MULTIPLE METHODS OF MULTIPLICATION

In school we are taught one standard algorithm for multiplication. It looks like this:

There are actually many more ways to multiply. Here are a few from different times in history and different parts of the world: ancient India, ancient Egypt, and renaissance Scotland. Try using these methods on a couple of mutiplication problems--you may find a method you like better than our usual one!

How to multiply like an Egyptian

1. Make two columns of numbers. In the left hand column, write the powers of 2, starting with 1 and doubling it over and over again. In the right hand column, write the larger of the two numbers you are multiplying next to the 1 in the left hand column. Double it over and over again.
2. Check the numbers in the left-hand column which add up to the smaller of the numbers you're mulitplying.
3. Add up the numbers in the right hand column which are next to the numbers you have checked. This sum is the answer to your multiplication problem.

Multiply like an Egyptian 53×72

This is 53×72, ancient Egypt-style:	Here's the same problem in our usual notation:
	1	72
	2	144
	4	288
	8	576
	16	1152
	32	2304
	72 + 288 + 1152 + 2304 = 3816

Why does this work?

You can write:

Now multiply:

This is the same as

1×72 + 4×72 + 16×72 + 32×72 = 72 + 288 + 1152 + 2304 = 3816

John Napier

John Napier was a Scottish baron, Laird of Merchiston, inventor, and mathematician.

Logarithms ("for the more easie working of questions in arithmetike and geometrie") are his most famous contribution to math.

Nearing the end of his life, John Napier, also developed an ingenious arithmetic trick - not as remarkable as logs, but very useful all the same. His invention was a method for performing arithmetic operations by the manipulation of rods, (also called “bones” because they were often constituted from ivory) and printed with digits. Napier’s rods essentially rendered the complex processes of multiplication and division into the comparatively simple tasks of addition and subtraction.

In 1617 Napier published Rabdologia. In it he explained how to use 'Napier's rods' which could be used to multiply numbers together where the calculator only needed to use addition. Napier dedicated the book to Alexander Seton, Earl of Dunfermline and explained in an introduction that he:

... was induced to publish a description of the construction and use of the numbersing rods because many of my friends, to whom I have already shown them, were so pleased with them that the rods are already almost common and are even being carried to foreign countries.

'Napier's rods' consisted of 10 rectangular blocks with multiples of a different digit on each of the four sides. For example the top of one of the rods on which multiples of 4 are given is shown here.

The other three sides of this rod had multiples of other digits. Note that, except for the top square, each square is divided by a diagonal, and when the digit 4 is multplied by 3 the resulting 12 is written with the 1 above a diagonal and the 2 below. All two digit numbers appear on the rods in the same way with the 10s digit above the diagonal and the unit digit below.

So, each "bone" is a list of the first nine multiples of a number between 1 and 9. There is an index "bone" for reference.

Napier's rods or bones consist of ten oblong pieces of wood or other material with square ends. Each of the four faces of each rod contains multiples of one of the nine digits, and is similar to [the one shown above], the first rod containing the multiples of 0, 1, 9, 8, the second of 0, 2, 9, 7, the third of 0, 3, 6, 9, the fourth of 0, 4, 9, 5, the fifth of 1, 2, 8, 7, the sixth of 1, 3, 8, 6, the seventh of 1, 4, 8, 5, the eighth of 2, 3, 7, 6, the ninth of 2, 4, 7, 5, and the tenth of 3, 4, 6, 5. Each rod, therefore, contains on two of its faces multiples of digits which are complementary to those on the other two faces; and the multiples of a digit and its complement are reversed in position.

Let's see how the rods were used...

To multiply 4x759 using the bones, line up the 7, 5, and 9 bones. Put the index bone next to them. Look at the 4 row only.

Add the numbers in the diamonds. You get 2 in the 1000's place, 8 + 2 = 10 in the 100's place, and 6 in the 1's place.

Answer: 2000 + 1000 + 30 + 6 = 3036.

What if you want to multiply a number by a two digit number, such as 64×759?

Multiply 4×759 as shown above.

Multiply 6×759. Use the same bones but look at the 6 row.

Multiplying 64×759 is the same as multiplying 60×759 + 4×759.

