1. Tally, Babylonian, Roman And Hindu-Arabic
Number Systems
Tally, Babylonian, Roman
And
Hindu-Arabic

2. 1 2 3 4 5 6 7 | || ||| |||| ||||| ||||| | ||||| ||
The number system we use today to represent numbers has resulted from innovations during various times in history to be one of the most concise efficient ways to represent numbers. This section looks at the developments that have taken place in number systems throughout the years.
Tally Systems
The tally system used one character (usually a dot (●) or a stick (|) to stand for each unit represented.
Our Number 1 2 3 4 5 6 7
Tally with | | ||
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Tally with ● ●
● ●
● ● ●
● ● ● ●
● ● ● ● ●
● ● ● ● ● ●
● ● ● ● ● ● ●
The advantage of a tally system is that is easy to understand. Some disadvantages are that it is difficult to write really big numbers (i.e. 6472) and it is hard to distinguish numbers right away:
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3. Egyptian Numeration Systems
The early Egyptians solved the problem of how to represent big numbers with a smaller number of symbols. Different symbols were assigned specific values. Writing down the number would mean to adding the values of the symbols together.
Symbol
Name
Value |
staff 1
heel bone 10
scroll
100
lotus flower
1,000
finger
10,000
fish
100,000
The symbols below represent the number 24,356
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What number is represented by the following symbols?
10,634
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This advantage of this system is that it did enable people to write large numbers in a short amount of space. The problem is that new symbols were introduced for bigger numbers and numbers like 99,999 used many symbols.

4. Symbol Value 1 10 Babylonian Numeration System
The Babylonians were able to make two important advancements in how numbers are expressed.
1. They used only two symbols, one to represent 1 and the other to represent 10. Later they introduced a third symbol that acted like 0.
2. They introduced the concept of place value. This has to do with where a symbol is positioned determines its value. If positioned in one place it would have a different value than in another place.
The system that was used was a base 60 system. The symbol furthest to the right represented ones. The symbols second from the right represented groups of 60. The symbols third from the right represented groups of 3600 (6060). The groups were initially separated by a space later by the symbol for 0.
Symbol
Value 1 10
The symbols below represent the number 697.
10+1=11
We have 11 groups of 60.
1160=660
30+7=37
We have 37 ones.
371=37
660+37=697

5. What do the following represent?
30+5=35
(260) + (20+4)=144
(3060)+(10+3)
1813
(23600)+(160)+(30+8)
7298
How do you write each of the following numbers?
347
34760 = 5 remainder 47
1571
157160 = 26 remainder 11

6. I V X L C D M 1 5 10 50 100 500 1000 Roman Numeral Base-ten Value
Roman Numeration System
The Romans devised a system that used an addition/subtraction method for writing numbers. They had only 7 letters that stood for numbers given in the table below. To limit the number of symbols the Romans said that a symbol could not be used more than 3 times.
Roman Numeral I V X L C D M
Base-ten Value 1 5 10 50
100
500
1000
To find the value of a Roman numeral start at the left adding the numerals that are of equal or lesser value as you move to the right. If you find a numeral of smaller value than the numeral to its right subtract it from the one to the right.
Example: M M D C C C L X V I I
MMDCCCLXVII
=2867 M CD XC IV
MCDXCIV
1000+( )+(100-10)+(5-1)= =1494

7. Base-Ten Place-Value System
The sleek efficient number system we know today is called the base-ten number system or Hindu-Arabic system. It was first developed by the Hindus and Arabs. This used the best features from several of the systems we mentioned before.
1. A limited set of symbols (digits). This system uses only the 10 symbols:0,1,2,3,4,5,6,7,8,9.
2. Place Value. This system uses the meaning of the place values to be powers of 10.
For example the number 6374 can be broken down (decomposed) as follows:
6 thousands
3 hundreds
7 tens
4 ones
6000
+ 300
+ 70
+ 4
61000
+ 3100
+ 710
6103
+ 3102
+ 7101
The last row would be called the base-ten expanded notation of the number 6374.

8. Write each of the numbers below in expanded notation.
= 810, 1,  10 + 51
= 8   100 b)
= 310 + 71 + 9(1/10) + 2(1/100) + 4(1/1000)
= 3    10-3
Write each of the numbers below in standard notation.
a) 6   100
= 600,
= 600,145
b) 7   10-3 = =

9. Base
Symbols
Place Values as Numbers
Place Values as Powers 2
0,1
… , 16, 8, 4, 2, 1
… , 24, 23, 22, 21, 1 3
0,1,2
… , 81, 27, 9, 3, 1
… , 34, 33, 32, 31, 1 4
0,1,2,3
… , 256, 64, 16, 4, 1
… , 44, 43, 42, 41, 1 5
0,1,2,3,4
… , 125, 25, 5, 1
… , 53, 52, 51, 1 6
0,1,2,3,4,5
… , 216, 36, 6, 1
… , 63, 62, 61, 1 7
0,1,2,3,4,5,6
… , 343, 49, 7, 1
… , 73, 72, 71, 1 8
0,1,2,3,4,5,6,7
… , 512, 64, 8, 1
… , 83, 82, 81, 1 9
0,1,2,3,4,5,6,7,8
… , 729, 81, 9, 1
… , 93, 92, 91, 1 10
0,1,2,3,4,5,6,7,8,9
… , 1000, 100, 10, 1
… , 103, 102, 101, 1
Writing Numbers in Other Bases
A number in another base is written using only the digits for that base. The base is written as a subscripted word after it (except base 10).
For Example: is a legitimate base four number “Read base four”
15424 is not a legitimate base four number not allowed 4 or 5

