Decimal to Binary Conversion
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The document discusses various numeric systems used in computers including: 1) Binary conversion which involves dividing a decimal number by 2 to get the binary representation. 2) Hexadecimal conversion which involves dividing a decimal number by 16. 3) Logic gates like AND, OR, and NOT that computers use to perform calculations. 4) How negative numbers are represented using 2's complement by "hijacking" the most significant bit. 5) How subtraction is performed by adding the normal representation to the 2's complement.
![DECIMAL TO BINARY CONVERSION
600 / 2 = 300 r 0 [least significant bit - lsb]
300 / 2 = 150 r 0
150 / 2 = 75 r 0
75 / 2 = 37 r 1
37 / 2 = 18 r 1
18 / 2 = 9 r 0
9 / 2 = 4 r 1
4 / 2 = 2 r 0
2 / 2 = 2 r 0
1 / 2 = 0 r 1 [most significant bit - msb]](https://image.slidesharecdn.com/hotx74hpqbwyghr1sjcn-signature-39e111ed8707507001f922358b32fbcd54b6778643bc81af86eceff3e5c429f7-poli-150905100051-lva1-app6891/85/Decimal-to-Binary-Conversion-1-320.jpg)
![Negative numbers
On paper, a negative number is represented by a dash – in front of
the number. A computer can only work with 0's and 1's. A dash is
not possible.
An 8-bit binary number (data type 'int' in C, 'byte' in Java) can be
used to represent negative numbers by 'hijacking' the MSB for use
as a 'sign bit'. With 8 bits you can represent numbers which range
from:
(Sign bit)+ ----->0] 0 0 0 0 0 0 0 (+0 in decimal)
(Sign bit)- ----->1] 0 0 0 0 0 0 0 (-0 in decimal)
(Sign bit)+ ----->0] 1 1 1 1 1 1 1 (+127 in decimal)
(Sign bit)- ----->1] 1 1 1 1 1 1 1 (-127 in decimal)
The first number represents whether the number is positive or
negative.
0 = + 1 = -](https://image.slidesharecdn.com/hotx74hpqbwyghr1sjcn-signature-39e111ed8707507001f922358b32fbcd54b6778643bc81af86eceff3e5c429f7-poli-150905100051-lva1-app6891/85/Decimal-to-Binary-Conversion-4-320.jpg)
Decimal to Binary Conversion
- 1. DECIMAL TO BINARY CONVERSION 600 / 2 = 300 r 0 [least significant bit - lsb] 300 / 2 = 150 r 0 150 / 2 = 75 r 0 75 / 2 = 37 r 1 37 / 2 = 18 r 1 18 / 2 = 9 r 0 9 / 2 = 4 r 1 4 / 2 = 2 r 0 2 / 2 = 2 r 0 1 / 2 = 0 r 1 [most significant bit - msb]
- 2. Decimal to Hexadecimal Conversion 5789 ÷ 16 = 361 r 13 or D LSB 361 ÷ 16 = 22 r 9 or 9 22 ÷ 16 = 1 r 6 or 6 1 ÷ 16 = 0 r 1 or 1 MSB 5789 in decimal = 169D in hexadecimal.
- 3. Numeric Systems used in Computers Input Output 0 = 1 Invert (NOT) 1 = 0 Inputs Output +---+ 0 + 0 = 0 | 1 | 0 + 1 = 1 0R | 0 | (Not OR) 1 + 0 = 1 | 0 | NOR 1 + 1 = 1 | 0 | +---+ Inputs Output +---+ 0 x 0 = 0 | 1 | 0 x 1 = 0 AND| 1 |(Not AND) 1 x 0 = 0 | 1 | 1 x 1 = 1 | 0 |-----NAND +---+ Inputs Output 0 + 0 = 00 0 + 1 = 01 ADD 1 + 0 = 01 1 + 1 = 10
- 4. Negative numbers On paper, a negative number is represented by a dash – in front of the number. A computer can only work with 0's and 1's. A dash is not possible. An 8-bit binary number (data type 'int' in C, 'byte' in Java) can be used to represent negative numbers by 'hijacking' the MSB for use as a 'sign bit'. With 8 bits you can represent numbers which range from: (Sign bit)+ ----->0] 0 0 0 0 0 0 0 (+0 in decimal) (Sign bit)- ----->1] 0 0 0 0 0 0 0 (-0 in decimal) (Sign bit)+ ----->0] 1 1 1 1 1 1 1 (+127 in decimal) (Sign bit)- ----->1] 1 1 1 1 1 1 1 (-127 in decimal) The first number represents whether the number is positive or negative. 0 = + 1 = -
- 5. For example:: 00000111 represents 7 in decimal 10000111 represents -7 in decimal So: 00000000 to 01111111 represents 0 to 127 and 10000000 to 11111111 represents 0 to -127 Notice that there are two representations for zero 00000000 and 10000000
- 6. Complementary numbers Complementary numbers are found by counting backwards from the highest number available. For an 8-bit number this is: 11111111 Counting backwards we get:: 11111111 = -0 11111110 = -1 11111101 = -2 11111100 = -3 11111011 = -4 This is known as the 1's complement of a positive number.
- 7. So a complementary number is an inverted 'normal' number, where a negative number is made by 'bitwise inverting' the positive number. 00000000 = +0 11111111 = -0 (silly) 00000001 = +1 11111110 = -1 00000010 = +2 11111101 = -2 Notice that 00000000 and 11111111 both represent zero, which is a 'waste' of a binary number.
- 8. To make better use of the complementary number range, it would make better sense to let 00000000 represent zero, and 11111111 to represent -1 instead of -0, which is silly. So take the 1's complement and add 1 to it. Now it is 11111111 that represents -1 rather than 11111110. This is shown here: 11111110 + 00000001 = 11111111 = (or represents) -1 11111101 + 00000001 = 11111110 = (or represents) -2 11111100 + 00000001 = 11111101 = (or represents) -3 11111011 + 00000001 = 11111100 = (or represents) -4 11111010 + 00000001 = 11111011 = (or represents) -5 11111001 + 00000001 = 11111010 = (or represents) -6 11111000 + 00000001 = 11111001 = (or represents) -7 1's complement + 1 = 2's complement So 1's complement + 1 is known as the 'true' or 2's complement.
- 9. So. To find a 2's complement representation of (say) the number -7 start with +7, ie. 00000111 Then invert (NOT) every bit to get the 1's complement. 11111000 Then add 1 to the number to get the 2's complement. 11111000 + 1 = 11111001 The MSB is a 1 so the number is negative, and this number now REPRESENTS -7 for calculation purposes.
- 10. Subtraction in Binary Computers can only move binary information, and add it together. They cannot directly perform multiplication, subtraction or division. Subtraction of two binary numbers can only be done in a computer by adding together the normal representation of a positive number to the 2's complement of the negative number. For example:: 45 - 20 = 25 This can be expressed as: +45 + (-20) = +25
- 11. +45 = 0010 1101 normal representation +20 = 0001 0100 normal representation -20 = 1110 1011 1's complement + 0000 0001 add 1 to give: -20 = 1110 1100 2's complement Note the process to get the 2's complement Now add: +45 = 0010 1101 normal representation -20 = 1110 1100 2's complement = (1)0001 1001 = 25 in decimal The "overflow" 9th bit above is ignored.