1. WEEK #2 NUMBER SYSTEMS, OPERATION & CODES (PART 1)
DKT 122/3 DIGITAL SYSTEM 1
WEEK #2
NUMBER SYSTEMS, OPERATION & CODES
(PART 1)

2. Numbers & Codes Numbering Systems Number Conversion Binary Arithmetic
Decimal numbering system (Base 10)
Binary numbering system (Base 2)
Hexadecimal numbering system (Base 16)
Octal numbering system (Base 8)
Number Conversion
Binary Arithmetic
1’s and 2’s Complements of Binary Numbers

3. Numbers & Codes (cont..) Signed Numbers Other Number Codes
Arithmetic Operations with Signed Numbers
Other Number Codes
Binary-Coded-Decimal (BCD)
ASCII codes
Gray codes
Digital Codes & Parity

4. Numbering Systems Decimal Binary Octal Hexadecimal 0 ~ 9 0 ~ 1 0 ~ 7
(base 10)
Binary
(base 2)
Octal
(base 8)
Hexadecimal
(base 16)
0 ~ 9
0 ~ 1
0 ~ 7
0 ~ F

5. Num. Systems (Characteristics)
The digits are consecutive (berturutan).
The number of digits is equal to the size of the base.
Zero is always the first digit.
When 1 is added to the largest digit, a sum of zero and a carry of one results.
Numeric values determined by the implicit positional values of the digits.

6. Numbering Systems (Cont.)
A B C D E F
Binary
Octal
Hex
Dec

7. Numbering System (Decimal)
Also called the Base 10 system
Have 10 digits : 0  9
The position for each digit in the decimal number indicates the magnitude of the quantity represented and can be assigned a weight

8. Numbering System (Decimal)
The weight for whole numbers are positive powers of ten that increase from right to left
For fractional numbers, the weights are negative powers of ten that decrease from left to right ….
Decimal point

9. Numbering System (Binary)
Also called the Base 2 system
The binary number system is used to model the series of electrical signals computers use to represent information
0 represents the no voltage or an off state
1 represents the presence of voltage or an on state

10. “There are 10 kinds of mathematicians.
Just think for a while..
“There are 10 kinds of mathematicians.
Those who can think binarily and those who can't...”
So, what is the meaning of this?
YOU FALL IN WHICH CATEGORY?

11. Significant Digits Binary: 11101101 Hexadecimal: 1D63A72A
Most significant digit (MSB)
Least significant digit (LSB)
Hexadecimal: 1D63A72A
Question: How many bits does the numbers represent?

12. Number Conversion
Any Radix (base) to Decimal Conversion

13. Binary to Decimal Conversion
Decimal value of any binary number can be found by adding weights of all bits that are 1 and discarding the weights of all bits that are 0

14. Solve this..
Convert the following binary numbers to decimal
(a) 10102
Answer : ?
(b)
Answer : ?
(c)
Answer : ?

15. Decimal to Binary Conversion
For whole number conversion, use the repeated division-by-2 process and record the remainder
For fractional number conversion, use repeated multiplication by 2 until the fractional product is 0 or until the desired number of decimal places is reached

16. Decimal to Binary Conversion
Remainder
Whole number
2 5 = 2
1 2 =
6 =
3 =
1 =
MSB LSB
2510 =

17. Decimal to Binary Conversion
Fractional number
Carry
x 2 =
x 2 =
x 2 =
0.5 x 2 =
The Answer:
MSB
LSB

18. Solve this..
Convert the following decimal numbers to binary
(a) 3910
Answer : ?
(b) 5810
Answer : ?
(c)
Answer : ?

19. Binary Arithmetics
Binary Addition
Binary Subtraction
Binary Multiplication
Binary Division

20. Binary Addition Four basic rules for adding binary digits (bits) are:
0 + 0 = 0 (Sum of 0 with a carry of 0)
0 + 1 = 1 (Sum of 1 with a carry of 0)
1 + 0 = 1 (Sum of 1 with a carry of 0)
1 + 1 = 1 0 (Sum of 0 with a carry of 1)

