the answer. It often good to estimate when you use the calculator to make sure you didn't push a wrong button. There needn't be anything complicated and fancy about estimating, though. All you have to do is replace the numbers with ones that are close to them, but that you can add quickly in your head and do the addition with those numbers and compare the answer to what you go on your calculator to see if it is in the same ball park with your answer. For example in this problem, the first number is around 3000 and the second number is second number is around 4000, so you should expect an answer around 7000, a little less since both of the numbers were under. If you got an answer of 27 on your calculator, you would know that you must have pushed a wrong button.

Subtraction

Like with addition probably the earliest method to do subtraction was to sort of make a model of the situation and literally take away such things as pebbles in a pile. But again this gets difficult when you have to deal with large numbers, so you have to use some kind of grouping system where certain objects or symbols represent more than one thing. To find out out how the ancient Egyptians would have done subtraction see Egyptian Subtraction. To find out how a medieval merchant would have done subtraction with a counting board see How to Add and Subtract with a Counting Board.

Using Hindu-Arabic numerals there are a couple of different methods that accomplish the same thing. Again like with addition it is similar to the counting board except that instead of physically removing counters you memorize the subtractions for small numbers. If you have already memorized your additions, though, this shouldn't be too difficult, because you can just think about what you need to add to get the big number from the small number, or if you want if you forget you can count it on your fingers or on a counting board. When I was in school the way I learned to do the subtraction that I showed on the counting board was like this.

This is pretty much like the counting board except without the spaces between the lines. You can't subtract 8 from 1, so you break up one of the tens from the tens column into ones by putting the 1 next to the 8, and then there is only 1 tens, so you cross out the 2 and write a 1 above it. Then you can subtract the 8 from the 11 to get 3. But you can't subtract 3 tens from 1 ten, so you have to again borrow, this time from the hundreds column, rewriting 4 hundreds as 3 hundreds and the 10 tens that are added to the tens column to make 11 tens. Then the 11 is big enough to subtract 3 from and when you do this you get 8. But the 3 hundreds aren't enough to take away 6 hundreds from, so you have to break up one of the thousands. You turn 4 thousands into 3 thousands and 10 hundreds to make a total of 13 hundreds, which is now big enough to subtract the 7 hundreds from. Now finally you are left with 3 thousands to subtract 2 thousands from, and that leaves you with 1 thousand, and a total of 1783 for your answer.

There is another method that at least seems to be a slight improvement on this method that people of my mother's generation and also people from private schools and other countries learned. If nothing else it is better adapted to paper and pencil calculation, because it avoids the crossing out. The idea is that instead of decreasing the top numbers when you borrow, you increase the bottom numbers, and you can indicate that in a way that is consistent with carrying for adding, which is to put a 1 next to the number. In fact often when people are taught this method they are taught to think of the subtraction a reverse addition asking themselves instead of "What do I get when I take away 8 from 11?", "What can I add to 8 to get 11?". To do this same problem using this modified method would go like this.

You can't take 8 from 1, so you borrow 1 from the tens imagining it to have to be carried in an addition problem, you carry it to the 3 to make the 3 a 4 and then instead of subtracting 3 from 11, you subtract 4 from 12, asking yourself what can you add to 4 to get 12. Since it is 12 and not 2, you have to carry 1 to the 6 and make it a 7. Then since you can't add anything to 7 to get 4, you have to find something to add to 7 and get 14 and again carry a 1 to the 2. Then you ask yourself what you can add to 3 to get 4.

There are some who like to argue about which of these two methods is best. Personally I don't think it is very important. The fastest way to do subtraction is the way that we are going to talk about next, which is to use a calculator or a computer. I think it is good to try both ways and learn something about how numbers work from them and use whichever way you feel most comfortable with. For myself, I suspect that the second method is indeed more efficient, but I learned the first method and am used to it. If I had some need for performing a lot of subtractions quickly without a calculator I might be inclined to learn to get used to the second method, but that is not likely to happen, so I doubt if I ever will.

And speaking of calculators, again, like with addition, nowadays you also have the option of doing it on a calculator or a computer. To do that you just key in the first number, then press the - button, then key in the second number, and then press the equal sign. Again as with addition, it is always good to estimate when you use a calculator, and also a good idea to do the computation more than once if it is important.

Multiplication

Multiplication is a more complicated idea than addition and subtraction. Primitive people probably didn't need it, but when people started to do the more complicated things that civilizations do like trading and buying and selling things, then it started to become a useful operation to have. If you know what the price of one item is and you want to know the price of several of them, then you have a use for a repeated addition operation. But even in ancient Egypt multiplication wasn't one of the fundamental operations. The two fundamental operations were addition and doubling and problems that involved multiplication were solved by a combination of these two operations. To find out out how the ancient Egyptians would have done multiplication see Egyptian Multiplication . Multiplying on a counting board is similar to Egyptian Multiplication. Read How to Multiply with a Counting Board to see an example of this.

