Numeration: Babylonian Numerals
The Babylonian system (3000 BC) is built around Base 60. Although the numeration system and
Babylonian Mathematics were very advanced, there are only two symbols in the Babylonian
number system. This is likely because the Babylonians did not have paper (they pressed reeds
into clay tablets).
Symbol
<
|
Value
10
1
The symbols for the Babylonian system are shown in your textbook, but we won't be able to
type them in that fashion. Instead, we will use the "less than" symbol to represent "ten," and a
vertical bar to represent "one."
To type the symbol for 10, you should use “shift-comma,” and to type the symbol for 1, you
should use a “capital letter i.”
Your first thought might be... with only those symbols, they wouldn't be able to write very
many numbers. Let's look at how Babylonian numerals would be written. These aren't all the
numbers. Instead, they are just examples of some numbers, so that you get the idea:
Hindu-Arabic
Babylonian
Hindu-Arabic
Babylonian
1
|
11
<|
2
||
12
<||
3
|||
13
<|||
4
||||
20
<<
5
|||||
24
<<||||
6
||||||
30
<<<
7
|||||||
37
<<<|||||||
8
||||||||
45
<<<<|||||
9
|||||||||
52
<<<<<||
10
<
59
<<<<<|||||||||
Remember in our Base 10 system, how the largest single digit that we have is 9 ?
Well, in the Base 60 Babylonian system, the largest single digit is 59, which is the last number
shown in the table above. In order to write numbers larger than 59, the Babylonians used place
value, in Base 60.
60^2 = 3600
60^1 = 60
60^0 = 1
three thousand
six hundreds
sixties
ones
In order to indicate that we have changed place value positions in the Babylonian system, we
will put a space between the place value positions.
Converting from Babylonian to Hindu-Arabic (the number system that we use):
An important thing to remember here is that we are converting from Base 60 to Base 10, so this
type of problem will be similar to other problems that we have worked on, with the added step
of using different symbols.
Example:
Write the Babylonian numeral
<<||| <|
as a Hindu-Arabic numeral
Since we have been given a base 60 numeral, we need to determine the place value positions for
that system. The space between the symbols that we were given indicates a change in the place
value position. Here, the numeral that we were given is made up of 2 digits, so we need to find
two place value positions. Then, we can put the Babylonain digits into their proper place value
positions. Also, we can rewrite those Babylonian digits as Hindu-Arabic numerals:
60^1 = 60
60^0 = 1
sixties
ones
<<|||
<|
23
11
In expanded form, the Babylonian numeral would be:
23 x 60^1 + 11 x 60^0
In order to rewrite this as a Hindu-Arabic numeral, all we need to do is simplify the expression
that we just created:
= 23 x 60 + 11 x 1
= 1380 + 11
= 1391
So, the Babylonian numeral <<||| <| would have the same value as the Hindu-Arabic
numeral 1391.
Another Example:
Write the Babylonian numeral | <<<<| <<||||||| as a Hindu-Arabic numeral.
Since we have been given a base 60 numeral, we need to determine the place value positions for
that system. The spaces between the symbols that we were given indicates the changes in place
value positions. Here, the numeral that we were given is made up of 3 digits, so we need to find
three place value positions. Then, we can put the Babylonain digits into their proper place value
positions. Also, we can rewrite the Babylonian digits as Hindu-Arabic numerals:
60^2 = 3600
60^1 = 60
60^0 = 1
three thousand
six hundreds
sixties
ones
|
<<<<|
<<|||||||
1
41
27
In expanded form, the Babylonian numeral would be:
1 x 60^2 + 41 x 60^1 + 27 x 60^0
In order to rewrite this as a Hindu-Arabic numeral, all we need to do is simplify the expression
that we just created:
= 1 x 3600 + 41 x 60 + 27 x 1
= 3600 + 2460 + 27
= 6087
So, the Babylonian numeral | <<<<| <<||||||| would have the same value as the HinduArabic numeral 6087.
Converting from Hindu-Arabic (the number system that we use) to Babylonian:
An important thing to remember here is that we are converting from Base 10 to Base 60, so this
type of problem will be similar to other problems that we have worked on, with the added step
of using different symbols.
