How Computers Represent Information

1
How Computers Represent Information

• Yong Choi
• School of Business
• CSU, Bakersfield

2
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.
• 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

3
Binary Numbers

• 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.

4
Advantages of Binary Numbers

• Simple
• Only two O and 1
• Unambiguous and clear signals
• Analog signal vs. Digital signal
• Flawless copies can be made
• Anything (English, Spanish, Chinese) can be
  represented with patterns of bits

5
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/toolbox/co
  lor.html

6
Review of Decimal Numbers

• Decimal numbers are based on powers of 10.
  Mathematically, the number 537 can be
  calculated..
• Similarly 9238 is
• 9 103 2 102 3 101 8 100

5 102 3 101 7 100
5 100 3 10 7 1
500 30 7
7
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

8
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 ..
9
Converting Decimal to Binary

• A little more work than binary to decimal
• 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
• 3 2 1 11 (thats 121 120)
• 5 4 1 101 (thats 122 021 120)

10
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
11
Converting Hex to Binary

• Hexadecimal table

Dec Bin Hex
0 0000 0
1 0001 1
2 0010 2
3 0011 3
4 0100 4
5 0101 5
6 0110 6
7 0111 7
Dec Bin Hex
8 1000 8
9 1001 9
10 1010 a
11 1011 b
12 1100 c
13 1101 d
14 1110 e
15 1111 f
12
Hex to Binary Example

• Hex to Binary just convert digits

2ac
2ac 001010101100
No magic!! - remember each hex digit 4 bits
13
Binary to Hex

• Just convert groups of 4 bits

101001101111011
1011
0101 ?
0111 ?
0011 ?
101001101111011 537b
14
Hex to Decimal
Dec Hex
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

• Just multiply each hex digit by decimal value,
  and add the results.

2ac
2 256
10 16
12 1
684
???? ???? ???? ????
163 162 161 160
4096 256 16 1
15
Decimal to Hex

• 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

16
Decimal to Hex
Dec Hex
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
684
2__
684/256 2
684/256 172
2a_
172/16 10 a
172/16 12 c
2ac
???? ???? ???? ????
163 162 161 160
4096 256 16 1
17
Exercise Questions

• Convert decimal value 110 to
• Binary
• Hexadecimal
• Convert binary value 1100111 to
• Decimal
• Hexadecimal
• Convert hexadecimal value 6CAD to
• Decimal
• Binary