Multiply 4554 by 10 and add it to 4×759=3036.

64×759=3036 + 45540 = 48576.

For example to multiply 4138 by 567 place four rods with the digits 4, 1, 3, 8 on top as shown alongside a strip containing the numbers 1 to 9 in squares

To multiply 4138 by 567 we now examine rows 5, 6 and 7 of the four rods (as indicated by the right hand strip). Looking at row 5, starting from the right, we write down the number obtained by adding the digits in the parallellograms as shown.

Next do the same with row 6, this time noting that we have to carry 1 when we add 8 + 4.

Place the numbers so that the second starts one place to the right of the first as shown

20690
 24828

Now do the same with row 7, again placing the result one place further to the right

20690
 24828
  28966

We now add the three numbers

20690
 24828
  28966
2346246

We obtain the product of 4138 and 567 is 2346246.

Now this may not seem a great saving in effort but, as we mentioned above, only addition is needed to carry out calculations. Napier's rods became quite popular and we used in Britain and on the Continent. The Rabdologia was translated into Italian and Dutch, and the original Latin text was republished in Leiden.

We have used the name 'Napier's rods' in this article, but often the calculating aid was called 'Napier's bones'. The name comes from the title of a work by William Leybourne publiahed in 1667 entitled The art of numbering by speaking rods: vulgarly termed Napier's bones. Why did Leybourne use the term 'speaking rods'? This was simply a mistranslation of Napier's Rabdologia. This comes from 'rabdos' meaning a 'rod' and 'logia' meaning 'collection'. However Leybourne thought it came from 'rabdos' meaning a 'rod' and 'logos' meaning 'word'.

Let us note that Napier also designed "square root rods" and "cube root rods" but these did not become so popular.

Often Napier's rods would come in a box which contained other tables and aids for calculation. For example one set made in 1679 came in a box 12 cms by 6.4 cms by 2.8 cms. The box had a hinged lid which had, on the outside, a table giving the interest at 6 per cent for one, two, three, six, and twelve months, for each £10 from 10 to 90, and for hundreds of pounds to £500. The bottom of the box had two tables on it, one giving the year, week day, age of the moon on 1 January for 1679 to 1693, and a "Perpetual Almanac" with the year beginning in March. The second table gave the time of high tide at various places relative to the age of the moon. Inside the lid of the box is a table for addition in 13 columns of eleven numbers, the first numbered downwards from 1 to 10, the next from 1 to 11 and so on to the thirteenth numbered from 12 to 22.

One version of Napier’s rods is displayed in the picture below:

The rods were extremely popular in Napier’s day and oddly, they constituted the Scottish mathematician’s chief claim to glory among his contemporaries. It is indicative of the poor knowledge of arithmetic at the time, that extensive use was made of Napier’s rods all over Europe, since even the simplest arithmetic operations were beyond the reach of most people’s abilities.

Thus, the rods circulated widely in basic, middling and deluxe versions. An expensive edition was also available, in which the rods were made out of ivory and they came in a carrying case of fine leather, with an addition table attached to the lid, which was included for good measure. Later on, the rods were replaced by cylinders (in which case all the rods from 0 to 9 were displayed on each cylinder) and fitted into wooden boxes.

Instead of placing the rods on a board equipped with a vertical index labeled from 1 to 9, all you had to do was to rotate them in their places in the box. Napier’s invention was employed extensively by people whose work depended on calculations and numbers, such as accountants, bookkeepers etc. The value of Napier’s rods is exemplified by the fact that they were still being used in primary schools in Britain in the mid-1960s to assist in teaching multiplication.

Logarithms

Napier's discussion of logarithms appears in Mirifici logarithmorum canonis descriptio in 1614. Two years later an English translation of Napier's original Latin text was published, translated by Edward Wright. In the preface of the book Napier explains his thinking behind his great discovery (we quote from the English translation of 1616 of the original Latin of 1614):

Seeing there is nothing (right well-beloved Students of the Mathematics) that is so troublesome to mathematical practice, nor that doth more molest and hinder calculators, than the multiplications, divisions, square and cubical extractions of great numbers, which besides the tedious expense of time are for the most part subject to many slippery errors, I began therefore to consider in my mind by what certain and ready art I might remove those hindrances. And having thought upon many things to this purpose, I found at length some excellent brief rules to be treated of (perhaps) hereafter. But amongst all, none more profitable than this which together with the hard and tedious multiplications, divisions, and extractions of roots, doth also cast away from the work itself even the very numbers themselves that are to be multiplied, divided and resolved into roots, and putteth other numbers in their place which perform as much as they can do, only by addition and subtraction, division by two or division by three.