Meanwhile even as early as the fall of Rome, in India they were using a number notation quite similar to the one we use nowadays and in the so called dark ages they were multiplying in a very different way not much different than the way you have probably learned in school. It looks a bit different, but it is basically the same idea. The method for multiplying that was popular in India in those days is called gelosia multiplication. It actually has some advantages over the method you have probably learned in school. Read Gelosia Multiplication to learn about it. Like the method you learned in school the disadvantage this method also has is that you have to either memorize the multiplication of small numbers or consult a times table. More on that later, but for those who have trouble with remembering the times table, here's one to consult.

When the Arabs conquered India and took over their number notation they also learned to multiply by this gelosia system.

I'm sure that there are still people who learn multiplication in a slightly different form, but the method I learned in school goes like this. All the same multiplications get done as in the gelosia method except that the promoting of the tens place digits to the next place is done instead by carrying. So to do the second example I did by the gelosia method it would go like this.

3 4 2

2 3 2

2 2 1

7695

x543

1 1

23085

30780

38475

4178385

You multiply 3 times 5 and get 15 and write down the 5 and carry the 1. Then you multiply 3 times 9 and get 27 and add the carried 1 to it and to get 28 and you write down the 8 and carried the 2. Now multiply 3 times 6 to get 18 and add the carried 2 to get 20 and write down the 0 and carry the 2. Then 3 times 7 is 21 and add the carried 2 to get 23, and since there are no more columns left to carry the 2 to, you simply write down the 23. Then you go to the 4 in the tens place, and because it really represents multiplying by 40, you indent one for your answers and continue like you did with the 3. (I have written the carrying numbers for the multiplying by 4 on the next line up from those for the 3, and those for the 5 on the top line.) After taking care of the 4 you go to the 5 and again indent another place because this is really multiplying by 500. After doing all of the multiplying now you add up to get the final answer of 4,178,385.

Both the gelosia method and this method are a bit more mysterious than the repeated addition of the ancient Egyptians technique or the counting board and it is not really known how the Indians discovered it. How does this sort of multiplication work?

It works by the same sort of breaking the multiplier down into smaller numbers that are easier to multiply that these other methods use except that you break down both of the numbers and don't use any doubling. I will tell you what is going on here in the last example and other multiplications are similar. We are multiplying 543 times 7695 by multiplying 7695 by 500, 40, and 3, and adding up the results. But in order to multiply 7695 by 500 we are breaking it up to and multiplying 7000, 600, 90, and 5 by 500 and adding them up. This same goes for multiplying it by 40 and 3. Each of these multiplications now can be done easily provided we know how to multiply the small numbers, because for each 0 we just have to shift the place over, like pushing the counters up one line on the board. So what we have to do is to multiply all possible combinations of the digits and then arrange them in a way so that they have the proper place value and line them up and add them all together. Both the arrangement of the gelosia and the arrangement in the method above take care of the place values very nicely.

In the gelosia method the product of the two ones place numbers, the two fives in this example are positioned to the far right so that they come out in the ones column. The products of the ones digit of one number and the tens digit of the other number end up in the next place over, which is right because between them they only have one 0, so the product should represent a number of tens. Then next diagonal over has the product of the ones and the hundreds, the tens and the tens, and the hundreds and the ones, each appropriately making products that represent hundreds, so that they are also in the right place. In the next diagonal over you have thousands times ones, hundreds times tens, tens times hundreds, and thousands times ones, and it pattern continues this way until in the far left diagonal you have only hundreds times thousands, the largest place value. But if it were just that simple you wouldn't need the diagonal lines and it would just look like rows of boxes. But the diagonals allow the tens place numbers from the multiplications to get promoted to one place up, so that if you write your number from left to right the tens place is automatically on the next diagonal over and gets added with the ones from higher place value multiplication.

In the method I showed above we multiply 7695 first by 3, then by 40, and then by 500, and add them up. When we multiply by 40 we pretend like we are multiplying by 4, but by indenting one place, it makes it multiplying by 40. Then when we multiply by 500 we pretend to be multiplying by 5, but by indenting by two places it becomes multiplying by 500. Each of these three multiplications consist of 4 multiplication. To multiply the 3 by 7695, we multiply it first by 5, then by 90, then by 600, and then by 7000, and add those up, but that is taken care of simply by multiplying by each of the digits, and doing the carrying. The same holds with the 40 and the 500, we are multiplying them by the 5, the 90, the 600, and the 7000, and adding those up by multiplying them by the appropriate digits, and putting the answer in the appropriate place for it to count the way we want it to count.

