Think like a Babylonian! With "centimal" base 100 arithmetic
Let's try to understand Babylonian sexagesimal (base 60) arithmetic by modifying our own Hindu Arabic system just a little! We'll use base 100, and work in a true floating point fashion, just like the Babylonians did (1900 - 1600 B.C.E.). But multiplication and addition don't work exactly as we are used to! Is this system superior to our own base 10 system? That is an interesting question.
This new arithmetical system can give us a deeper understanding of the completely remarkable, far ahead of its time, mathematics of an ancient Mesopotamian civilization, now buried under the sands of Iraq.
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Video Content:
00:00 Introduction
3:11 Digits
05:39 Converting from centimal to decimal
15:26 How to evaluate products of digits?
19:15 Tables of squares
25:00 Addition of centimal numbers requires alignment
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0:11
hello everyone i'm Norman Wildberger we're looking at the sociology of pure mathematics
0:17
especially now at number systems and trying to position the way we think about numbers in terms
0:23
of sort of a broader picture of historically what's been going on and i've been telling you
0:28
about the Egyptians and the Babylonians and i want to um clarify a little bit that the Babylonian
0:35
situation that's very interesting and it's not so easy for a sociological type person to get a
0:43
hang of what's going on there because there are these um these obstacles that first of all we're
0:48
we're talking about entirely different culture and the way they approach arithmetic is very
0:53
very different but even more so the fact that they're working in this base 60 system means that
0:59
most of us don't have a lot of experience with arithmetic so i'm going to propose a variant of
1:04
our arithmetic today which will be a great aid i am sure in being able to understand more clearly
1:13
how the Babylonians thought at least this is my attempt at doing that okay so i'm going
1:18
to introduce what i call sentimental or base 100 arithmetic so this is a new a new sort of
1:25
point of view but it's not just changing the base i'm also going to go back to a an old babylonian
1:31
point of view so let me remind you that with our ordinary hindu arabic numbers which of
1:36
course a base 10 system here's a big number if i want to read this number i'm going to look at it
1:43
you know here here's ones here's thousands there's millions there's billions there's trillions
1:51
after trillions comes quadrillions so i would read this as 67 quadrillion 149 trillion 25 billion
2:02
488 million 665 thousand and 313 and when you when you hear that you start to realize that actually
2:13
our number system is not just a base 10 system it's also partially a base 1000 system
2:19
because each of these sort of triples here is in some sense a digit between 1 or between 0 and 999.
2:29
so we have sort of two levels going on here the base 10 level and this larger base 1000 level
2:35
which is usually not really acknowledged but you can see it very clearly here so what i'm
2:42
going to propose today is in some sense a kind of a compromise between base 10 and base 1000.
2:46
we're going to look at base 100 okay but i'm going to approach it in a in a true floating point uh
2:53
fashion just like the old Babylonians from around 1900 to 1600 bc would have proceeded with their
2:59
sexismal system so by getting our head around this system it'll be much easier for you to appreciate
3:06
the remarkable base 60 system that they had okay so base 100 means we're going to have to have
3:12
digits from 1 to 99. the old Babylonians did not have a special digit representing zero so that's
3:21
actually excluded in our world just digits from 1 to 99 what we call 99. now this is the way that
3:30
a Babylonian scribe would approach that problem well we use the two symbols that the Babylonians
3:36
used a vertical stroke representing one and a wedge opening to the right representing ten
3:41
so then here would be the digits one through nine there'd be 10 this would be 11 12 13
3:48
14 and so on up to 1920 all the way then to 30 then eventually to 90. right in the sexagesimal
3:58
world you're not allowed to use nine tens the maximum number of tens that ever appear is five
4:05
but this is sentimental so it's a little bit bigger so we can go all the way up to uh 99 okay
4:10
so these would be the 99 digits that a Babylonian scribe would want to write down in for our system
4:19
but this is too onerous okay so it's a lot easier for us to use the standard Hindu Arabic numbers
4:26
that we are familiar with 1 through 99 although you know might be easier for us but for somebody
4:32
who's not familiar with this this system it would be probably harder and maybe this would be sort of
4:39
simpler and easier to understand okay so then what is a sentimental number well a sentimental number
4:45
is just a string of these digits put together uh separated by commas and we'll use commas because
4:54
that's a little bit more in keeping with what i showed you in our first slide where we had
4:59
these groups of three decimal digits separated by commas representing thousands millions
5:05
billions etc so i want you to think about this as being some kind of variant of that instead of
