- Recall:
(why a computer must use binary numbers)
- Computer memory consists of
electronical switches
that can be in
the (1)
on state
(representing the
digit 1) or
the (2)
off state
(representing the
digit 0)
Simpler stated:
- A computer can only "write" the digits 0 and 1 !!!
- A series of
binary digits (bits) forms
a binary number
Example:
10001001 01010010
- A byte =
a group of 8 bits
A byte is the smallest addressable (= identifiable) memory unit
Since computer memory can only store binary numbers, one could wonder how a computer can store non-numerical information - like letters, pictures, sounds, etc...
At the heart of the representation system used by the computer is the binary number system used by the computer to represent values using binary numbers
We will now study the following question:
- What (value) does a binary number represent ???
- Computer memory consists of
electronical switches
that can be in
the (1)
on state
(representing the
digit 1) or
the (2)
off state
(representing the
digit 0)
- In the following webpages, we
discuss how to
assign values to
binary numbers
I.e., we will study the binary number system
(Some of you may have studied this topic in high school before)
- Number System:
(From: Wikipedia:
click here )
- A numeral system =
a writing system
for expressing numerical values
A number system is a mathematical notation for representing numbers using digits or other symbols in a consistent manner.
- A numeral system =
a writing system
for expressing numerical values
- There are many different
number systems possible.
Two number systems are in common use:
- The decimal number system:
- The decimal number system is
based on the
number 10
- The decimal number system uses
10 digits:
0 1 2 3 4 5 6 7 8 9
- Humans are trained from childhood (in elementary school) to use the decimal number system
- The decimal number system is
based on the
number 10
- The binary number system:
- The binary number system is
based on the
number 2
- The binary number system uses
2 digits:
0 1
- Computers
uses the binary number system
because
a computer can
only "write" (= remember)
2 different digits:
- Computer memory consists of switches
- A switch can only represent 0 (= off state) and 1 (= on state)
- The binary number system is
based on the
number 2
- The decimal number system:
- The decimal (deca = 10) number system uses
10 digits to
represent
10 different values.
The values that we need to represent in the decimal (= 10) number system are:
( ) (•) (••) (•••) (••••) (•••••) (••••••) (•••••••) (••••••••) (•••••••••)
The values are represented by the number of dots between the brackets ( ) !!!
- The symbols
used to represent these
10 values are:
( ) (•) (••) (•••) (••••) (•••••) (••••••) (•••••••) (••••••••) (•••••••••) 0 1 2 3 4 5 6 7 8 9
All of you have learned to recognized these 10 symbols and associate the assigned value when you were kids (in elementary schools !!!)
(That's not the case for someone that is borned and raised inside - e.g., the Amazon jungle !!!)
- A decimal number is
made up of
a number of decimal digits
In elementary school, you have learned that:
- The value of
a digit depends on its
position in the
(decimal) number
- The value of a digit is multiplied by 10 for each position to the left
Example:
2 represents the quantity (••) 3 represents the quantity (•••)
23 represents the quantity: (•• •••) •• •• •• •• •• •• •• •• •• ^ | ten of (••) If you count the total number of •, you will get: twenty three •'sNote:
- Many text book explains
the value represented by 23 as
follows:
23 = 2 × 10 + 3What these text book meant is:
- Take the value that
is represented by
2 and
put down
ten times
as many of them:
Value represented 2 is: •• Put down ten times: •• •• •• •• •• •• •• •• •• •• - Then add the
value that is
represented by
3
•• •• •• •• •• •• •• •• •• •• ••• (= 3) - Count all the • to find the value that is represented by 23
- Take the value that
is represented by
2 and
put down
ten times
as many of them:
- The value of
a digit depends on its
position in the
(decimal) number
- The binary number system "works"
in a similar manner as
the good old
decimal number that you
are familiar with;
except that
we use 2 values and
2 (different) digits:
- The binary (bini = 2) number system
uses
2 digits to
represent
2 different values.
- The values that we need to
represent in the
binary (= 2) number system are:
( ) (•)
The values are represented by the number of dots between the brackets ( ) !!!