Example:
Write the Hindu-Arabic numeral 381 as a Babylonian numeral.
Since we are looking to find a base 60 numeral, we need to determine the place value positions
for that system. This time, we have to find enough place value positions to make sure that we
will be able to convert the number that we were given (381). The way to do this is to continue
finding place value positions until you have reached a number greater than the one you are
trying to convert:
60^2 = 3600
60^1 = 60
60^0 = 1
three thousand
six hundreds
sixties
ones
Since we have reached a number in the place value positions (3600) that is greater than the
number that we are trying to convert (381), we know that we have enough place value
positions. Also, we won't need any 3600s as we try to construct 381 in the base 60 system.
60^2 = 3600
60^1 = 60
60^0 = 1
three thousand
six hundreds
sixties
ones
Now, we start the process of constucting the number 381 in the base 60 system:
Ask yourself the question: How many 60s will go into 381?
This is just division, and it will tell you the number of 60s that will be in our answer.
Since 60 will go into 381 six times, we write that down in our place value chart as part of our
answer.
60^1 = 60
60^0 = 1
sixties
ones
6
We are trying to construct the number 381, and so far we have accounted for 360 (which is 6 x
60).
Subtraction (381 – 360) tells us that there is still 21 remaining to be put into our place value
position chart.
That 21 will go into our "ones" place, completing our place value position chart.
60^1 = 60
60^0 = 1
sixties
ones
6
21
Finally, we need to rewrite these digits in Babylonian numerals:
60^1 = 60
60^0 = 1
sixties
ones
6
21
||||||
<<|
Now, we have our answer.
So, 381 in Hindu-Arabic numerals is rewritten as |||||| <<| in Babylonian numerals.
Note:
Remember to put a space between the place value positions when typing these numerals on
quizzes or exams.
Another Example:
Write the Hindu-Arabic numeral 1052 as a Babylonian numeral.
Since we are looking to find a base 60 numeral, we need to determine the place value positions
for that system. This time, we have to find enough place value positions to make sure that we
will be able to convert the number that we were given (1052). The way to do this is to continue
finding place value positions until you have reached a number greater than the one you are
trying to convert:
60^2 = 3600
60^1 = 60
60^0 = 1
three thousand
six hundreds
sixties
ones
Since we have reached a number in the place value positions (3600) that is greater than the
number that we are trying to convert (1052), we know that we have enough place value
positions. Also, we won't need any 3600s as we try to construct 1052 in the base 60 system.
60^2 = 3600
60^1 = 60
60^0 = 1
three thousand
six hundreds
sixties
ones
Now, we start the process of constucting the number 1052 in the base 60 system:
Ask yourself the question: How many 60s will go into 1052?
This is just division, and it will tell you the number of 60s that will be in our answer.
Since 60 will go into 1052 seventeen times, we write that down in our place value chart as part
of our answer.
60^1 = 60
60^0 = 1
sixties
ones
17
We are trying to construct the number 1052, and so far we have accounted for 1020 (which is 17
x 60).
Subtraction (1052 – 1020) tells us that there is still 32 remaining to be put into our place value
position chart.
That 32 will go into our "ones" place, completing our place value position chart.
60^1 = 60
60^0 = 1
sixties
ones
17
32
Finally, we need to rewrite these digits in Babylonian numerals:
60^1 = 60
60^0 = 1
sixties
ones
17
32
<|||||||
<<<||
Now, we have our answer.
So, 1052 in Hindu-Arabic numerals is rewritten as <||||||| <<<|| in Babylonian numerals.
Note:
Remember to put a space between the place value positions when typing these numerals on
quizzes or exams.
Another Note:
Please keep in mind, as you stuggle with this Base 60 system, that a Babylonian would likely
struggle with our Base 10 system. Having grown up in the culture of Base 60, this system would
be as natural to them as ours is to us.
Also, you work in Base 60 pretty frequently, probably without realizing it.
For example, you will have 90 minutes to complete your exams in this class. This is something
that we very easily translate to 1 hour and 30 minutes. Which is 1 x 60 + 30 x 1. Well what do
you know... Base 60!