Babylonian Number System

1
Babylonian Number System
2
Your text will give you background information on
this number system and tell how it works. This
lecture reviews how the system works and
demonstrates how to convert from Babylonian to
our decimal system and vice versa. A calculator
will be helpful.
3
(No Transcript)
4
(No Transcript)
5
It may appear then that the Babylonian system is
an Additive System. This is not the case.
Position matters in interpreting Babylonian
numbers. It is a Positional System.
6
(No Transcript)
7
If the symbols occur on the far right, they
represent how many ones there are. There are 24
ones.
8
If the symbols occur on the far right, they
represent how many ones there are. There are 24
ones.
If there are more symbols positioned a bit to the
left, they represent how many sixties there are.
There are 13 sixties.
9
If the symbols occur on the far right, they
represent how many ones there are. There are 24
ones.
If there are more symbols positioned a bit to the
left, they represent how many sixties there are.
There are 13 sixties.
If there are more symbols positioned even further
to the left, they represent how many 3600s there
are. There are 5 three thousand six hundreds.
10
(No Transcript)
11
(No Transcript)
12
(No Transcript)
13
(No Transcript)
14
18,804
Look this over for a bit to make sure you see how
it works.
15
Determine the value of this Babylonian number.
16
Determine the value of this Babylonian number.
17
Determine the value of this Babylonian number.
18
Determine the value of this Babylonian number.
19
Determine the value of this Babylonian number.
20
Determine the value of this Babylonian number.
21
Try this one on your own.
22
Try this one on your own.
3 x 3,600
10,800

23
Try this one on your own.
60s
1s
24
Try this one on your own.
60s
1s
21 x 60

1260
25
Try this one on your own.
26
Try this one on your own.
10 x 3,600
36,000

27
Whats wrong with this number?
28
Whats wrong with this number?
The ones place asks for 64 ones. 60 should be
represented by an extra in the 60s place. The
number is written this way
3 x 60 4 x 1 184

29
Suppose that you wanted to write the number 3604.
You would put a
in the 3600s place
and
in the ones place.
30
Suppose that you wanted to write the number 3604.
You would put a
in the 3600s place
and
in the ones place.
But what about the 60s place?
?
31
This was a problem for the Babylonians. So at
some point they invented a place holder. This
is there place holder symbol
32
This was a problem for the Babylonians. So at
some point they invented a place holder. This
is there place holder symbol
With the place holder, they could clearly write
the number 3604 as
33
So what is this number?
34
So what is this number?
60s
1s
36 x 60

2160
35
What do we use for a place holder in our number
system?
36
What do we use for a place holder in our number
system?
0
We use the number zero. For instance, in the
number 307, there are 3 hundreds, no tens and 7
ones. In our system zero is not just a place
holder but is also a number.
37
What do we use for a place holder in our number
system?
38
Spend a few moments trying to write the number
125 in Babylonian.
39
Spend a few moments trying to write the number
125 in Babylonian.
If you have an answer, click again. If you dont,
please try, its not that hard!
40
Spend a few moments trying to write the number
125 in Babylonian.
There are clearly no 3600s in 125 however there
are two 60s. So put two wedges in the 60s
place.
60s
1s
41
Spend a few moments trying to write the number
125 in Babylonian.
There are clearly no 3600s in 125 however there
are two 60s. So put two wedges in the 60s
place. Two 60s make 120. There are five ones
left.
60s
1s
42
Now try to write 824 in Babylonian. It may help
to first put a heading for each place value as
done below.
43
Now try to write 824 in Babylonian. It may help
to first put a heading for each place value as
done below.
Again there are no 3600s in our number. 824
divided by 60 is 13, remainder 44.
44
Now try to write 824 in Babylonian. It may help
to first put a heading for each place value as
done below.
Again there are no 3600s in our number. 824
divided by 60 is 13, remainder 44.
So we have 13 in the 60s place and 44 in the
ones place.
45
Now try to write 8903 in Babylonian.
46
Now try to write 8903 in Babylonian.
8903 is greater than 3600, so the 3600s place
must be used. There are two 3600s in 8903.
8903 divided by 3600 is 2, remainder 1703.
47
Now try to write 8903 in Babylonian.
8903 is greater than 3600, so the 3600s place
must be used. There are two 3600s in 8903.
8903 divided by 3600 is 2, remainder 1703.
There are twenty eight 60s in 1703. 1703
divided by 60 is 28, remainder 23. The 23
remaining are ones.
48
Now try to write 111,637 in Babylonian.
49
Now try to write 111,637 in Babylonian.
111,637 is greater than 3600, so the 3600s place
must be used. There are thirty one 3600s in
111,637. 111,637 divided by 3600 is 31,
remainder 37.
50
Now try to write 111,637 in Babylonian.
111,637 is greater than 3600, so the 3600s place
must be used. There are thirty one 3600s in
111,637. 111,637 divided by 3600 is 31,
remainder 37. There are no 60s in 37, so the
placeholder must be used. The 37 remaining are
ones.
51
What place value comes after 3600?
52
What place value comes after 3600?
In our decimal system, the first place value is
one, then 1 x 10, or 10, then 10 x 10 or 100,
then 100 x 10 or 1,000, then 1,000 x 10 or
10,000, then 10 x 10,000 or 100,000, ...
In Babylonian the first place value is one, then
1 X 60 or 60, then 60 x 60 or 3600. What comes
next?
53
In Babylonian the first place value is one,
then 1 X 60 or 60, then 60 x 60 or 3600, then
3600 x 60 or 216,000, then 216,000 x 60 or
12,960,000, and so forth for ever.
54
An advantage of a place value system for writing
numbers is that you can write as large a number
as you like without having to invent new symbols.
We will only use the first three place values in
writing Babylonian numbers. For more practice,
do your exercises.