Survey of Math - MAT 140
Page: 1
Our System and Early Positional Systems
Number - a quantity
Numeral - a symbol that represents a quantity.
Our numeration system is based on the Hindu-Arabic symbols and a Base of 10 (probably because we have 10
ngers...what would happen if we counted with our toes also??). For short we will just call our system, H-A.
In H-A we can Expand our numeration to better understand it. For instance, the number
3; 507 D 3
1000 C 5
100 C 0
10 C 7
1: Notice that each place could be replaced:
3 1000 C 5
3 103 C 5
100 C 5
102 C 5
10
101
C 7 1
C 7 100
So as we work from Right to Left, the value of the next place value is 10 times the previous one. To understand our
place value system better we will look at Ancient Methods and how they worked.
1
Babylonian Numeration
The Babylonian's wrote most of their history in Clay Tablets using a "stick" with a thin triangle as its end. In this way
they wrote their only numerals:
1
10
This is all they had in symbolic form. But the way that they used them was impressive.
To write 59, they would write 5-10's and 9-1's
I wonder how they would write 61?
To accomplish this they used a base of 60 (like 60 seconds in a minute and 60 minutes to an hour. I wonder if the
Babylonian's had anything to do with this??) So how much time passes after 61 seconds? Well that would be one
minute and one second. Thus in Babylonian they would write it like this:
Notice that there is more space between these to "1's", representing Two Place Values. The rst place value (starting
on the far Right) is always the One's and the second (the one on the Left) is the 60's place. That means there is one
60 and one 1, in other words 61:
Copyright 2007 by Tom Killoran
Survey of Math - MAT 140
Page: 2
Example 1 What number in H-A does this Babylonian Number represent?
There are Two Place Values (because of the one Gap). In the First Place the numeral is representing 11 (one ten and
one one) and in the Second Place value the numeral is representing 23 (two tens and three ones). Thus:
23
600 s
in expanded form: .23
60/ C .11
11
10 s
1/ D 1391:
So this Babylonian Numeral is the Number 1391:
1.1
Subtraction Notation
Now they realized after time that some of their numbers could become quite long (like 59). So they also developed a
method to write a number using subtraction. Their symbol of subtraction is:
so to represent 59 would be simple as
6-10's minus 1.
D 59
Example 2 What quantity is represented by the following?
There are Four Place Values in this numeral.
1
603 D 216; 0000 s
20 3 D 17
602 s D 36000 s
10 C 4 D 14
600 s
40
2 D 38
10 s
So the number in H-A would be:
1
1.2
603 C 17
602 C .14
60/ C .38
1/ D 278 ; 078
Converting H-A into a Babylonian Numeral
Changing "Our" Numbers into Babylonian is a little harder. Above all we needed to know was how many place values
in "Their" system and multiply their "digits" to their respective place value and add these up. Thus to go backward,
we will have to use opposite operations. But the most important thing to realize is that we are still interested in how
many place values there will be in Their system!!!
Copyright 2007 by Tom Killoran
Survey of Math - MAT 140
Page: 3
Example 3 What is 29; 495 in Babylonian?
First we will have to nd out how many place values this number will be in the Babylonian system. To do this construct
the place values starting with 1, until you get to one that is Too Big:
x
216000's 3600's 60's
1's
For the numeral 29; 495; we will need Three Place Values in Babylonian Base 60. Starting with the largest we need
to nd out how many 3600 are in 29; 495.
This question is answered by long division:
8
3600Þ29, 495
-28,800
695
This means that there are 8; 36000 s for the third place in the Babylonian Number, with a remainder of 695; that will
be "absorbed" in the next place value...
by nding out how many 600 s are in 695
11
60Þ695
-660
35
Which would be 11 for the 60's place and 35 for the 10 s place. Thus
8
3600Þ29, 495
29; 495 D
-28,800
695
11
60Þ695
-660
35
29; 495 D .8
60/ C .35
3600/ C .11
35
1/
or in Babylonian:
1.2.1
Finding Whole Answer and Remainder using a Calculator
For the next one I am going to use a calculator technique to nd the remainder, but rst lets look at a simple example:
237
27
8
27Þ237
-216
21
=
8 21
27
Copyright 2007 by Tom Killoran
Survey of Math - MAT 140
Page: 4
21
left. The 21 is the number that I am interested in
27
21
obtaining, so if we multiply by the denominator (what we divided by to begin with)
27 D 21 we would have the
27
remainder.
Now if we removed the whole number 8; we would have
Try this: On a calculator nd
237
D 8: 777 777 77
27
From this subtract the Whole Number part .8/ (just type in Minus and 8):
AN S
8 D 0: 777 777 77
Then multiply by what we originally divided by .27/
AN S
27
and what you get is: 21 the remainder.
Example 4 What is 1; 438; 279 as a Babylonian Number?
Again we must rst concern ourselves with How Many Place Values in Their System?