Unlike the logarithms used today, Napier's logarithms are not really to any base although in our present terminology it is not unreasonable (but perhaps a little misleading) to say that they are to base 1/e. Certainly they involve a constant 10⁷ which arose from the construction in a way that we will now explain. Napier did not think of logarithms in an algebraic way, in fact algebra was not well enough developed in Napier's time to make this a realistic approach. Rather he thought by dynamical analogy. Consider two lines AB of fixed length and A'X of infinite length. Points C and C' begin moving simultaneously to the right, starting at A and A' respectively with the same initial velocity; C' moves with uniform velocity and C with a velocity which is equal to the distance CB. Napier defined A'C' (= y) as the logarithm of BC (= x), that is

y = Nap.log x.

Napier chose the length AB to be 10⁷, based on the fact that the best tables of sines available to him were given to seven decimal places and he thought of the argument x as being of the form 10².sin X.

At first, Napier called logarithms "artificial numbers" and antilogarithms "natural numbers". Later, Napier formed the word logarithm to mean a number that indicates a ratio: logos meaning proportion, and arithmos meaning number. Napier chose that because the difference of two logarithms determines the ratio of the numbers for which they stand, so that an arithmetic series of logarithms corresponds to a geometric series of numbers. The term antilogarithm was introduced in the late 17th century and, while never used extensively in mathematics, persisted in collections of tables until they fell into disuse.

Napier did not use a base as we now understand it, but his logarithms were, up to a scaling factor, effectively to base 1/e. For interpolation purposes and ease of calculation, it is useful to make the ratio r in the geometric series close to 1. Napier chose r = 1 - 10^-7 = 0.999999. Napier's original logarithms did not have log 1 = 0 but rather log 10⁷ = 0. Thus if N is a number and L is its logarithm as calculated by Napier, N = 10⁷(1 - 10^-7)^L. Since (1 - 10^-7)¹⁰⁷ is approximately 1/e, this makes L/10⁷ approximately equal to log_1/e N/10⁷.

The fact that Nap.log 1 does not equal 0 is a major difficulty which make Nap.logs much less convenient for calculations than our logs. A change to logs with log 1 = 0 came about in discussions between Napier and another mathematician, Briggs who read Napier's 1614 Latin text and, on the 10 March 1615 wrote in a letter to a friend:

Napper, lord of Markinston, hath set my head and hands a work with his new and admirable logarithms. I hope to see him this summer, if it please God, for I never saw a book which pleased me better or made me more wonder.

In fact Briggs did make the difficult journey from London to Edinburgh to see Napier in the summer of 1615 (would he have dreamed that now it takes 4 hours by train, rather than at least 4 days by horse and coach in those times). A description of their meeting was told by John Marr to William Lilly who writes the following :

Mr Briggs appoints a certain day when to meet at Edinburgh; but failing thereof, Merchiston was fearful he would not come. It happened one day as John Marr and the Lord Napier were speaking of Mr Briggs, "Oh! John," saith Merchiston, "Mr Briggs will not come now"; at the very instant one knocks at the gate, John Marr hastened down and it proved to be Mr Briggs to his great contentment. He brings Mr Briggs into my Lord's chamber, where almost one quarter of an hour was spent, each beholding other with admiration, before one word was spoken. At last Mr Briggs began, -"My Lord, I have undertaken this long journey purposely to see your person, and to know by what engine of wit or ingenuity you came first to think of this most excellent help unto astronomy, viz. the Logarithms ...

Briggs had suggested to Napier in a letter sent before their meeting that logs should be (in our terminology) to base 10 and Briggs had begun to construct tables. Napier replied that he had the same idea but :-

... he could not, on account of ill-health and for other weighty reasons undertake the construction of new tables.

