The numeral system and arithmetic operations
Unlike the Egyptians, the mathematicians of the Old Babylonian period went far beyond the immediate challenges of their official accounting duties. For example, they introduced a versatile numeral system, which, like the modern system, exploited the notion of place value, and they developed computational methods that took advantage of this means of expressing numbers; they solved linear and quadratic problems by methods much like those now used in algebra; their success with the study of what are now called Pythagorean number triples was a remarkable feat in number theory. The scribes who made such discoveries must have believed mathematics to be worthy of study in its own right, not just as a practical tool.So , here we are going to explain to you about number system.
NUMBER SYSTEM…
Introduction to number system
Number system is a basic symbol to represent a set of quantities .There are many types of number system but in this topic we are going to focus only on three number system, which are:
• binary,
• decimal and
• hexadecimal;
1.1 NUMBER
SYSTEM BASE…
1.1.a) What is a base for number
system types ??
- Most of the numbering system will have a base.
- Base ~ maximum number that can be represented on the
single
digit or number.
TABLE
1-1: Types of Number system
SYSTEM
|
BASE
|
POSSIBLE
DIGITS
|
Binary
|
2
|
0
1
|
Octal
|
8
|
0
1 2 3 4 5 6 7
|
Decimal
|
10
|
0
1 2 3 4 5 6 7 8 9
|
Hexadecimal
|
16
|
0
1 2 3 4 5 6 7 8 9 A B C D E F
|
Binary number
- · Base 2.
- · The number consist only two digit 0 and 1 only.
- · The weight structure of binary number is :
Binary
Points
2 n – 1 …
23 22 21 20 . 2-1 2 -2
… 2 - n
·
The least significance bit (LSB)
and most significance bits (MSB) can be identified
based on the size of binary number itself.
MSB LSB
1000001110
- · Base of 10.
- · Value of the assigned weight composed by 10 digits (0 – 9).
- · Position weight structure will determined the positive values and negative values.
- · Base of 16.
- · Value of the assigned weight composed by 16 digits (0 until F)
- · The digits is suitable to present in fours bit number.
1.1 NUMBER
SYSTEM CONVERSION…
In this section, you will learn how
number system (binary, decimal, hexadecimal) will be convert. For your information, there are many
ways or method that can be used to convert those number system. But in this
blog, we will show what method that we used which exactly the systematic method
to convert the number system. We are going to show how number system is converted using repeated – division by base 2, base 10, and base 16.
TABLE 1-2 : Number
System Conversion
BINARY
|
DECIMAL
|
HEXADECIMAL
|
0000
|
0
|
0
|
0001
|
1
|
1
|
0010
|
2
|
2
|
0011
|
3
|
3
|
0100
|
4
|
4
|
0101
|
5
|
5
|
0110
|
6
|
6
|
0111
|
7
|
7
|
1000
|
8
|
8
|
1001
|
9
|
9
|
1010
|
10
|
A
|
1011
|
11
|
B
|
1100
|
12
|
C
|
1101
|
13
|
D
|
1110
|
14
|
E
|
1111
|
15
|
F
|
Example 1-1
Convert decimal
number 51.3125 to binary number.
Weight
|
26
|
25
|
24
|
23
|
22
|
21
|
20
|
2-1
|
2-2
|
2-3
|
2-4
|
Value
Represented
|
64
|
32
|
16
|
8
|
4
|
2
|
1
|
0.5
|
0.25
|
0.125
|
0.0625
|
Binary
(*)
|
0
|
1
|
1
|
0
|
0
|
1
|
1
|
0
|
1
|
0
|
1
|
51 – 32 = 19
19 – 16 = 3
3 – 2 = 1
1 - 1 = 0
0.3125 – 0.25 = 0.0625
0.0625 – 0.0625 = 0
Therefore, the conversion of 51.312510
= 110011.01012
or, you may try this method too.
A systematic method for
number conversion
Example 1-2
2
|
51
|
||
2
|
25
|
1
|
|
2
|
12
|
1
|
|
2
|
6
|
0
|
|
2
|
3
|
0
|
|
2
|
1
|
1
|
|
1
|
|||
0.625 x
2 = 1.25 1
0.25 x 2 = 0.5 0
0.5 x 2
= 1.0 1
(the answer should be read from bottom to top)
Therefore, the conversion of 51.312510 = 110011.01012
prepared by: nurul izzati nor rusham (B031310238)
prepared by: nurul izzati nor rusham (B031310238)
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