Transcript Number Systems - Muskingum University

Number Systems
Tally, Babylonian, Roman
And
Hindu-Arabic
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
Tally with |
1
|
2
||
3
|||
4
||||
5
6
7
||||| ||||| | ||||| ||
●
●●
●●●
●●●●
●●●●●
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:
|||||||||||||||||||||||
24
23
||||||||||||||||||||||||
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
|||
|||
What number is represented by the
following symbols?
10,634
||
||
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.
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
0
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
What do the following represent?
(260) + (20+4)=144
30+5=35
(3060)+(10+3)
(23600)+(160)+(30+8)
1800+13
7200+60+38
1813
7298
How do you write each of the following numbers?
347
1571
34760 = 5 remainder 47
157160 = 26 remainder 11
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
Base-ten Value
I
1
V
5
X
10
L
C
D
M
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:
MMDCCCLXVII
M
M D
C L X V I I
C
C
1000+1000+500+100+100+100+50+10+5+1+1=2867
MCDXCIV
M
CD
XC
IV
1000+(500-100)+(100-10)+(5-1)=1000+400+90+4=1494
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
61000
3
610
+ 300
+ 3100
2
+ 310
+ 70
+ 710
1
+ 710
+4
+4
+4
The last row would be called the base-ten expanded notation of the number 6374.
Write each of the numbers below in expanded notation.
a) 82,305
= 810,000 + 21,000 + 3100 + 010 + 51
= 8104 + 2103 + 3102 + 5100
b) 37.924
= 310 + 71 + 9(1/10) + 2(1/100) + 4(1/1000)
= 3101 + 7100 + 910-1 + 210-2 + 410-3
Write each of the numbers below in standard notation.
a) 6105 + 1102 + 4101 + 5100
= 600,000 + 100 + 40 + 5
= 600,145
b) 7103 + 3100 + 210-2 + 810-3
= 7000 + 3 + .02 + .008
= 7003.028
Base
Symbols
2
0,1
3
0,1,2
4
0,1,2,3
5
Place Values as Numbers Place Values as Powers
… , 16, 8, 4, 2, 1
… , 24, 23, 22, 21, 1
… , 81, 27, 9, 3, 1
… , 34, 33, 32, 31, 1
… , 256, 64, 16, 4, 1
… , 44, 43, 42, 41, 1
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
…,8 ,8 ,8 ,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
3
2
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:
10324 is a legitimate base four number “Read 1-0-3-2 base four”
15424 is not a legitimate base four number not allowed 4 or 5
Base
Four
Base
Ten
Dienes
Blocks
Base
Four
Base
Ten
Dienes
Blocks
14
1
1 unit
214
9
1 unit
2 longs
24
2
2 units
224
10
2 units
2 longs
34
3
3 units
234
11
3 units
2 longs
104
4
304
12
114
5
1 unit
1 long
314
13
1 unit
3 longs
124
6
2 units
1 long
324
14
2 units
3 longs
134
7
3 units
1 long
334
15
3 units
3 longs
204
8
1004
16
1 long
2 longs
3 longs
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, ….
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.
23675
2105
12334
111
23676
2115
13004
112
23677
23678
2125
2135
13014
13024
113
114
23679
2145
13034
115
23680
2205
13104
116
23681
2215
13114
117
Notice that
when the
numbers
convert
they stay in
the same
order.
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
13024 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=
0
3  16 =
48
1  64 =
+ 64
114
Lets convert some of these other numbers to base 10.
20123
2748
21=
2
41=
4
13=
3
87=
56
09=
0
2  64 =
+ 128
2  27 =
+ 54
188
59
2748 = 2×82 + 7×81 + 4×80
246710 = 246 r 7
remainders
quotients
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.
quotients
Converting a number to a different base
remainders
20123 = 2×33 + 0×32 + 1×31 +2×30
593 = 19 r 2
24610 = 24
r6
193 = 6 r 1
2410 = 2
r4
63 = 2 r 0
210 = 0
r2
23 = 0 r 2
2467
20123
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
29
(1024) (512)
28
27
(256) (128)
Change the base 2 number
1100112 to a base 10
(decimal) number.
1100112
11 =
12 =
04 =
08 =
116 =
132 =
1
2
0
0
16
32
51
26
(64)
25
(32)
24
(16)
23
(8)
22
(4)
21
(2)
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
47 = 1011112
20
(1)
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
Convert T3E12 to base 10.
E1 = 11 1 = 11
312 = 3 12 = 36
T144 = 10144 = 1440
1477
Write the base 16 number A2D16 in expanded form and convert it to base ten.
In expanded form A2D16 is:
A2D16
A×162 + 2×161 + D×160
10×162 + 2×161 + 13×160
D1 = 13 1 = 13
216 = 2 16 = 32
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 32045 to base 16.
1st convert 32045 to base 10
32045
4×1
0×5
2×25
3×125
=
4
=
0
= 50
= 375
429
2nd convert 429 to base 16
429  16 = 26 remainder 13 = D
26  16 = 1 remainder 10 = A
1  16 = 0 remainder 1 = 1
We get the following: 32045 = 1AD16