21. Examples (a) 100 + 10 1 0 0 + 1 0 (Answer) 1 1 0 (b) 111 + 11 1 1 1 +
Perform the following binary additions:
(a)
1 0 0
1 0 +
1 1 0
(Answer)
(b)
1 1 1 +
1 1
(Answer)

22. Solve this.. Perform the following binary additions: (a) 11 + 01
Answer : ?
(b)
Answer : ?
(c) :
Answer : ?

23. Binary Arithmetics
Binary Addition
Binary Subtraction
Binary Multiplication
Binary Division

24. Binary Subtraction 0 - 0 = 0 1 - 1 = 0 1 - 0 = 1
Four basic rules for subtracting binary digits (bits) are:
0 - 0 = 0
1 - 1 = 0
1 - 0 = 1
= 1 (0 – 1 with a borrow of 1)

25. Examples (a) 101 – 011 1 0 1 - 0 1 1 (Answer) 0 1 0 (b) 110 – 101
Perform the following binary subtractions:
(a) 101 – 011
1 0 1
0 1 1 -
0 1 0
(Answer)
(b) 110 – 101
1 1 0
1 0 1 -
0 0 1
(Answer)

26. Solve this.. (a) 101 – 100 (b) 1110 - 11 (c) 1100 - 1001:
Perform the following binary subtractions
(a) 101 – 100
Answer : ?
(b)
Answer : ?
(c) :
Answer : ?

27. Binary Arithmetics
Binary Addition
Binary Subtraction
Binary Multiplication
Binary Division

28. Binary Multiplication
Four basic rules for muliplying binary digits (bits) are:
0 x 0 = 0
0 x 1 = 0
1 x 0 = 0
1 x 1 = 1

29. Examples Multiply 111 and 101: 1 1 1 x 1 0 1 1 1 1 0 0 0 1 1 1
(Answer)

30. Solve this.. (a) 11 x 11: (b) 110 x 111: (c) 1101 x 1010: Answer : ?

31. Binary Arithmetics
Binary Addition
Binary Subtraction
Binary Multiplication
Binary Division

32. Binary Division Example: 1 0 (Answer) 1 1 1 1 0 1 1 0 0 0
Division in binary follows the same procedure as division in decimal
Example:
Perform the binary divisions of 110  11
1 0
(Answer)
1 1
1 1 0
1 1
0 0 0

33. Solve this.. Divide the binary numbers as indicated: (a) 100  10
Answer : ?
(b) 1100  100:
Answer : ?

34. 1’s Complement Example: 1 1 0 1 0 0 1 0 1 Binary number
Changing all the 1s to 0s and all the 0s to 1s
Example:
Binary number
’s complement

35. Find the 1’s complements of the numbers
Step 1:
Find the 1’s complements of the numbers
Binary number
’s complement
Step 2:
Add ‘1’ to the 1’s complements
’s complement
Add 1
’s complement

36. Solve this..
Determine the 2’s complement of each binary number:
(a)
Answer : ?
(b)
Answer : ?

37. Signed Numbers Left most is the sign bit
0 is for positive, 1 is for negative
Sign & magnitude
= +25
sign bit magnitude bits

38. Sign-Magnitude Numbers
The left-most is the sign bit and the remaining bits are the magnitude bits …
Sign bit
31 bits for magnitude
0 = positive
1 = negative

39. Signed Numbers (Cont.) 1’s complement
The negative number is the 1’s complement of the corresponding positive number
Example
+25 is So, -25 is
2’s complement
The positive number – same as sign magnitude and 1’s complement
The negative number is the 2’s complement of the corresponding positive number.
+25 is So, -25 is

40. END Solve this.. Express +19 and -19 (as an 8-bit number) in
i. sign magnitude
ii. 1’s complement
iii. 2’s complement
Answer : ?
Answer : ?
Answer : ?
END