Divide like a Babylonian! With our new "centimal" arithmetic (base 100)

Division is the most difficult of the four basic arithmetical operations. Here we shed light on the Old Babylonian approach to division in their sexagesimal or base 60 system, by translating their arithmetic to the new base 100, or centimal" . We have some critical things to say about our current number systems and the logical weakness in dealing with "repeating decimals".

The role of regular numbers is stressed, along with the importance of a "centimal reciprocal table". We also look at multiplication / division tables and their role.

A big thanks to all my Subscribers, Members of the Wild Egg Maths channel, and Patreon supporters!

Video Content:

00:00 Intro, centimal arithmetic and division

3:59 Example of difficulties with the division operation

8:59 A logical treatment of operations

12:36 Centimal number arithmetic

17:43 The centimal reciprocal table

24:58 Centimal combined multiplication/division tables

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0:10

hello everyone i'm Norman Wildberger and we're talking about the sociology of pure mathematics

0:14

and in particular number systems and i'm trying to sort of expand your potential thinking in this

0:20

direction by going back historically looking at the Egyptians the Babylonians and now i've

0:25

introduced this newfangled sentimental kind of arithmetic it's a variant of our

0:32

Hindu Arabic system which however parallels more closely the way the Babylonians thought

0:38

about arithmetic so it's our chance to practice thinking like Babylonians and today we're going to

0:44

tackle the most difficult of the four fundamental operations addition multiplication subtraction and

0:52

the most difficult one is always division it's almost the heart of the the matter even from

0:59

earliest times to even modern computational systems and still division is much more

1:05

problematic expensive etc than the other operations so let's let's have a look at

1:13

some of the issues here so suppose that we would just ask a primary school student what's

1:19

24 divided by 6 well the answer is 4. and how do they know that well they can look it up on a

1:24

multiplication table so they know that 4 times 6 is 24 so that means that 24 divided by 6 is

1:31

going to be 4. so that's a sort of reversing of what's on a multiplication table a little

1:37

bit more sophisticated 24 divided by 5. so that's not on a multiplication table something else has

1:44

to be done to appreciate that the answer is 4.8 that's a decimal number and that's typically done

1:51

in by using something called long division which you may or may not have learned as a student in my

1:58

day yes we had to learn long division but i think perhaps students these days don't spend so much

2:04

time on that okay here is a very different kind of problem it looks very much the same what's

2:10

24 divided by seven well if you perform this long division okay and it is a little bit of

2:17

an elaborate process but you start performing this long division you see that it doesn't seem to end

2:23

it sort of keeps on going okay it doesn't finish like this one did so you end up with three point

2:29

four two eight five and you're keeping on doing this process five and then that's seven and then

2:34

one and then four then two and then eight and then five and then seven and then one and then

2:39

four and two okay so maybe you get up to this stage here and then you look at it and say well

2:43

actually wait a minute there's a there's a kind of a pattern here we're getting this four two eight

2:48

five seven one and then it repeats four two eight five seven one it's a repeating kind of decimal

2:54

so it doesn't end okay we're we're going outside this realm of finite decimal arithmetic and uh

3:03

sort of invoking these new kinds of objects called repeated decimals so we might be inclined to

3:09

rewrite this uh capturing that repeating thing by writing it as 3.428571 with a bar over it say to

3:17

represent this repeating aspect so this requires us to perform a long division and notice a pattern

3:26

okay that's not that onerous when the pattern only consists of half a dozen numbers but if

3:32

the pattern involves a half a billion numbers uh then saying what does it mean to notice a pattern

3:42

what how do you notice a pattern in general that's actually you know hugely challenging

3:47

to to understand so this particular division is a lot more complicated than these two divisions

3:54

even though on the face of it they look very parallel so here is a more complicated example

4:03

that shows us just the difficulties that we are likely encountering with this division operation

4:09

suppose we take this number one followed by one two three four five six seven eight nine zeros and

4:16

then a one and we're dividing it by this number which is 26 i guess that's ones thousands millions

4:24

billions so 26 billion seven hundred ninety eight million seven hundred thousand four hundred twenty

