1. Binary
Negative Integers

2. Negative Integers Sign and magnitude One’s complement Two’s complement
Binary Coded Decimal (BCD)

3. Sign and Magnitude
The method used in decimal to represent negative numbers is sign and magnitude.
- 25
Sign Magnitude/value
This system is available in decimal where: 1 ≡ negative sign

4. Sign and Magnitude The method is as follows:
Convert the value or magnitude to binary and represent using eight or ten bits or as you are instructed
Change the leftmost bit to 1 if the number is negative

5. Example:
Represent the following decimal as binary, using sign and magnitude:
NEGATIVE 1010
NEGATIVE 2510

6. -10 to Binary using 8 bit Sign and Magnitude
Convert 10 to binary
1010
Use 8 bits to represent
Change to negative:
-1010= 2 10
Remainder 5 1

7. -25 to Binary using 8 bit Sign and Magnitude
Convert 25 to binary
11001
Use 8 bits to represent
Change to negative:
-2510= 2 25
Remainder 12 1 6 3

8. One’s Complement The method is as follows:
Convert the value or magnitude to binary and represent using eight or ten bits or as you are instructed
Find the complement by changing all the 0’s to 1’s and all the 1’s to 0’s

9. Example:
Represent the following decimal as binary, using ONE’S COMPLEMENT:
NEGATIVE 1010
NEGATIVE 2510

10. -25 to Binary using 8 bit One’s Complement
Convert 25 to binary
11001
Use 8 bits to represent
Change to negative: 1 0 and; 0 1
-2510= 2 25
Remainder 12 1 6 3

11. Two’s Complement The method is as follows:
Convert the value or magnitude to binary and represent using eight or ten bits or as you are instructed
Find the One’s complement by changing all the 0’s to 1’s and all the 1’s to 0’s
Add one to the new value

12. Example:
Represent the following decimal as binary, using TWO’S COMPLEMENT:
NEGATIVE 1010
NEGATIVE 2510

13. -25 to Binary using 8 bit Two’s Complement
Convert 25 to binary
11001
Use 8 bits to represent
Find one’s complement
Add one to the answer
-2510= 2 25
Remainder 12 1 6 3 1 +

14. Binary Coded Decimal
B.C.D.

15. Format Each digit is converted separately using four (4) bits each. 25
Remainder 1
2= 0010 25 2 5
Remainder 1
5=0101
0010
0101

16. Format
Decimal positioning is kept 25
= 0010
0101 2 10

17. Negative BCD Use Sign and Magnitude where the signs are:
+ Positive = 1110 2
- Negative = 1111 2

18. Positive and Negative
+25 =
0101 2
-25 =
0101 2

19. Convert the following numbers from decimal to binary using BCD format:
Steps:
Convert each digit to binary
Write sign (if necessary)
Write answer in decimal order
Convert the following numbers from decimal to binary using BCD format:

20. Binary
Real Numbers

21. Real Numbers Real numbers are numbers containing fractions.
There are two ways real numbers are represented in binary.
They are:
Fixed-point numbers
Floating-point numbers

22. Fixed-point Numbers
Decide the number of places after the point because the point is not stored among the digits.
Convert the whole number to binary
Convert the fraction to binary:
Multiply the fraction by two and record the any resulting whole number
Repeat until you get the set amount of places after the point

23. Fixed-point Numbers
Convert to binary with 4 places after the point.
The answer is therefore: 2 4 R 1
=100
0.2 x 2 =
0.4
0.4 x 2
0.8
0.8 x 2
1.6
0.6 x 2
1.2
=0011

24. Floating-point Numbers
The number of places after the point varies.
Data is represented in the following parts:
A sign
A fractional part (example 0.345) or mantissa
The base
An exponent

25. Standard Form
Change to standard form:
345
-45.6

26. Floating-point Numbers
Decimal Example:
This is equal to writing a number in standard form 3
345 = x 10
Exponent
Sign
Mantissa
base 2
-45.6 = x 10

27. Floating-point Numbers
Binary Example: Binary number
The mantissa is a binary fraction
The sign bit : 1 for negative and 0 for positive
This exponent uses sign and magnitude 1
111
1010
Sign
Exponent
Mantissa

28. Floating-point Numbers
IEEE Standard uses 32 and 64bits, but for simplicity we will use only 8 bits as follows:
The sign – 1 bit
1 means negative; 0 means positive
The Exponent – 3 bits
Sign and magnitude. Leftmost bit is the sign
The Mantissa – 4 bits
A fraction

29. From Decimal: 3¾
Convert the decimal to binary (maintain the whole and fraction parts).
Normalise the mantissa
Convert the resulting exponent
Insert the sign bit
Write the number in SEM format
3 ¾ to binary retaining decimal format: 11.11
Normalised mantissa as if in standard form: .1111x22
The exponent : 2 = 011
The number is positive, so the sign = 0
RESULT:

30. Let us Calculate: Binary Example: 111110102 The mantissa : 0.625
The sign bit : - (negative)
The exponent : -3
RESULT: X 2-3
2-1
2-2
2-3
2-4 1 .5
0.125 1
111
1010
Sign
Exponent
Mantissa - -3
0.625 =

31. Let us Calculate: Binary Example: 111110102 The mantissa is: 0.625
The sign bit : - (negative)
The exponent : -3
RESULT: X 2-3 = = =

32. Characters ASCII (American Standard Code of Information Interchange)
EBCDIC (Extended Binary Coded Decimal Interchange Code

33. Parity Bit
To maintain data integrity a special signal bit is sometimes used. This is a parity bit. Instead of the regular eight bits that make up the byte, nine bits are used.
If he number of “1” bits is odd then the parity is set to 1 so that the number of 1”s is always even
If the number of “1” bits is even the parity is set to “0”.

34. The END