- We use the symbols "0" and "1"
to represent these
2 values:
( ) (•) 0 1
- The binary (bini = 2) number system
uses
2 digits to
represent
2 different values.
- A binary number is
made up of
a number of binary digits
Example:
0 1 100 10101Just like the decimal number system:
- The value of
a digit
in a binary number
also depends on its
position in the
(binary) number:
- The value of a binary digit is multiplied by 2 for each position to the left
Example:
1 represents the quantity (•) 1 represents the quantity (•)
11 represents the quantity: (• •) • ^ | 2 of (••) If you count the total number of •, you will get: three •'s Therefore: 11 in the binary number system represents the value three - The value of
a digit
in a binary number
also depends on its
position in the
(binary) number:
- Recall:
- Computer memory consists
of electronical switches that
can be switched
on or
off
- The "on"-state
represents the
binary digit (bit) 1
The "off"-state represents the binary digit (bit) 0
- A group of 8 bits (switches)
form a byte
A byte is the unit of computer memory in todays computers
- Computer memory consists
of electronical switches that
can be switched
on or
off
- Notice:
-
We can consider
a group of
bits as
a binary number !
Example:
00000000 is the binary number (= a number is th
-
We can consider
a group of
bits as
a binary number !
- Binary number:
- A binary number = a
number in the
binary number system
A binary number must consist of only binary digits !!!
Examples of binary numbers:
0 = 00 = 000 = 0000 Note: leading 0's can be omitted 1 = 01 = 001 = 0001 10 11 100 10101 ....
- A binary number = a
number in the
binary number system
- In contrast,
humans are
trained (= conditioned/"brainwashed")
to use
decimal digits
(= 0, 1, 2, 3, 4, 5, 6, 7, 8, 9) and the
decimal number system
Examples of numbers in the decimal number system:
4 = 04 = 004 = 0004 Note: leading 0's can be omitted 9 = 09 = 009 = 0009 10 11 100 10101 ....
- We can apply the binary number rule
above:
- The value of a binary digit is multiplied by 2 for each position to the left
to find the (decimal) value that is represented by a binary number:
Given a binary number: dndn-1...d1d0 where di are the digits in the binary number The decimal value represented by the binary number is: dn×2n + dn-1×2n-1 ... + d1×21 + d0×20Example:
Binary number: 10101 After numbering the digits and its position: d4d3d2d1d0 1 0 1 0 1 We can use the formula to find the decimal valure represented by 10101: d4×24 + d3×23 + d2×22 + d1×21 + d0×20 Using d4=1, d3=0, d2=1, d1=0, d0=1, we get: 1×24 + 0×23 + 1×22 + 0×21 + 1×20 = 1×16 + 0×8 + 1×4 + 0×2 + 1×1 = 21 (decimal)
- Here are a short list
of different values and their
representation in the
decimal number system
and the binary number system:
Representation in the Representation in the Value Decimal number system Binary number system ----------------------------------------------------------------- Zero 0 0 One 1 1 Two 2 10 Three 3 11 Four 4 100 Five 5 101 Six 6 110 Seven 7 111 Eight 8 1000 Nine 9 1001 Ten 10 1010 Eleven 11 1011 Twelve 12 1100 Thirteen 13 1101 Fourteen 14 1110 Fifteen 15 1111Do you get what this T-shirt is saying:
It actually says:
- There are two types of people
in the world: those who understand binary and those who don't.
(You have to read 10 as a binary number !!!)
- There are two types of people
in the world: those who understand binary and those who don't.
- Notice that:
- Some representations
(such as 10)
can occur in
different number systems
- The same representation
in different number system
represents
different values !!!
• 10 in the decimal number system represents the value ten • 10 in the binary number system represents the value two
- Some representations
(such as 10)
can occur in
different number systems
-
To avoid this ambiguity,
we use the following
subscript notation
to indicate
the number system used:
• 10(10) = 10 in the decimal number system • 10(2) = 10 in the binary number system