Lets see, starting from one and multiplying by 60 to get to the next place we get:
x
12960000's 216000's 3600's 60's
1's
where 604 D 12; 960 ; 000 is WAY too large. So this Babylonian Numeral will have 4-place values in it:
603 ; 602 ; 60; 1
So we start with the 603 s place: (this will be shown by calculator method)
1438279
D 6: 658 699 074
603
Next subtract the 6 away from answer to get:
AN S
and multiply by what you divided by:
AN S
6 D 0: 658 699 074
603 D 142 279
the remainder for the next place value is 142279:
Repeating the process again for the 602 s and the remainder above we get:
142 279
D 39: 521 944 44
602
AN S 39 D 0: 521 944 44
AN S 602 D 1879
Copyright 2007 by Tom Killoran
Survey of Math - MAT 140
Page: 5
Repeating again for the 60's
1879
D 31: 316 666 67
60
AN S 31 D 0: 316 666 67
AN S 60 D 19
The quotients of each division are as follows: 6; 39; 31; 19 so we get:
1; 438; 279 D 6
603 C 39
602 C .31
60/ C .19
1/
or in Babylonian:
2
Mayan Numeration
The Numerals of the Mayan Culture were:
H A
Mayan
1
5
0
where they wrote their numbers vertically, with the highest place values at the top. What is amazing is that they where
one of the rst cultures to have a symbol for zero, which looks like an Empty Bowl. They used it mostly as a place
holder, but the concept was there for more stringent mathematics.
The number 17 in Mayan would be:
2-1's and 3-5's
The largest single digit that they would write would be 19 because their Place Value system was "sort-of" based on 20
(counting your toes and ngers), but there is a strange occurrence in their third place value. It should be:
202 s
200 s
10 s
but the clergy used this system more like a calendar, where the Left most was "years" the middle was "months" and
the Right was "Days".
To make this work with a 360 days a year calendar, they changed the 400's place to the 18
that their place value system became:
3600 s 200 s 10 s
and then they continued each successive place by multiplying by 20 again.
Copyright 2007 by Tom Killoran
20 D 360s place. So
Survey of Math - MAT 140
Page: 6
Example 5 What numeral does this Mayan Number represent?
There are Three Place Values in this Number, thus we get:
.13
360/ C .0
13
360
0
20
16
1
20/ C .16
D
1/ D 4696
Example 6 What does this Mayan Number represent in H-A (Hindu-Arabic)?
There are Four Place Values in this number, so we have at the top the 360
nally the 1's
3
3
2.1
.360
20/ C .13
360/ C .7
200 s , then the 360's, then 20's, and
20/
.360
13
360
7
20
0
1
20/ C .0
D
1/ D 26 420
Converting H-A (ours) to Mayan (theirs)
This has the same pattern as converting to Babylonian. First nd out How Many Place Values in Their System,
then divide and nd remainders...
Copyright 2007 by Tom Killoran
Survey of Math - MAT 140
Page: 7
Example 7 Convert 6; 138 to Mayan.
Find the number of place values in the Mayan System, we again start with one and build each larger place value:
x
7200's
360's
20's
1's
so there are 3 Place Values, 3600 s ; 200 s ; 10 s :
Next nd out how many 360's are in 6138
6138
D 17: 05
360
AN S 17 D 0:05
AN S 360 D 18
then nd out how many 20's are in the remainder of 18
18
D 0:9
20
in other words, there are no 20's in 18 (it is smaller than 20). Thus the value of the 20's is ZERO
Then the 18 just become how many ones are left:
18
Thus:
6138 D .17
360/ C .0
20/ C .18
1/
20's
1's
Example 8 Convert 54; 680 to Mayan:
Places:
x
144000's 7200's
360's
therefore there are 4 Place Values for this number:
How many 7200's are in 54680?
54680
D 7: 594 444 444
7200
AN S 7 D 0:594444444
AN S 7200 D 4280
Copyright 2007 by Tom Killoran
Survey of Math - MAT 140
Page: 8
How many 360's are in the remainder 4280?
4280
D 11: 888 888 89
360
AN S 11 D 0:88888889
AN S 360 D 320
How many 20's are in the remainder 320?
320
D 16:0
20
Since it went in evenly, there are no ones in this number:
0
Putting it all together:
54 680 D .7
3
7200/ C .11
360/ C .16
20/ C .0
1/
Creative System:
Lets say that the aliens have invaded, and the army was on vacation. The aliens numeration system has to be gured
out. What they give us is that their Base is
and the "digits" are f|; }; ~; •; zg
So how would we Expand their number: z|~•
There are 4 Place Values where
numeral in expanded form as:
is the base, thus we will need ,
z|~• D z
3
C |
2
3
;
C .~
Copyright 2007 by Tom Killoran
2
;
1
/ C .•
;
1/
0
D 1 , so we can write the
Survey of Math - MAT 140
Page: 9
Problems:
Exercise 1 Convert to H-A numeral:
Exercise 2 Convert to H-A numeral:
Exercise 3 Write the number 66; 487 in Babylonian Numerals.
Exercise 4 Write the number 91; 869 in Mayan Numerals
Copyright 2007 by Tom Killoran
Survey of Math - MAT 140
Page: 10
Answers:
Solution 1 Convert to H-A numeral:
.30
1/ .3600/ C .42/ .60/ C 7 D 106; 927
Solution 2 Convert to H-A numeral:
D 13 .7200/
D 5 .360/
D 0 .20/
D 18 .1/
13 .7200/ C 5 .360/ C 0 .20/ C 18 .1/ D 95 ; 418
Solution 3 Write the number 66; 487 in Babylonian Numerals.
x
216000's 3600's 60's
1's
66487
D 18: 4686111
3600
2/AN S 18 D 0:4686111
3/AN S 3600 D 1687
1/
cont:::
18, 28, 7=
Solution 4 Write the number 91; 869 in Mayan Numerals
x
144000's 7200's
360's
20's
1's
91869
D 12:75958333
7200
Ans 12 D 0:75958333
AN S 7200 D 5469
cont:::
12, 15, 3, 9. Vertically Top to Bottom:
Copyright 2007 by Tom Killoran