At their meeting Napier suggested to Briggs the new tables should be constructed with base 10 and with log 1 = 0, and indeed Briggs did construct such tables. In fact Briggs spent a month with Napier on his first visit of 1615, made a second journey from London to Edinburgh to visit Napier again in 1616 and would have made yet a third visit the following year but Napier died in the spring before the planned summer visit.

How to find the logarithm of a number, using four figure log tables (below)

To find the log of a number, you first need the two parts of the answer - the index or characteristic (the part before the decimal point) and the mantissa (the decimal part after the decimal point). For example, the index of 2.6742 is 2 and the mantissa is .6742.

Finding the index of a logarithm

This is easy if you follow these rules:
For numbers with more than 1 digit before the decimal point (10 and above), the index will always be one less than the number of digits before the decimal point. For example, the index of 245.211 is 2, the index of 24521.1 is 4 and the index of 24.5211 is 1.
For numbers with just 1 digit before the decimal point (1 to 9), the index will be 0. For numbers less than 1, then index will be one more than the number of zeros between the decimal point and the first significant figure. For example, the index of 0.245211 is (bar) 1, the index of 0.00245211 is (bar) 3 and the index of 0.000245211 is (bar) 4.
 
Finding the mantissa of a logarithm
Use the numbers at the far left of the table to give the first two significant figures of the number. If there are more than two digits in the number, follow across the table - the column headings give the third digit of the number. The numbers given in the log table represent the mantissa part of the answer.

For Example:

To find log 45, follow down the numbers on the far left of the table until you reach 45. The number in the column headed 0 will be the mantissa - the index is 1 (see above) - so log 45 = 1.6532.

To find log 456, follow down the numbers on the far left of the table until you reach 45 and take the number in column 6. The index is 2, so log 456 = 2.6590.

Using the Mean Difference Column for numbers with four digits

For numbers with four digits, simply add the number in the mean difference column (right hand 9 columns) to the answer you would get if you just ignored the fourth digit.

For Example:

To find log 4567, take the log of 456 (2.6590) and add the number in column 7 of the mean differences (7) to the mantissa. ie.

log 4567 = 3.6590 + 0.0007 = 3.6597.
 

Finding logs of numbers smaller than one

This is simple - just find the log of the four significant figures of the number, and and take away the index.

For Example:

To find the log of 0.004567, take log 4.567 = 0.6597 and subtract 3 (the index).

So log 0.004567 = (log 4.567) - 3 = -2.3403.

The log tables

Use of Logs and Antilogs

Use Of Common Logs

Common Logarithms are powers to the base 10. Natural logs are exponents to the base "e". "e" = 2.71

In working with common logs the following procedure is followed:

1. Convert the number whose log is being determined to scientific notation

2. Determine the log of the principle number in front of the exponential part of the notation using a log table.

   There are 4 place, 5 place, and 6 place log tables found in most math and science books located usually in the Appendix.

3. Determine the log of the exponential part. That is easy because the log of the exponential part is simply the exponent of 10. For example:

   log 10⁷ = 7

   log 10³ = 3

   log 10^-2 = -2

4. Add the log from step 2 to the log in step 3 for the final log of the number.

The difficult part is step 2. You have to have a log table to get it. Lets assume that you have a 4-place log table. Here is how you would determine the log. Let's say you wanted to know the log of 4.18. In the log table there will be rows of four digit numbers.

Each row will be labeled at the extreme left of the row with a two digit numbers separated by a decimal. Then there will be columns of four digit numbers each c olumn headed by a single digit from 0-9.

Each of the four digit numbers in the table has a decimal to the left most digit which does not show but should be placed in there. So if we want the log of 4.18, we would go to the left most end of the rows and go down until you reach the row labeled with 4.1.

We would move to the right in this row. Then we would go to the top of the columns and move over to the column labeled with an 8. We would then proceed downward in that column.

If you extend from the 4.1 across to the right and extend from the top of the column marked 8 downward until the two extensions intersect, at the point of intersection will be a four digit number actually called the mantissa of the final log determination. To the left of this four digit number will be a decimal or one should be placed there.