The big disadvantage with any of the systems of multiplying using the Hindu-Arabic numerals and place values is the one I mentioned earlier, which is that you have to either carry around a times table for multiplication of the numbers from 1 through 9 or memorize them. So every year a new group of children gets initiated into the times table memorization game, and since it is considered an essential part of an education, it mostly gets done. It is only at most 45 multiplications to learn, so it can't be much worse than learning the alphabet, so most of us just suffer and get through it. But some people don't take to it as well as others and it becomes a major stumbling block for further mathematics learning, and they might even come to think of themselves as stupid, which is definitely not true. If anybody thinks they are stupid for not knowing their times table they might think about the fact that the ancient Greek mathematicians, who discovered so many clever things probably didn't know their time tables either. Also even those who do successfully memorize the times table are seldom perfect about it unless they really use it a lot. Most people confuse some of them occasionally leading to careless errors in calculations. So I'm going to now give a few tips for dealing with this matter.

Avoid the whole matter and do multiplication with an abacus or counting board or even the method of repeated doubled used on them.
Consult a times table for each product that you need. You probably won't be able to do that on a test, but there would be nothing to stop you from recreating one yourself on your scratch paper for use for the duration of your test, and you could make it up without too much difficulty by successive adding.
A less extreme version of 2. for those who know most of them is if you forget one of them, get it by adding to a previous one, so for example if you forget that 6 times 8 is 48 and you know that 6 times 7 is 42, you just need to add another 6 to 42 to get your answer.
Just Do It! I tend to take this attitude about things that need to be memorized in general and if it works for you it is a good one to use. The attitude is that you don't have to be good at memorizing, have a photographic memory or use any kind of fancy tricks in order to memorize things. You just have to spend as much time doing it as you are spending procrastinating it. One brute force method that works for me is flash cards. I make them up and them play games with them. The main one I use is the elimination game. You go through the cards and when you get one right you put it down and if you get it wrong you put it at the back. Then you keep going through until you run out of cards. Then maybe another time during the day you do the same thing and hope to get more of them right the first time around. If you get them all right the first time through try and see if you can do it twice in a row. If you can do that then you pretty much have got it.
Another possibly better way to memorize something is to learn it from use. One way to do this with the times table would be to use a table to look up your multiplication's, but turn it upside down and only consult it when you need to. Hopefully pretty soon you will get tired of turning it over and memorize them. If that doesn't work you can make the looking up a little harder by for example requiring yourself to work them out by adding or by one of the tricks I am going to mention below.
There are some tricks for some of the times table entries. The twos are pretty easy because most people know how to count by twos. The threes are harder, but there is a trick to partly check yourself. The sum of the digits should be 3, 6, or 9. For fives there is a pattern too. The even ones end in 0 and the odd ones end in 5 and for the even ones you can half it and multiply by 10 by adding a 0. For the odds you can subtract 1 and then half and add a 5. So example with 6 times 5 half of 6 is 3, so the answer is 30 and for 7 times 5, 7-1=6, half of 6 is 3, so the answer is 35. No special trick for 6 except that I was taught to remember that 6 times 8 is 48 rhymes. 7s and 8s are equally stupid, but you don't have to learn many of them because 9s are easy. 9s are easy because the sum of the digits is always 9 and they have a nice pattern to them where each successive one is one bigger in the tens place and one smaller in the ones place and the tens place is always one smaller than the multiplier. So if you just subtract one from the multiplier to get the tens place and subtract that from 9 to get the ones place you have your answer. Then once you have everything but the 7s and 8s all you have left is 7 times 7, 7 times 8, and 8 times 8.
When the Hindu-Arabic place value system was being introduced into Europe many of the arithmetic teachers even in those earlier days recognized that time table memorization might be a stumbling block, and one might well imagine that particularly with the adult beginners that they would have that it would be. So they came up with a number of tricks to make it easier. One I particularly like is a finger trick for multiplying numbers between 6 and 9 provided you know how to multiply numbers up through 5. You hold up the number of fingers for how much your first number is over 5 in one hand and how much your second number is over 5 in your other hand. So for example if you are multiplying 7 times 8 you hold up two fingers with your right hand and 3 fingers with your left hand. Then you count the number of finger up and that is your tens place and you multiply the numbers of fingers not up and that is your ones place. So in the example of 7 and 8, there are a total of 5 fingers up and 2 fingers down on your right hand and 3 fingers down on your left hand, so you multiply 2 times 3 to get 6 for your ones place and the answer is 56. The only problem with this method is that you have to be careful with the smaller ones. 6 times 6 gives you two for the tens place and 16 for the ones place, so you have to be able to figure out that twenty 16 is the same as 36 and 6 times 7 gives you thirty 12, which you have to realize is really 42.
Another trick they used was for multiplying a number smaller than 5 by a number bigger than 5 subtract the bigger number from 10 and multiply that by the smaller number and then subtract that from the smaller number with a 0 after it. So for example to compute 4 times 8 compute 4 times 2 is 8 and subtract 8 from 40 to get 32.
If you use the above two tricks you only need to memorize the times table up through 5 and the twos and fives are easy. If need be the 4s can be done by doubling twice, so that only leave 3 times 3 to memorize. And for 3 times 3 if you don't know it already all you have to remember is:

