Binary and Hexadecimal Numbers

Bits and Bytes
- Storage locations in digital computers are built using electical circuits that can either have a high voltage or a low voltage.
- 0 represents a low voltage
- 1 represents a high voltage
- A byte is 8 bits together
- A word is a group of bytes

Denary
136 represented in denary is (1x100) + (3x10) + (6x1)

Coverting Binary to Denary
- Create a Binary table
- Write in the binary number from right to left
- Add up the values where a '1' is in the column.

Example:  01110101
128     64     32     16     8     4     2     1
  0        1       1       1      0     1     0     1

= 64+32+16+4+1
= 117

Converting Denary to Binary
- Create a Binary table
- Go from left to right making sure each number in the table is less than the number before
- Fill gaps with '0'

Example: 51
64     32     16     8     4     2     1
0        1       1      0     0     1     1

Adding Binary Numbers
- Put binary numbers into column method

Example:
     10011
     01011
=   11110  = 30

Rules of Adding Binary

0 + 0 = 0 (no carry)

0 + 1 = 1 (no carry)

1 + 0 = 1 (no carry)

1 + 1 = 10 (0 and carry 1)

1 + 1 + 1 = 11 (1 and carry 1)

Multiplying Binary

Example:

 110

x 11

= 1100

Binary Multiplication Method

1.Start on the second row at the left.

2.For each 1 you find you create a row below the bar with the number at the  top followed by the number of places to the right of the 1 you are considering.

3.Finally add all the rows below the bar – using binary addition rules

 

Converting Denary Negatives into Binary

- Find Binary equivalent to denary

- Change 0's to 1's and 1's to 0's

- Add one to the result

 

Example: -13

16     8     4     2     1

0       1     1     0     1 (binary equivalent)

1       0     0     1     0 (change 0's and 1's vice versa)

1       0     0     1     1 (add one)

 Converting Binary Negatives into Denary

- Put into binary table

- Convert 1's to 0's and 0's to 1's

- Add one to the result

-Convert to denary and place a negative sign in front

Example:  0100111

64     32     16     8     4      2     1

0        1       0      0     1      1     1 (binary in table)

1        0       1      1     0      0     0 (change 0's to 1's vice versa)

1        0       1      1     0      0     1 (add one to result)

= -89

Subtracting Binary

- Convert number to be subtracted  into a negative number and add them.

Hexadecimal

- Hexadecimal goes from denary 1 to denary 16.

- Hexadecimal goes from 1,2,3,4,5,6,7,8,9,A,B,C,D,E,F,10

Converting Denary to Hexadecimal

- Convert to binary

- Take each group of 4 and convert into hex

Example: 213

128     64     32     16     8     4     2     1

 1         1       0       1      0     1     0     1

1101 = 13 = D

0101 = 5 = 5

= D5

Decimals

- Create a binary table with a decimal place after 1

e.g.  32      16      8      4      2      1     .     0.5     0.25  etc.

       

Example

48.75

32     16     8     4     2     1     .     0.5      0.25

 1       1      0     0     0     0     .      1          1