Let me take an example from the beginning:

Let's determine the log 25,600

1. Convert number to standard scientific (exponential) notation. That would be

   2.56 × 10⁴

2. Determine the log of 2.56 as described above. If you do this you will arrive at a four digit number (mantissa) which is .4082

3. Determine the log of the exponential part:

   log 10⁴ = 4

4. Add the two together for the final answer:

   4 + .4082 = 4.4082

Negative Logs

It is possible to have negative logs as well. For example, what would be the log of

0.00314.

1. Convert to standard exponential notation:

   3.14 × 10^-3

2. Determine the log 3.14 from the table:

   log 3.14 = .4969

3. Determine the log 10^-3 = -3

4. Add the two from step 2 and step 3 together:

   .4969 + (-3) = -2.5031

Often they leave negative logs in the first form that is

.4969 -3

Calculation Using Logs

You can calculate with logs in the following manner

Multiplication of numbers

To multiply numbers using logs simply:

1. Convert each number to a log
2. Add the logs (because when we multiply numbers we add their log forms)
3. Determine the antilog for the product

Division of Numbers

1. Convert the denominator and the numerator into log forms
2. Subtract the log form of the denominator from the log form of the numerator to get a difference
3. Determine the antilog of that difference in step 2 for the final quotient

Raising A Number To A Power

1. Convert the number to a log form
2. Multiply the log form by the exponent (power)
3. Determine the antilog of step 2 for the final answer

The last step in all the above computations using logs involves knowing how to determine the antilog of a logarithm. This is easy if the log is positive.

1. Separate the whole number part of the log form (called the characteristic) from the decimal part(called the mantissa the four digits with the decimal to the left most position)
2. Determine the antilog of the decimal part (mantissa). This is done by going back to the four place log table and browse through the mantissas located throughout the table. Find the mantissa that comes closest to the one you are determining. Then placing your finger on that mantissa extend your finger to the left until it reaches the beginning of the row where the two digit numbers separated by a decimal are. That two digit number is the first two digits in the number. Now take you finger and proceed upward from the found mantissa until you reach the head of the column it was in. That single digit heading that column is the third digit in the antilog.
3. Determine the antilog of the whole number part of the log form (sometimes called the characteristic). That is the simple part because that would be 10 raised to the power equal to the whole number part. So if the characteristic is 3 then the Antilog of 3 = 10³
4. Multiply the antilog in step 2 with that in step 3 for the final antilog.

Let's take an example:

Antilog of 3.8734

1. Separate the mantissa from the characteristic 3 and .8734

2. Determine the antilog of .8734.

   If you don't find a mantissa in the table that is .8734 find the one that comes closest. In my four place log table the closest that to .8734 that it has is one that is .8733. Move from there to the left to the front of the row where we find the digits 7.4

   Moving your finger from the .8733 upward to the top of the column gives the single digit 7

   So antilog of .8734 = 7.47

3. Determine the antilog of the characteristic 3 which would be

   Antilog 3 = 10³

4. Multiply the two together for the final antilog

   7.47 × 10³

   or 7470

   So the Antilog 3.8734 = 7.47 × 10³ = 7470

Determining the Antilog Of A Negative Log

Determining antilogs of negative log forms can be a bit tricky since a Negative log form has a negative mantissa. All of the mantissas in the 4 place log table are positive mantissas so one has to take the additional step of converting the negative log to a form that has a positive mantissa.

For example:

Antilog of -4.5611

1. Note the characteristic which in this case is a four. Add the next whole number to that log form:

         -4.5611
         +5
   

   but if we just add 5 that would change the log so we have to subtract 5 as well:

   -4.5611
   +5      
   -5
   

   If I do this I will get:

   .4399 - 5

   The above log form is equivalent with the original log form but its mantissa is positive and can now be looked up in the table as we did before.

2. Determine the antilog of the positive mantissa

   In looking in the table there is no .4399 but the closest mantissa is

   .4393

   so we will take that one and move back to the beginning of that row and also upward to the top of the column it is in and we get:

   2.75

3. Now we determine the antilog of -5 which would be 10^-5

4. Multiply the two together for the final antilog result:

   2.75 × 10^-5

More examples using log tables

Multiplication using log tables

1. Firstly we will look in the log table to find the value of log(4138):

   • the characteristic of 4138 is 3
   • the mantissa of 4.138 is 0.6160 +0.0007 = 0.6167.