3 6 9 the goose drank wine.
The monkey chewed tobacco on the street car line.
The line broke.
The monkey got choked.
And they all went to heaven in a little green boat.

But even with all that help multiplication is hard no matter how you do it. So lots of attempts have been made to find methods and devices to make it easier. Some of them use quite complicated techniques that are beyond the scope of this article. If you go to a good museum of science you can see all kinds of fascinating calculating devices. It can be fascinating to try to figure out how particularly some of the analogue devices work. A slide rule, for example, is an analogue computing tool that using logarithms (something you may learn about in a more advanced course) to turn multiplication into addition and then does addition by adding lengths.

When I was in 8th grade 1967-68 I had a teacher who was very old and conservative and in a reaction against the trends of the so called new math he was determined to make sure that we learned the 'basics' of calculation and not mess around with the conceptual approach of the new math. When I was feeling rebellious against him I used to claim that we wouldn't have to be able to compute when we grew up, because by then everyone would have a computer. Of course I had no idea how true this would be, since in those days computers were big and expensive enough that even the school couldn't afford one, and if I had any kind of picture in my mind of what this future might be like it was something like a great big box at the wall that would be hooked up to a big computer elsewhere like a terminal, and even at that I didn't imagine it to be capable of much more than arithmetic. But the times indeed have changed! And now we don't have to worry anymore about coming up with clever ways to multiply, because calculators and computers are so fast that they don't even have to be clever to do it. So now we come to the best method for multiplying in our modern technological world and also the simplest. To multiply two numbers on the calculator just like with adding and subtracting, just key it in they way it is written. Key in one number, then the times button, then key in the other number, and then press the equal button and you will have the answer displayed. But as before with addition and subtraction, it is always a good idea to estimate by working out the multiplication with simpler numbers that are close to the ones you want to multiply.

Division

The ancient Egyptians and other early civilizations needed division in order to share things. To find out out how the ancient Egyptians would have done division see Egyptian Division. The process on an abacus or a counting board would be fairly similar and you can probably figure it out yourself.

There are a number of more efficient methods for division using Hindu-Arabic numerals and even today long division is not taught the same way in all countries. Division is difficult no matter how you do it, so people have been quite creative with it. The method that I learned in school and probably most of you have learned as well would do the division I did in Egyptian Division like this.

You estimate how many times 56 goes into 196 by asking yourself how many times 5 goes into 19, but often this doesn't work. In this case it does, and we get 3. You write down the 3 above the 6 and then multiply this by the 5 to get 168. You now subtract to get 28 and bring down the 0 to get 280. Now you try to figure out how many times 56 goes into 280 by asking yourself how many times 5 goes into 25. This time the estimate of 5 works, so you write down the 5 and multiply it by the 56 to get 280 and subtract to get 0 and you are done, and 35 is the answer.

So what's going on here? Well, it is closer to what the ancient Egyptians did than you might think. When you ask yourself how many times 56 goes into 196 what you are actually doing is trying to find out what multiple of 10 times 56 you can take out of it just like we did Egyptian style when we found out that 30 times 56 gets pretty close. We are finding out that if you are giving out 1960 loaves of bread to 56 people, you can give out 3 boxes of 10 to each and still have some left over. By placing the answer to 3 times 56 under the 6 instead of to the far right we are really subtracting 30 times 56 from 1960 to see how many loaves of bread will be left after we give out 30 to each person. After we do that and see that the answer before bringing down the 0 is less than 56, so we know that we can't give out any higher number of 10-boxes, because there are less than ten times 56 loaves left, so now we have to see how many single loaves we can give out, so we bring down the 0 for that. We then make a guess of 5 and check it and since it comes out even we know that we can give out 30+5 or 35 loaves to each person.

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