Assignment Due March 12,2005

 

Basic Concepts
When we first learned about numbers, we were taught that, in the decimal system, things are organized into columns:

H T O
1 9 3

"H" is the hundreds column, "T" is the tens column, and "O" is the ones column. So the number "193" is 1-hundreds plus 9-tens plus 3-ones.
(1 x 100) + (9 x 10) + (3 x 1)
The ones column = 100. The tens column = 101. The hundreds column = 102.
So: 193 can be seen as:

102 101 100
1 9 3

The number 193 is really {(1x102)+(9x101)+(3x100)} => (1 x 100) + (9 x 10) + (3 x 1).
The decimal system uses the digits 0-9 to represent numbers.
The binary system works under the exact same principles as the decimal system, only it operates in base 2 rather than base 10. In other words, instead of columns being

102 101 100

They are:

22 21 20

Instead of using the digits 0-9, we only use the digits 0 and 1.
The binary system uses the digits 0 and 1 to represent numbers.
The first column we fill is the right-most column, which is 20, or 1. For each additional power of 2 we need to use an extra column to the left.
The number 5 in binary is 101:

22 21 20
1 0 3

This is: 1x22) + (0x21) + (1x20) = (1 x 4) + (0 x 2) + (1 x 1). = 4 + 0 + 1

Converting From Binary to Decimal

Example:
What is the decimal value of the binary number 11001101?

Solution:

1 Power of 2 27 26 25 24 23 22 21 20
2 Decimal 128 64 42 16 8 4 2 1

3 Binary 1 1 0 0 1 1 0 1
4 Multiply 128 64 0 0 8 4 0 1

Sum the product in row # 4

  • 128 + 64 + 0 + 0 + 8 + 4 + 0 + 1
  • 205

This conversion can also be done without the chart:

1111010

  • (1x 26) +(1x 25) +(1x 24) +(0x 23) +(1x 22) +(1x 21)
  • 64 + 32 + 16 + 8 + 0 + 2 + 0
    = 122

Exercises:

Convert the following binary numbers to decimal:

1.10
2.111
3.10101
4.11110
5.11001
6.101011
7.1100010
8.1111000
9.11111010
10.11101110

Imsg
Discussion

Blogging

E-mail

 

 

   The binary system