How Computers Represent
Information
Yong Choi
School of Business
CSU, Bakersfield
Digital Storage


Humans: decimal (or base 10) number system
Computer: binary (base 2) number system



On (“high” or 1)
Off (“low” or 0)
Each digit in a binary number is known as a bit
and can have only one of two values, 0 or 1.
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

One bit can store a number from 0 to 1
Two bits can store from 0 to 3
n bits can store numbers from 0 to 2n
Binary Numbers
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
Programmers often need to read numbers in
the computer because the number of bits in
calculation affects the accuracy and size
limitations of numbers manipulated by the
computer.
So, it is useful to understand how the binary
number system is used within the computer.

In Java, programmer can declare a variable to be
short (16 bits) and or long (64 bits) integer
depending on the anticipated size of the number
being used and required accuracy in calculations.
Advantages of Binary Numbers

Simple

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Unambiguous and clear signals

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Only two O and 1
Analog signal vs. Digital signal
Flawless copies can be made

Anything (English, Spanish, Chinese) can be
represented with patterns of bits
Hexadecimal Numbers

Important!! Because it is commonly used as a
shorthand notation for binary numbers.


Each hexadecimal number exactly represent 4
binary bits.
Most computers store and manipulate
instructions and data using word sizes that
are multiple of 4 bits.


Colors in VB can be specified as a six-digit
hexadecimal number.
http://www.mrs.umn.edu/committees/wwwac/tool
box/color.html
Review of Decimal Numbers


Decimal numbers are based on powers of 10.
Mathematically, the number 537 can be
calculated..
5 • 102 3 • 101 7 • 100
5 • 100
3 • 10
7•1
500
30
7
Similarly 9238 is
9 • 103 + 2 • 102 + 3 • 101 + 8 • 100
Converting Binary to Decimal

Easy, just multiply digit by power of 2


Binary numbers are based on power of 2.
Just like a decimal number is
represented
So, 101 binary is
1 • 22 + 0 • 21 + 1 • 20 = 5
or
1•4 + 0•2 + 1•1=5
Binary to Decimal Example
7
6
5
4
3
2
1
0
27
26
25
24
23
22
21
20
128 64
32
16
8
4
2
1
What is 10011100 in decimal?
1
0
0
1
1
1
0
0
128 + 0 + 0 + 16 + 8 + 4 + 0 + 0 = 156
27 *1 + 26 *0 ……………..
Converting Decimal to Binary
A little more work than binary to decimal

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Find largest power-of-two smaller than decimal
number
Make the appropriate binary digit a ‘1’
Subtract the power of 2 from decimal
Do the same thing again
Some examples
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3 = 2 + 1 = 11 (that’s 1•21 + 1•20)
5 = 4 + 1 = 101 (that’s 1•22 + 0•21 + 1•20)
Decimal to Binary Example

Convert 28 decimal to binary
28/2 = 12 ……. 0
12/2 = 10 ……. 0
10/2 = 5 ……… 0
5/2 = 4 ……….. 1
4/2 = 2 ……….. 0
2/2 = 1 ……….. 0
Answer: 1001000
Converting Hex to Binary

Hexadecimal table
Dec
Bin
Hex
Dec
Bin
Hex
0
0000
0
8
1000
8
1
0001
1
9
1001
9
2
0010
2
10
1010
a
3
0011
3
11
1011
b
4
0100
4
12
1100
c
5
0101
5
13
1101
d
6
0110
6
14
1110
e
7
0111
7
15
1111
f
Hex to Binary Example

Hex to Binary – just convert digits
2ac
0010 1010 1100
2ac = 001010101100
No magic!!
- remember each hex digit = 4 bits
Binary to Hex

Just convert groups of 4 bits
101001101111011
0101 0011  0111 1011
5
3
7
b
101001101111011 = 537b
Dec Hex
Hex to Decimal

Just multiply each hex digit by
decimal value, and add the results.
2ac
2 • 256

163
4096
+ 10 • 16 + 12 • 1 = 684

162
256

161
16

160
1
0
0
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
a
11
b
12
c
13
d
14
e
15
f
Decimal to Hex
1.
2.
3.
4.
Analogous to decimal  binary.
Find largest power-of-16 smaller
than decimal number
Divide by power-of-16. The integer
result is hex digit.
The remainder is new decimal
number.
Do the same thing again
Dec Hex
Decimal to Hex
0
0
1
1
2
2
3
3
4
4
5
5
6
6
2a_
7
7
8
8
2ac
9
9
10
a
11
b
12
c
13
d
14
e
684
2__
684/256 = 2
684/256 = 172
172/16 = 10 = a
172/16 = 12 = c

163
4096

162
256

161
16

160
1
Exercise Questions
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Convert decimal value 110 to
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Convert binary value 1100111 to
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Binary
Hexadecimal
Decimal
Hexadecimal
Convert hexadecimal value 6CAD to
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Decimal
Binary