5:10
grouping things by groups of three with commas we're separating with groups by groups of two
5:17
all right but it's a true floating point system so you should not think of this as being the ones and
5:24
this is the hundreds and this is 100 squareds etc that's only one possibility we're not saying where
5:29
the decimal point or rather the sentimental point is here right that's just a sentimental number
5:35
that can be interpreted in a number of different ways so now that we have sentimental numbers
5:39
which are just these strings of sentible digits separated by commas we can maybe convert them to
5:47
our more familiar decimal Hindu Arabic numbers but there's a number of ways of doing that depending
5:52
on where we think of the decimal place as being so that's not specified in the sentimental world
6:00
so this sentinel number 5 43 92 17 could be interpreted well one way would be to think of
6:07
this as being like ones this is hundreds this is hundred squares this is a hundred cubes so then
6:13
we would just get five four three nine two one seven so that's directly just translatable that
6:20
would be five million four hundred thirty nine thousand two hundred and seventeen but we could
6:25
also perhaps think of the the the sentimental point if you like as being there which case
6:31
this would be sort of representing the integer part and this would be representing a fractional
6:35
or sentimental part then it would correspond to i guess 5 54 392 0.17 or maybe we think of
6:47
the sentimental point as being right there and then then the interpretation will be 543.9217
6:55
okay now one of the advantages in having this flexibility where we not actually
7:03
thinking about having a sentimental point is that multiplication becomes simpler
7:09
because we don't have to worry about where to put a final central point because there is no
7:13
final sentimental point so let's let's talk about multiplication so suppose we want to multiply
7:20
these two sentimental numbers okay 23 17 times 41 52. how are we going to find out what that is
7:31
well what we're going to think about is doing essentially a number of multiplications of
7:39
just single digits so these digits here this 52 and this 17 if we multiply them
7:47
we're going to get 8 84 okay which corresponds in in our language to
7:54
52 times 17 equals 884 okay but as a centimeter scribe we would say 52 times 17 equals 884
8:05
another product 52 times 23 well just that product by itself
8:10
is 11 96 41 times 17 is 697 and 41 times 23 is 943. so these are just perhaps you might
8:23
like to think of them as sort of entries in this hundred by hundred multiplication table
8:29
that records all possible products of the digits here and digits here actually for us it's not
8:36
really a hundred by hundreds it's more a 99 by 99 because actually zero is not allowed as a digit
8:45
of course these just correspond to familiar or easily checkable things in the hindu arabic world
8:52
so 41 times 17 is actually 697 so you know it's the same kind of information as we usually have
8:59
so that makes it a lot easier to understand what's going on so how are we going to multiply these two
9:05
numbers well we have the individual products of the digits so we are going to kind of wanting to
9:11
put them together so that's our inclination is to you know figure out how to do it on the page but
9:16
almost surely that's not how they did it they did not do their computations on their clay tablets
9:21
they did their computations somewhere else and we don't really know so the best guess is probably
9:27
that they use something like an abacus okay so here's a more modern abacus but we can still
9:32
just use it for illustration purposes so what we're going to do is think about putting a number
9:38
uh on the abacus sort of from the right so because you know we're starting with the smaller things
9:43
here they're going to move over here so for example 2317 we'd represent that with a okay
9:49
a seven on the last column okay maybe these are five let's say these are ones okay so we would
9:54
move one of those up and that's a five and then two of these up okay so that would be like seven
9:59
so i've recorded a seven and then in the next uh the next uh you know digit of one i would
10:06
put a one there no if i just wanted to record 23 uh 17. so there's a 17 and then i was let's uh
10:13
that's five sorry that's one there and then three there whoops three there and there's two okay
10:21
so that might that might be a representation of 23 17 sort of roughly something like this
10:26
okay so in order to make this product they wouldn't have to actually put these things down
10:32
because these would be uh you know on on in front of them we want to write down the the result so
10:39
let's say we multiply 52 by 17 first so we figure out somehow maybe we've memorized it maybe we look
10:45
it up somehow but we have to know what the basic digit multiplication is so in this case it's 884
10:53
or 884 but we'll just think of it as 884. so we have to record an 884 on the last three of these
11:00
columns so we put a 4 on this one and an 8 on this one 8 on this one and then we do another
11:05
digit multiplication say 41 times 17 and or maybe 52 times 23 that's that 1196 so now we have to put
11:14