   Result: log(4138) = 3.6167

2. Next we will look in the log table to find the value of log(567):

   • the characteristic of 567 is 2
   • the mantissa of 5.67 is 0.7536

   Result: log(567) = 2.7536

3. Next we calculate log₁₀(4138 × 567) = log₁₀(4138)+log(567):

   3.6167 + 
   2.7536
   6.3703
   

   log₁₀(4138 × 567) = 6.3703

4. Now we calculate the antilog of 6.3703.

   • For mantissa 6, the index is 6
   • the closest characteristic value in the table to 3703 is 3692 that coresponds to the antilog value 234 and also because 3703 - 3692 = 11 we can find the fourth digit 6 in the mean difference column (right hand 9 columns). So we get 2.346
   • The result is 4138 × 567 = 2.346 × 10⁶ = 2346000

Comparing to the exact result we got using the Napier's bones (2346246) we are pretty close.

The error is:

100 × (2346246 - 2346000)/2346246 = 0.01% 100 × 246/2346246 = 0.01%

Division using log tables

The good news we can also divide numbers using log tables.

For example :

log10(4138 : 567) = log10(4138)- log(567):
3.6167 - 
2.7536
0.8631

The antilog of 0.8631 is 7.296 or 7.297. So 4138 : 567 = 7.297.

The square root of 2 using log tables

The cube root of 2 using log tables

The Logarithmic Scale

Soon afterwards Gunter introduced his rule, in 1622, William Oughtred simplified things further by taking two Gunter's lines and sliding them relative to each other thus eliminating the dividers.

He noticed that there is a easyer way to add distances than using a divider.

To measure the length of a segment of right-hand side we normally use a scale either in cm, or in milimeters. If we have 2 identical rules we can use them to make additions and subtractions. It will be admitted that each division line in mm corresponds to a number.

For example, to add 34 + 48 = 82 we lay out the 2 rules one against the other as indicated in in the figure.

We place the division 0 of the rule C above division 34 of rule D. It results the division 48 of the rule C is placed opposite to dvision 82 of rule D. 82 represents the sum we are looking for.

Based on this observation, Oughtred was the first to see that a simpler and more sophisticated method of multiplication and division could be achieved by placing two logarithmic rules side by side and using the position of the numbers relative to each other to calculate the desired results.

The first slide rule

Use the mouse to darg the rules and the cursor to perform some calculations. For example drag the upper rule and align the division 1 (on the left of the upper rule) over the division 2 on the bottom rule (be careful to not place it by mistake over the division 1 2).

Next drag the red hair cursor to align over the division 3 on the upper rule and read the result under the cursor on the bottom rule. It must be 6. This way we calculated 2×3=6.

Next let the cursor as is, positioned over division 6 on the bottom rule and drag the upper rule until its division 4 is aligned under the red hair cursor (being this way also aligned over the division 6 on the bottom rule).

Now read on the buttom rule the value pointed by the division 1 on the upper rule. This should be 1 5. This way we divided 6 by 4 obtaining the result 1.5. (6 ÷ 4 = 1.5).

Actually the result we obtained was 1 5 not 1.5

Placing the decimal point is our responsability, depending the operands values.

We would obtain the same 1 5 result not only dividing 6 by 4 but also .6 by 4, 60 by 4 and 600 by 4, etc.

The slide rule doesn't have a clue what is the magnitude of the operands. It gives us only the result digits and is up to the slide rule operator to decide where to place the decimal point depending on the the operands values.

William Oughtred was one of the world's great mathematicians.

Oughtred's most important work, Clavis Mathematicae (1631), included a description of Hindu-Arabic notation and decimal fractions and a considerable section on algebra. He experimented with many new symbols including the "×" symbolfor multiplication and :: for proportion as well as the abbreviations "sin" and "cos" for the sine and cosine functions..

Like all Oughtred's works it was very condensed containing only 88 pages.

He also developed the circular slide rule, which operated in the same fashion as a linear slide rule, except that it makes use of an inner and an outer ring Oughtred published his renowned work Circles of Proportion and the Horizontal Instrument in 1632.