number representation. representing numbers n numbers are represented as successive powers of a...
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Representing numbers
Numbers are represented as successive powers of a base, or radix
The powers increment upwards to the left, starting with zero to the far right
Representing numbers
Numbers are represented as successive powers of a base, or radix
The powers increment upwards to the left, starting with zero to the far right
For any base, b:
b0b1b2b3etc. . . .
Representing numbers
We typically use a base of 10 (presumably because we have ten fingers), the decimal system
100101102103etc. . . .
ones column
tens column
hundreds
column
thousands
column
Representing numbers
For any base b, there are b digits In base 10, there are 10 digits – 0 through 9
Representing numbers
For any base b, there are b digits In base 10, there are 10 digits – 0 through 9
100101ones column
tens column
When we want to represent a value greater than the highest digit, we have to make a change in the appropriate column to the left
1234567891 70892 0
etc. . . .
Representing numbers
In any base b, if we have n digits, the range of values we may represent is
bn
If we have two digits in base 10, b=10 and n=2
With two digits, we can represent 10 , or 100, values – 0 through 99
2
Representing numbers in computers Computers may store and transmit numbers
in the form of circuits A circuit has two states: ON and OFF
Representing numbers in computers Computers may store and transmit numbers
in the form of circuits A circuit has two states: ON and OFF Computers are therefore able to represent
numbers in a system that has two digits
Representing numbers in computers Computers may store and transmit numbers
in the form of circuits A circuit has two states: ON and OFF Computers are therefore able to represent
numbers in a system that has two digits Base two, the binary system, fits this
description
Representing numbers in computers Computers may store and transmit numbers in
the form of circuits A circuit has two states: ON and OFF Computers are therefore able to represent
numbers in a system that has two digits Base two, the binary system, fits this description The binary number system has the digits 0 and 1
Binary numbers
Binary numbers have only two digits But that distinction aside, the system of
representing numbers is exactly the same as in the decimal system:
20212223etc. . . .
ones column
twos column
fours column
eights column
24sixteen
s column
Binary numbers
With only two digits, a sequence of binary numbers changes columns more quickly than a series of decimal numbers:
Decimal Binary 0 01 12 103 114 1005 1016 1107 1118 1000
etc. . . .
Essential terminology
When a binary number is used by a computer, a single digit is called a bit (short for binary digit)
10110010
bit
Numbers (computer words) are often stored in sequences of eight bits, called a byte
byte
A sequence of four bits is called a nibble
nibble
Essential terminology
The bit that represents the lowest power is called the least significant bit
10110010
least significant bit
The bit that represents the highest power is called the most significant bit
most significant bit
Essential terminology
Larger numbers need to be represented with two or more bytes (16 bits form a two-byte word)
10110010
least significant byte
The byte that represents the lower powers of the number is called the least significant byte
most significant byte
10110010
The byte that represents the higher powers of the number is called the most significant byte
Bit resolution
A computer system is often referred to as an “n bit system,” meaning it represents numbers with n digits
Bit resolution
A computer system is often referred to as an “n bit system,” meaning it represents numbers with n digits
In a binary system, b=2, so this is a statement to describe the resolution of the system
Bit resolution
A computer system is often referred to as an “n bit system,” meaning it represents numbers with n digits
In a binary system, b=2, so this is a statement to describe the resolution of the system
An 8-bit system can represent 2 , or 256, values8
Bit resolution
A computer system is often referred to as an “n-bit system,” meaning it represents numbers with n digits
In a binary system, b=2, so this is a statement to describe the resolution of the system
An 8-bit system can represent 2 , or 256, values A 16-bit system can represent 2 , or 65,136 values
8
16
Hexadecimal notation
Base 16, the hexadecimal system, is often used in computer parlance Since our Arabic number system does not have digits to represent values
greater than 9, alphabetic characters are used:
Decimal Hexadecimal
0 01 12 23 34 45 56 67 78 89 910 A11 B12 C13 D14 E15 F
Hexadecimal notation
Hexadecimal notation is a convenience A four-bit nibble can be expressed as one hexadecimal bit
Hexadecimal notation
Hexadecimal notation is a convenience A four-bit nibble can be expressed as one hexadecimal bit An eight-bit byte can be expressed as two hex bits
10100110
A 6
To convert to decimal, multiply the most significant nibble by 16, then add the least significant nibble:
(10 * 16) + 6= 166