1196 we have to add 1196 to what we already have with our 884 now that involves moving over two
11:23
because where we're moving over a digit so we have to basically move over a digit there's actually
11:29
commas here which i perhaps should put but i'm not putting them okay so that means that this
11:34
1196 is going to be added and we're not going to use these last two columns here we're going to add
11:41
we're going to add that you know in these columns here these four columns here
11:45
so we can take the existing stuff and add 1196 with whatever abacus technology we've learned
11:52
okay and then that's going to be the total up to that point and then we do the next digit maybe 41
11:58
times 17 which is 697. so then we would have to add 697 again that actually would be aligned to
12:03
the same same one as before so i guess it would be still those four of these three in this case
12:09
so we would be adding something to these three columns to add 697 and then finally the 41 times
12:15
23 that's 943 so we have to add 943 again but again we have to sort of move over the right
12:20
number of columns two two so we have to move over to two so we sort of be adding uh from here okay
12:29
so we could do that and it would you know if you're skilled with an abacus of course as
12:36
in previous generations many Chinese, Japanese, Korean you know people students would have learned
12:42
you know they can compute with an advocate much faster than we can compute uh by by hand or with
12:49
pen and paper so it's a very powerful system there's no reason why the Babylonians could
12:53
not have you know mastered the use of something like that although we don't know exactly what
12:58
they're doing so i'd like to compare that with the same calculation in a more familiar Hindu Arabic
13:05
framework this is what you might find in some primary school classroom where the way we do it
13:11
is we also multiply digits but now the digits are just from the 10 by 10 multiplication table which
13:16
we've memorized so 2 times 7 you have to know what that one is so it's 14 there's a 4 there
13:21
and 1 comes up so you have to do all these individual products 4 times 4 16 individual
13:26
multiplications have to be sort of put into this okay but the the individual multiplications are
13:32
smaller because they're just sort of single digit multiplications so here we only have four
13:38
digit multiplications but any one of them is more onerous of course we end up getting the same total
13:45
maybe i should also have mentioned that once we when we get these digits we've done this on the
13:49
calculator okay but this is my attempt at doing it by by hand using paper the way we've been taught
13:57
you know you have to add up the columns and so on but they wouldn't have done that they would have
14:03
just you know done it on their abacus or counting board or whatever and then looked at the final
14:08
result and then just transcribed the final result so notice that when i go from this to the final
14:14
result i have to modify this 0 1 because 0 1 is not a digit in our system we have to just
14:20
replace it with 1. so the final answer is 9 62 184 and here's the Hindu Arabic equivalent and okay
14:28
which one is more work well they're pretty close to being comparable but you see that they look
14:34
quite different and so i think it's an interesting computational question here for those of you maybe
14:40
with a computer science uh programming background can we say anything about the relative merits of
14:46
these things right there's a trade-off here we're doing a lot of small calculations and here we're
14:51
doing fewer big calculations so there's probably scenarios where each of these maybe has advantages
14:59
but i think it's interesting to think about this uh sort of from a computational point of view
15:06
could you could you get some computer architecture which runs not on a sort of decimal
15:13
or binary equivalent uh you know sort of arithmetic but that that jumps up to sentimental
15:19
or some kind of binary equivalent maybe they already do this that could be please let me know
15:24
if that's the case all right but a big question here is how do you actually multiply these digits
15:32
do you memorize a 99 by 99 multiplication table well this is the same issue that we
15:37
have with the Babylonian story there it's 60 or 59 by 59 but it's still
15:43
quite onerous and i remind you that we don't really know how they do their multiplications we
15:49
don't really know but we think that we have some ideas and one of the ideas is to use
15:56
this particular relation which was well known to them that if you have two numbers m and n
16:01
and you're interested in their product m times n then one way of getting at it is through this
16:06
identity that it's equal to n plus m over 2 all squared minus n minus m over 2 all squared so
16:15
for example with 41 times 23. we're going to use this rule we calculate this is the average so you
16:22
add them up which is 64 and divide by 2. so that's 32 this is 32 squared and this one is
16:30
the difference divided by 2. okay the difference is 20 no the difference is 18 and divided by 2 is
16:39
is 9 so it's 9 squared 32 squared minus 9 squared 32 squared well you have to look that up or
16:46
let's say you know that that's 10 24 9 squared is 81. you take the difference you get 943.