4:30

seven so in that base a thousand way that we like to write numbers okay what is the result of this

4:37

division well again you can get your computer perhaps or you can do it longhand also you get

4:42

a long division process going and it starts 3.731524230938-036-299-874-937

4:55

and at that point maybe my computer stops although i know that the algorithm itself doesn't stop

5:01

just that i run out of memory or or time or or energy or something so we could extend this but

5:08

it's not at all clear what the pattern is in fact it's very likely that the repeating aspect of this

5:16

answer is not evident for uh many miles or from at least for many uh you know maybe

5:24

millions of digits this is here what do we say 26 billion so when you think about long division you

5:31

realize that there's sort of 26 billion possible remainders so in principle you could cycle through

5:37

a lot of those remainders before you got kind of a repetition so in principle this really could go on

5:44

for a long time before you even are in a position to to notice some kind of repeating pattern

5:53

so uh that's computationally uh basically almost impossible unless you have some you know super

6:00

computer i suppose that you're at your disposal now we have another number system floating

6:07

around in the modern world that young people learn about actually also in primary school

6:13

and that's a theory of fractions or rational numbers which you very likely hated when you

6:19

were in school and perhaps for some reasonable reason actually because you know arithmetic

6:27

with fractions is fraught with difficulties it's fraught with a logical subtleties okay

6:35

that can be surmounted but to do so requires a fair amount of sophistication and in fact the

6:43

practical applications of fractions as i pointed out other places

6:46

is ultimately rather limited when you go past sort of kitchen uh cooking recipe kinds of applications

6:54

uh in in science and in engineering and practical real life applications fractions are very quickly

7:02

dispensed with and we typically move to a decimal or even approximate decimal kind of framework

7:09

nevertheless okay one thing that we do learn as students is that fractions provide a kind of

7:16

pleasant in sort of theoretical environment in which division becomes

7:21

dramatically simplified one might almost say tautological because the way we would rescue this

7:31

seemingly impossible division is to say well that this number divided by this number is the fraction

7:39

one zero zero zero zero one over this number two six seven nine eight seven zero zero zero

7:44

four two seven that fraction is the answer that you get after you've performed this operation

7:55

now a babylonian scribe would look at this and say well that's hilarious that's hilarious

8:05

are you serious you know this is this is the way you're going to do arithmetic you're basically

8:09

just going to repeat the question in a slightly different form and say oh now we have an answer

8:15

it's not really so much different from the way my analyst friends do real number arithmetic i

8:20

remind you where you have a question like what's cosine of 5 plus log 7 and the answer is oh it's

8:29

cosine 5 plus log 7. that's the way we do real number arithmetic there's some parallels here so

8:39

this is uh clearly unsatisfactory from the Babylonian point of view

8:44

the Babylonian scribe would say well you haven't actually performed the division that's still

8:49

waiting to be done this is just a placeholder for an operation which you have not yet performed

8:57

so we might say okay yeah our Babylonian scribe friend has a point

9:02

um maybe we'll stick with repeating decimals we'll we'll say that the results of divisions

9:10

are repeating decimals so for that we need an arithmetic of repeating decimals which surely is

9:15

not that much more onerous than just our regular theory of arithmetic with regular decimal numbers

9:23

well unfortunately ladies and gentlemen the situation is somewhat more

9:28

delicate than that in fact i claim that a logical treatment of operations with

9:35

so-called repeating decimals is pretty well missing in modern pure mathematics

9:41

we don't have proper definitions of these operations at the level of repeating decimals

9:48

and we don't have proofs of the laws of arithmetic that we would want for those operations dealing

9:57

with repeating decimals so what i'm saying here is that if we're working in the world of repeating

10:03

decimals like these kinds of things okay there's x which is three point four two eight five seven

10:07

that's repeated let's say and y is twenty one zero four and then the three one nine is repeated okay

10:16

the question is what are actually the algorithms for computing these various

10:21

operations say on on two typical repeating decimals so what's the algorithm for obtaining

10:27

x plus y so x plus y should be another repeating decimal okay so you it's just a finite thing so

