Arithmetic for computer
NUMBER SYSTEM
A number system is a basic symbol
to represent a set of quantities.
1.1~ Basic types of
number system
1.2 ~ Number system base:
The maximum number that can be represented on
the single digit or number is called a base.
Ø DECIMAL
NUMBER
Arithmetic for computer
NUMBER SYSTEM
A number system is a basic symbol to represent a set of quantities.
1.1~ Basic types of
number system
1.2 ~ Number system base:
The maximum number that can be represented on
the single digit or number is called a base.
Ø DECIMAL
NUMBER
ü Most widely used by modern civilizations.
ü The prefix “deci-” stands for 10
ü Base of 10 because:
· There are 10 symbols that
represent quantities (0, 1, 2, 3, 4, 5, 6, 7, 8, 9)
· Each place value in a
decimal number is a
power of 10
ü The value of assigned weight is composed
by 10digits starting from 0 until 9.
ü The positive and negative values are
determined by their position weight structure(power of a number):
ü most basic number system that most machines (and
electricaldevices) use to communicate.
ü The hexadecimal number system is used as an intermediary
system in computers, such as arepresentation of memory addresses or a
representation of colors
1.3 ~
Number system conversion :
Converting Binary Numbers to Decimal
•Step1
– Starting with the 1s place,
write the binary place value over each digit in the binary number being converted
:
•Step2
– Add up all of
the place values that have a “1” in the binary number
*EXAMPLE
Convert the following binary number to decimal number :
Converting decimal Numbers to binary
• There are two
methods that can be used to convert decimal numbers to binary:
v Repeated
subtraction method
v Repeated division
method
•
Repeated
subtraction method
As an explanation of the repeated subtraction method, let’s
convert the decimal number 204 to binary.
Step 1:
Step 3:
Step 4:
Step 5:
* EXAMPLE
Convert the following decimal number
to binary using repeated subtraction method :
a) 18910
Repeated
division method
Step1:
• Divide the decimal number you’re
trying to convert by 2 in regular long division until you have a final
remainder.
Step2:
• Use the first remainder as the LEAST
SIGNIFICANT DIGIT of the binary number.
Step3:
•Divide the quotient you got from the
first division operation until you have a final remainder.
Step4:
•Use the remainder as the next digit
of the binary number.
Step5:
• Repeat Steps 3 & 4
as many times as necessary until you get a quotient that can’t be divided by 2.
Step6:
•Use the last remainder (the one that
can’t be divided by 2) as the MOST SIGNIFICANT digit.
* EXAMPLE
Convert the following decimal number to binary using repeated
division method
a) 33910
b) 59.62510
Converting Binary Numbers
to Hexadecimal
•Step1:
– Starting
with the LEAST SIGNIFICANT digit, mark off the digits in groups of
4.
– For example, to convert 110101001011 to hexadecimal, mark off the digits in
groups of
four.
• 1101 | 0100 | 1011
•Step2:
– Convert each
group of four digits to its hexadecimal character.
•HelpfulHint
– The last
group on the left can have anywhere from 1 to 4 binary digits group.
– If it will
help you see the pattern, you can fill in enough leading zeroes to make the
last group on the left have four digits.
– For example,
1 1 0 | 0 1 1 1 | 1 0 0 1 could be written
as
0 1 1 0 | 0 1 1 1 | 1 0 0 1
0 1 1 0 | 0 1 1 1 | 1 0 0 1
* EXAMPLE
Convert the following binary number to hexadecimal:
a) 11111012= 011111012
b) 10011111001.1012=010011111001.01002
Converting Hexadecimal to Binary Numbers
• Converting hexadecimal numbers to binary is just the reverse operation of converting binary to hexadecimal.
• Just convert
each hexadecimal digit to its four-bit binary pattern. The resulting set of 1s
and 0s is the binary equivalent of the hexadecimal number.:
* EXAMPLE
Convert the following hexadecimal to binary number:
a) AF025E16
b) 3B5D2A716
Converting Hexadecimal to Decimal
• Step 1
– Starting with the 1s place,
write the hexadecimal place value over each digit in the hexadecimal being converted
• Step 2
– Add up all of
the place values that have a value in the hexadecimal
* EXAMPLE
Convert the following hexadecimal to decimal:
a) 3DA416
b) 123D.216
Converting Decimal to Hexadecimal
• There are two methods to
choose from: –
Do a decimal-to-binary conversion and then a binary-to-hexadecimal
conversion.
– Do a direct
conversion using the repeated division method. • Since
this is a conversion to hexadecimal, 16 is the divisor each time.
decimal-to-binary conversion and
then a binary-to-hexadecimal conversion
* EXAMPLE
Convert 13610 to
hexadecimal
Step 1:
convert it to binary number
Step 2:
convert the binary number to hexadecimal
direct conversion
using the repeated division method
* EXAMPLE
Convert 83610 to
hexadecimal
Two's complement is a mathematical operation on binary numbers AND allows the use of binary arithmetic
operations on signed integers, yielding the correct 2's complement results.
1st and 2nd Complement of Binary Numbers
ü They are important because they permit the
representation of negative numbers
ü The method of 2nd complement
arithmetic is commonly used in computers handle negative numbers.
ü Assume that a microprocessor have 8 register bits,
figure below show the sign bit or the most significant bit ( MSB ) if the MSB
bit is 0 , then the number is positive (+) . Conversely, if the MSB is 1 , then
the number is negative (-). The others remaining 7 bits are represent as the
magnitude numbers.
1st Complement of Binary Numbers
v The 1st complement of a binary number is found by changing
all 1s to 0s and all 0s to 1s by using parallel inverters (NOT circuits).
v Example :
2nd Complement of Binary Numbers
v
The 2nd complement of a binary number is found by adding 1 to
LSB of the 1s complement.
v Example :
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