16:56
now here's a way of thinking about that here's m and here's n let's suppose that
17:00
m is smaller than n okay so the average is just the number that's in between them that's equally
17:05
far from either of them and this n minus m over 2 that represents well this whole
17:13
difference is n minus m so n minus m over 2 is is that separation or that separation
17:19
so another way of thinking about it is that this second one is the the separation between
17:24
either of these numbers and this average okay so for example once we found that the average is 32
17:31
up here then the next one we just look at the difference between 41 and 32 that's 9 or the
17:37
difference here that's also 9. okay so we don't actually have to do this calculation so much
17:43
but what do you do when you are multiplying say an odd number by an even number so this works fine if
17:48
you have two odd numbers or two even numbers what if you have one of each say 29 times 84. well what
17:55
you could do is you could because what you don't want to really apply this formula because then you
18:00
get you know when you add these two numbers you can get an odd number and then you have
18:04
to divide it by 2 and square it so that's that's more problematic so you could pull out a factor
18:08
of 2 here write it as 2 times 29 times 42 and then you can in fact pull out another factor
18:14
of 2 write it as 4 times 29 times 21. and now you have two odd things so you could do the same thing
18:21
as the rule suggests you take the average which is the number that's sort of equally close to both
18:27
of them that's 25 okay the difference between 25 in either of these is 4 so 25 squared minus
18:34
4 squared is 625 minus 16 which is 609 and then we have to multiply by 4 we get 24 36
18:44
so what this rule allows us to do is to reduce the calculation of a product with calculation of two
18:55
squares okay so if we're on top of squares or if we you know have some table or resource that tells
19:04
us quickly what squares are then we can actually use this quite efficiently to calculate products
19:15
okay so now we come to the table of squares which is an essential table
19:20
for the old babylonians will also be important for us as modern sentimental
19:24
arithmeticians so we have numbers down here and they're squares adjacent numbers and they're
19:31
squares numbers and they're squares 1 squared is 1. 2 squared is 4. 3 squared is 9. 4 squared is
19:37
16 5 squared is 25 6 squared is 36 7 squared is 49 a squared is 64. 9 squared is 81 10 squared equals
19:45
well not a hundred because there isn't really a hundred it's just one okay that's the way the
19:54
floating point point of view is towards 101 they're the same thing they're just
20:00
represented by a single vertical stroke what about 11 squared well 11 squared is 121
20:08
1 comma 21. 12 squared is 144. 13 squared is 169. 14 squared is 196. 15 squared is 2 25.
20:19
16 squared is 2 56 17 squared is 289 18 squared is 3 24 19 squared is 361 and 20 squared is 4.
20:31
these will be a little bit less familiar to most people everyone's memorized these in
20:36
primary school these are a little bit less familiar but actually it's very useful to
20:41
know what what 17 squared is et cetera so let's do one more column 21 squared is 441
20:48
22 squared is 484 23 squared is 529 24 squared is 576. 25 squared is 625.
20:57
26 squared is 676. 27 squared is 729 28 squared is 784. 29 squared is 841 and 30 squared is 9.
21:13
of course this is information that you can immediately you know
21:17
value this is the same as saying 24 squared is 576. it's the same information in fact 576 is
21:24
often stated as 576 right with recognizably the same thing okay and here's 30s and 40s i want
21:34
to draw a particular attention to the the last ones here so once we get to 46 squared it's 2116
21:42
which ordinarily we would say 2116 but it's just 21 16. and 47 squared is now 22 9
21:55
but there's no zero here because 0 9 is not a digit 9 is the digit similarly 48 squared is 23
22:03
- 49 squared is 24 1. so if we wanted to convert this to decimal we would say okay this is 2401
22:12
we would sort of implicitly put a zero in there but that's not needed and 50 squared is 25.