10:33

you should be able to write it down got finite input here finite input here ultimately and

10:38

there's going to be a finite output so how do you get that finite output from the finite inputs

10:44

and you can repeat that question for each of the four operations

10:50

and we're assuming that repeating decimals x and y are given well there is in fact a way of doing

10:56

this that in some sense is known to primary school students because they know or they're taught at

11:03

some point that repeating decimals are pretty well equivalent to fractions exactly because when you

11:12

do the long division of a fraction the way a Babylonian might want you to do then you're

11:17

going to get a repeating decimal so what you can do is there's a kind of a way of going backwards

11:22

so given a repeating decimal like this or like this you could convert those back into fractions

11:29

and then you could do the operations at

11:32

the level of the fractions and then convert back to a repeating decimal

11:38

although admittedly that's kind of a cheating right that that means you're going actually

11:42

outside this numerical system that you're trying to establish and saying okay we're

11:46

going to actually use some different arithmetic in order to substantiate this arithmetic that

11:52

we want to create over here in the world of repeating decimals in particular if you're a

11:58

real number enthusiast and presumably you have a theory of real number or infinite decimal

12:04

arithmetic which includes repeating decimals just as special cases then you should just be able to

12:11

take that existing theory and simply specialize it to this very special and relatively simple case

12:20

anyway this is a very subtle question i doubt if there's more than a handful of people

12:28

who really know how to do this and my guess is in fact that there's probably nobody knows how

12:34

to do it properly so this is going to be a really interesting theme for us just how convoluted is

12:41

the current landscape of number systems and does it really work to what extent does it really work

12:47

and to what extent are there all kinds of logical holes which are just more or less ignored or

12:55

we pretend that these are sort of elementary considerations which are beneath our consideration

13:00

so that's a very interesting sociological question obviously because arithmetic is you

13:06

know at the heart of mathematics education and if there are really fundamental issues

13:11

with the logical structure of this of this subject then that that has a lot to do with

13:17

the education of young people as well as more advanced applications theoretically so i want to

13:25

contrast our rather jumbled landscape of numerical systems with the Babylonian

13:32

system which was i think fair to say a much tighter and um more logically consistent

13:39

but they paid a price for that okay which maybe to us at first seems like uh

13:46

a weakness but when we look more carefully we see that this is just a result of them being more

13:52

careful more clear more logical about the way we do things okay so in order to try to explain this

14:00

this uh this issue i want to move to this sentinel point of view that i've introduced

14:05

in my last video which allows us to see the way the Babylonians thought in a more familiar way

14:12

because it's much closer to our base 10 system so let me remind you that we have sentimental

14:16

numbers which are basically base 100 numbers digits from 1 to 99 so here is a sentimental

14:23

number 23 17 here's another one 4152 and i've explained that the product is going to be another

14:28

sentinel number in this case 962 184 and the the question of the absolute values of these numbers

14:36

or where the decimal or sentimental points are is sort of irrelevant there is no central point okay

14:42

so these numbers are sort of floating freely the the the actual sentimental point or decimal point

14:49

is sort of freely floating when we talk about addition and subtraction however in order for

14:56

things to be sort of clear we need an alignment of of things so in this context here we have these

15:05

digits of this sentimental number and these digits of this central number sort of aligned the 43 on

15:11

top of the 58 the 81 over the 92. when we align them like this then addition becomes possible it's

15:18

just like pretty well the usual addition although you know we would probably better think about

15:25

the way the Babylonians did it in terms of an abacus but anyway we can sort of understand it

15:30

this way and subtraction same kind of thing so these are the three relatively simple operations

15:36

and now we come to division which is the much more complicated operation so how does that go

15:46

well the answer is it's the same kind of answer that a Babylonian would have given is that if you

15:51

want to do division exactly exactly you have to limit yourself you can't divide by any old number

16:01

division exactly only works in the sentimental world when you're dividing by a sentimental

16:08

regular number what does that mean that's a number which is a product of twos

16:15

and fives it's exactly those numbers that it turns out that we can divide exactly by in the