10. Base Four Ten Dienes Blocks 14 1 24 2 34 3 104 4 114 5 124 6 134 7 204
1 unit 24 2
2 units 34 3
3 units
104 4
1 long
114 5
124 6
134 7
204 8
2 longs
Base
Four
Ten
Dienes
Blocks
214 9
1 unit
2 longs
224 10
2 units
234 11
3 units
304 12
3 longs
314 13
324 14
334 15
1004 16
1 flat
Notice that the numbers in go in order just like in base 10 but only using the symbols 0, 1, 2, 3. In base 4 numbers are grouped in blocks 1, 4, 16, ….

11. Notice that when the numbers convert they stay in the same order.
We can use this different number system to illustrate what it is like to try to learn to count. Give the three numbers that come before and the three numbers that come after each of the numbers below.
23678
23675
2135
2105
13024
12334
114
111
Notice that when the numbers convert they stay in the same order.
23676
2115
13004
112
23677
2125
13014
113
23679
2145
13034
115
23680
2205
13104
116
23681
2215
13114
117
Converting a number to base 10
This process is a combination of multiplication and addition. You multiply each digit by its place value and add up the results. Convert to base 10.
In expanded form this number is given by:
13024 = 1×43 + 3×42 + 0×41 + 2×40
13024
2  1 = 2
0  4 =
3  16 = 48
1  64 =
+ 64
114

12. Lets convert some of these other numbers to base 10. 20123 2748
2  1 = 2
1  3 = 3
0  9 =
2  27 =
+ 54 59
4  1 = 4
8  7 = 56
2  64 =
+ 128
188
2748 = 2×82 + 7×81 + 4×80
20123 = 2×33 + 0×32 + 1×31 +2×30
Converting a number to a different base
To convert a number from base 10 to a different base you keep dividing by the base keeping tract of the quotients and remainders then reversing the remainders you got. The examples to the right first show how to convert a base 10 number 2467 to base 10. Then how you convert 59 to base three. (Notice 59 agrees with what we got for the base three number above.
remainders
remainders
quotients
quotients
246710 = 246 r 7
24610 = r 6
2410 = r 4
210 = r 2
593 = 19 r 2
193 = r 1
63 = r 0
23 = r 2
2467
20123

13. Base Two
The important details about base 2 are that the symbols that you use are 0 and 1. The place values in base 2 are (going from smallest to largest):
210
(1024) 29
(512) 28
(256) 27
(128) 26
(64) 25
(32) 24
(16) 23
(8) 22
(4) 21
(2) 20
(1)
Change the base 2 number to a base 10 (decimal) number.
Change the base 10 (decimal) number 47 to a base 2 (binary) number.
47  2 = 23 remainder 1
23  2 = 11 remainder 1
11  2 = 5 remainder 1
5  2 = 2 remainder 1
2  2 = 1 remainder 0
1  2 = 0 remainder 1
11 =
12 =
04 =
08 =
116 = 16
132 = 32 51
47 =

14. Base 12 and 16
For bases that are larger than 10 we need to use a single symbol to stand for the "digits" in a number that represent more than 10. This is because if you use more than one symbol the place values will get off. In particular, bases 12 and 16 are sometimes very useful.
In base 12 the digit 10 is represented with a letter T and the digit 11 is represent with a letter E.
In base 16 the letters A, B, C, D, E, F represent the digits 10, 11, 12, 13, 14, 15 respectively.
Base
Symbols
Place Values as Numbers
Place Values as Powers 12
0,1, 2, 3, 4, 5, 6, 7, 8, 9, T, E
… , 144, 12, 1
… , 122, 121, 1 16
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F
… , 256, 16, 1
… , 162, 161, 1
T3E12
E1 = 11 1 =
312 = 3 12 =
T144 = 10144 = 1440
1477
Convert T3E12 to base 10.

15. Write the base 16 number A2D16 in expanded form and convert it to base ten.
In expanded form A2D16 is:
A× ×161 + D×160
10× × ×160
A2D16
D1 = 13 1 =
216 = 2 16 =
A256 = 10256 = 2560
2605
Converting from Base to Base
If we wish to convert from one strange base to another we do this by "going through" base ten. In other words, for example if we want to convert from base 5 to base 16, first convert base 5 to base ten then convert that base ten number to base 16.
Example, Convert to base 16.
1st convert to base 10
2nd convert 429 to base 16
32045
429  16 = 26 remainder 13 = D
26  16 = 1 remainder 10 = A
1  16 = 0 remainder 1 = 1
4×1 =
0×5 =
2×25 =
3×125 = 375
429
We get the following: = 1AD16