22:23
okay and here's the second half of the table of squares i'm not going to go through it all but
22:28
here it is for you to admire 51 squared is 26 1 52 squared is 27 4. we have that same phenomena that
22:35
i already mentioned 53 squared is 28 9 and then 54 squared is 29 16 and so on 60 squared is 36
22:45
the 60s the 70s the 80s down to the 90s 96 squared is 92 16 97 square is
22:54
94 9 98 squared is 96 4.99 squared is 98.1 and 100 why isn't this 100 well
23:03
100 is equal to 1 in our system and so what's 100 is the same as one squared one squared is one
23:13
so involved here we we're going to memorize this table that would involve memorizing 99 things
23:20
and it's probably you know interesting to stare at it and see if there's any patterns i mean
23:26
there's some things that are that are sort of repeating the 96 the 69 these these things that
23:31
are that are happening here at the end are there's some some repetition um a square never ends in in
23:39
three for example right you never see the final number b3 then the final numbers are always
23:44
rather predictable in fact either the same across each of these columns the final numbers
23:48
are always 1 four nine six five and then or six nine same ones we just had six nine four one
23:58
um so yeah that's the table of squares and so you could imagine that it would
24:04
be possible for a young person to memorize this there's really 99 things to to memorize
24:08
although some of them are quite long so it would be definitely some some serious
24:13
amount of work to memorize this then it's an interesting question okay suppose that you got
24:18
some young people and you trained them this way you you got them to memorize this table
24:22
maybe you got them to perform digit calculations with using this device
24:33
you'd have to still teach them how to use some mechanical device say an abacus to
24:40
to deal with this sentimental arithmetic
24:43
and so then it's an interesting question well you know what would the result be how would
24:47
that compare with our current system is this really so outlandish or are there actually
24:53
potentially interesting reasons why we might want to uh to go this way okay so that's multiplication
25:00
what about addition well we've already seen some addition because in the course of our
25:04
doing our multiplication we've had to add these various products of digits that we've obtained
25:09
and what i did there is i i put them in in certain positions with respect to each other and i'm sort
25:14
of suggesting that we're going to end up using our abacus or counting board or whatever it is to to
25:20
perform those additions because that's reasonably easy to do with such a mechanical device
25:25
but from a theoretical point of view it's important to realize that if you have two
25:29
sentimental numbers say 44 37 2 and 98 95 asking what is the sum that's an ambiguous question
25:38
because we don't know the relative sizes of these things so it's not so much a question of knowing
25:46
where the sentimental points are in each of them it's more a question of knowing how the two
25:52
sentimental numbers align where we maybe think of putting one on top of the other so for example one
25:59
possible alignment might be this one here 44 37 to 98.95 maybe the 98 is comparable with the 44
26:07
and the 95 is comparable with the 37. okay then we can perform the operation with an abacus very
26:16
simply but on on the boards much the same 2 7 plus 5 is 12. you raise to put the 1 up there
26:23
you get a 13 you have to put a 1 there you get a 13 you have to put a 1 there you get 14 and
26:29
you get this so you get 1 43 32 2. let me do this one over here which is just a little bit different
26:36
okay so two other things to add 92 86 57 plus 4 13 45 and let's do the same thing so 5 plus 7 is 12.
26:47
i'll put a 1 up there 4 plus 5 is 9 plus 1 is 10. so we put a 0 there put a 1 up there we get
26:56
7 plus 3 is a 10. so we put a 0 there at 1. 8 plus 2 is 10 so we put a 0 there
27:04
and put a 1 up there and then we get a 7 and a 9. okay so we would put commas like this
27:11
but this really is not in sentimental form for two reasons first of all 0 2 is not a digit and
27:22
0 0 is not a digit so what we would want to do is we would want to rewrite this
27:28
so there would be a 97 comma okay let's not let's think about what we're going to
27:34
put in the middle here but this final one is just going to be two all right
27:42
so what are we going to put in the middle well the answer is we're not going to put anything in the
27:47
middle because we don't have a digit representing zero the pap loans didn't have a digit for
27:52
representing zero they just left a space of course they didn't have commas they but they just had
27:56
these digits arranged in a line and there were spaces between them so if there was a sufficient
28:03
amount of space they would know that there's going to be that's representing uh an absence of
28:09
of a digit okay so okay that might look a little bit strange to us but hey that's just a it's
28:15
just a space are our computers are capable of making spaces okay so there's an introduction
28:22
to sentimental arithmetic but we still have some ways to go because we still have to answer the the
28:30
serious question how do you do division and that's where things get interesting so i'll talk about
28:37
that in my next video i hope you'll join me for that i'm Norman Wildberger thanks for listening.