16:25

Babylonian world base 60 60 has factors two three and five so almost the same except that there's

16:34

a crucial third number the number three which is included in potential factors of regular numbers

16:44

and this makes a huge difference it means that in the sexagesimal world there are a lot more regular

16:51

numbers than in the sentiment or base 10 world which means that their capacity of doing division

16:58

is greatly increased over our capacity whether we're working in base 10 or working in base 100

17:07

this is an interesting point to get your head around the idea that first of all the number

17:14

system that you're working in does matter at least when it comes to division and that if you're

17:20

concerned with division and you want to have a an effective kind of as effective as possible

17:27

well then the babylonian system is a step up above our systems whether it's base 10 or base 100

17:38

and this is a disconcerting realization but it's the way it is so at the heart of the babylonian

17:44

approach to division was their reciprocal table and we've already had a look at it

17:50

in the sentimental world there's also a reciprocal table but it's simpler it's more familiar and it's

17:57

also smaller than the base 60 system so it's easier for us to get our heads around what's

18:02

happening here this is the reason one of the reasons i'm trying to advocate this sentimental

18:07

point of view because it brings into focus much more easily the role of the reciprocal tank

18:14

so the central reciprocal table base 100 here are the numbers n and here are their reciprocals n bar

18:21

and n bar n and bar and bar and we're going basically through the numbers from

18:28

2 to 99 or or 100 and once we get to 100 it's basically one and going a little bit beyond

18:35

because that's also what happened in the in the Babylonian case so it's basically a

18:41

list of regular numbers or at least small regular numbers single digit regular numbers okay where a

18:47

digit means something between 1 and 99. so these are the numbers which are multiples of twos and

18:56

fives from one up to up to one so the reciprocal of two is fifty because two times fifty is well

19:04

we would say a hundred but a hundred is the same as one in their floating point system four times

19:10

twenty-five is one five times twenty is one eight times twelve fifty is one so the reciprocal of

19:17

eight is twelve fifty reciprocal of ten is ten reciprocal of sixteen is six 25 reciprocal of

19:24

20 is 5 well i guess that's just the same as this one just reversed the reciprocal of 25 is 4. it's

19:30

really the same as that the reciprocal of 32 is 3 12 50. we get a number with three digits

19:37

reciprocal of 40 is 250 the reciprocal of 50 is 20. reciprocal of 64 is 156.25 reciprocal of 80

19:45

is 125. the reciprocal of one is one reciprocal of 1.25 is 80 the reciprocal of 128 is 78 12 50.

19:57

so if we were sentimental scribes we would want to memorize this table

20:04

and in some sense you're probably already familiar with aspects of this you probably know that

20:09

dividing by 8 is the same as multiplying by 0.125 another example 57 divided by 16 so we look for 16

20:23

in our reciprocal table look at the reciprocal of 625 so that's the same thing as 57 times 625

20:30

okay and there you have to do some multiplication perhaps with an abacus to get 3 56 25.

20:39

now if we were going to convert this into our sort of fixed point floating system

20:46

so 57 divided by 16 is going to be a little bit more than three so we would want to put

20:50

our decimal point right there and interpret this as being 3.5625 so i want to stress to you that

20:57

we can extract from these kinds of equations equations that are familiar to us in the hindu

21:03

arabic world right so this one here is really saying 57 divided by 16 is equal to 3.5625

21:11

that's what makes this a little bit attractive is that we can quickly appreciate what's going on

21:18

because 100 is not so different from 10 in terms of a base so here's an example of a multiplication

21:25

division table in the sentimental framework i'm just going to look at multiples of 3 12 50. why

21:35

3 12 50 well if you look on our reciprocal table that i just showed you 3 12 50 is the reciprocal

21:40

of 32 so that's one of the entries so following a Babylonian tradition we should create a

21:47

a table of multiples of 3 12 50. so times 1 is 3 12 50 times 2 is 6 25 times 3 is 9 37 5 and you

21:59

can have a look down to 10 and then it makes sense to go with further multiples of 10. so times 20 is

22:09

6250 times 30 is 93.75 times 40 is 125 times 50 is 156 25 times 60 times 70 times 80 times 90 is

22:25

81 25. so why would we want to have multiples of 3 1250 is it because

22:34

this is a multiplication that's likely to you know to pop up at us in in applications not so

22:40

much it's more indirect than that it's that it's the division by 32 that we're more interested in

22:47

okay the division by 32 in this Babylonian setting translates to a multiplication by the reciprocal

22:57

okay so we end up wanting to multiply by 3 12 50 if we say

23:01

taking 67 divided by 32 that's the same as 67 times 3 12 50. and then we could use this table to

23:08

pretty easily calculate that because the 67 is like 60 60 right here okay so 60 times

23:17

the reciprocal three 1250 is 187.50 there it is right there and seven times the same thing seven

23:25

is over here seven times the three twelve fifty is twenty one eighty seven fifty so now we have to

23:32

add these two but this is one of these cases where you have to add but in order to add you have to

23:37

align them okay you have to know which ones sort of go with which ones and in this case the it's

23:44

the 87 that goes with the 21 okay so put that over that 50 over the 87 and the 50 is sticking out

23:53

and then you do the addition and you get 2 9 37 uh five no you get 50 yes sorry

24:04

i think it should be a 50 here okay so 67 divided by 32 is 2 9 37 50. if we were going to convert

24:18

this into a hindu arabic form well the decimal point would be right there because 67 divided by

24:22

32 is going to be roughly 2. so it's going to be 2 .9 2.09 because that's really an o line 2.09375

24:34

but that 5 is really 50 okay so in in the sentimental point of view i need that 0 over there

24:43

so that's a single multiplication division table that's parallel to the kinds of things that the

24:50

Babylonians had that illustrates that division was an important usage for this kind of table

24:58

all right so we might aspire to creating a sensible combined multiplication division

25:03

table which is parallel to what the old Babylonians had where we could basically

25:07

combine all the various multiplication division tables and so let's agree that

25:12

multiples that we're going to be talking about are say multiples times 2 times 3 all the way up to 10

25:17

and then all them also multiples of 20 30 40 up to 90 because that's sort of the standard

25:22

multiplication division table by itself and then what we want to do is we want to

25:29

list multiples of pretty well everything that's on the standard sentimental reciprocal table that

25:38

table of reciprocals and in the babylonian times there would have been some additional

25:43

entries occasionally so um i'm going to just show you here uh briefly just some slides of what i've

25:52

cooked up to be a combined multiplication division table in the sentiment story

25:58

and so here you can see some various tables of various multiples of these various things and

26:06

overall i would say that this is still going to be less it's going to be comparable to a Babylonian

26:15

combined multiple multiplication table but maybe somewhat okay so

26:22

let's just give another illustration so suppose we have 57 divided by 16 how would we do that

26:28

well we would look this up 16 the reciprocal of 16 is 625 then we would write that 57 as 50 and 7

26:37

look up the various multiples of 625 on the corresponding

26:42

multiple division table they are 3 12 50 and 43.75 combine them in the appropriate way in this case

26:48

the 12 over the 43 the 50 over the 75 to get 3 57 350 625 and i'd like to point out that uh um

26:59

Jay Freiburg who's a sort of world authority on Babylonian texts and has done all kinds of uh

27:07

amazing work in this direction uh in a great book in 2007 called a remarkable collection

27:12

of Babylonian mathematical texts he really does i think you know point out this direction that that

27:18

there's a a reasonable chance that division was an important you know ingredient and the motivation

27:24

for these uh tables which have been priorly called multiplication tables so this is not just our idea

27:30

that they should be called division tables so i hope this little insight into the complexities

27:40

of division intrigues you but there's another very important sort of numerical aspect that i want to

27:48

talk about also in a sentimental point of view i want to shed a little bit of light on the way

27:54

the old Babylonians thought about square roots taking square roots and i'm going to draw upon

28:00

some the insights of my colleague Danielle mansfield for that and and give you a sort

28:04

of a sentimental version of that very interesting story i'm Norman Wildberger thanks for listening.

28:19

you