1. Binary Numbers

2. Outcome Familiar with the binary system
Binary to Decimal and decimal to binary
Arithmetic and logic operation in binary system
Logic gates
Half Adder and Full Adder
Hexadecimal system

3. Reading Goldsmiths Study guide: mathematics for computing

4. The Decimal Number System (con’t)
The decimal number system is also known as base 10. The values of the positions are calculated by taking 10 to some power.
Why is the base 10 for decimal numbers?
Because we use 10 digits, the digits 0 through 9.

5. The Binary Number System – base 2
The decimal number system is a positional number system with a base 10.
Example: 5623
5623 = = 5 x x x x 100
5000
600 20 3
5 x 103
6 x102
2 x 101
3 x 100

6. The Binary Number System
The binary number system is also known as base 2. The values of the positions are calculated by taking 2 to some power.
Why is the base 2 for binary numbers?
Because we use 2 digits, the digits 0 and 1.

7. The Decimal Number System - base 10
The decimal number system is a positional number system with a base 10.
Example: 1011
10112 = = 1 x x x x 20 = 1110
1000
000 10 1
1 x 23
0x22
1 x 21
1x 20

8. Why Bits (Binary Digits)?
Computers are built using digital circuits
Inputs and outputs can have only two values
True (high voltage) or false (low voltage)
Represented as 1 and 0
Can represent many kinds of information
Boolean (true or false)
Numbers (23, 79, …)
Characters (‘a’, ‘z’, …) ASCII, UNICODE
Pixels
Sound
Can manipulate in many ways
Read and write
Logical operations
Arithmetic …

9. Base 10 and Base 2 Base 10 Base 2 Each digit represents a power of 10
= 5 x x x x x 100
Base 2
Each bit represents a power of 2
101012= 1 x x x x x 20 = 2110

10. Converting from Binary to Decimal
X 20 = 1
X 21 = 0
1 X 22 = 4
20 = X 23 = 8
21 = X 24 = 0
22 = X 25 = 0
23 = X 26 = 64
24 =
25 = 32
26 = 64

11. Converting from Binary to Decimal (con’t)
Practice conversions: Binary Decimal

12. Converting From Decimal to Binary (con’t)
Make a list of the binary place values up to the number being converted.
Perform successive divisions by 2, placing the remainder of 0 or 1 in each of the positions from right to left.
Continue until the quotient is zero.
Example: 4210
42/2 = and R = 0
21/2 = and R = 1
10/2 = and R = 0
5/2 = 2 and R = 1
2/2 = 1 and R = 0
1/2 = 0 and R = 1 =

13. Example 1710
We repeatedly divide the decimal number by 2 and keep remainders
17/2 = and R = 1
8/2 = 4 and R = 0
4/2 = 2 and R = 0
2/2 = 1 and R = 0
1/2 = 0 and R = 1
The binary number representing 17 is

14. Converting From Decimal to Binary (con’t)
Practice conversions: Decimal Binary

15. Fractional Numbers Decimal
= 4 x x x x x 10-2
Binary
= 1 x x x x x x 2-2
= / ¼
= =

16. Binary Fractional to decimal nNumbers (cont)
Example1
= 1 x x x x x x 2-2
= / ¼
= =
Example 2:
= 1 x x x x x 2-2
= / ¼ =
Example3: = 1 x x x x x 2-3
= ¼ /8 =

17. Fractional numbers 4 2 1 Examples: 7.7510 = (?)2
Conversion of the integer part: same as before – repeated division by 2
7 / 2 = 3 (Q), 1 (R)  3 / 2 = 1 (Q), 1 (R)  1 / 2 = 0 (Q), 1 (R) = 1112
Conversion of the fractional part: perform a repeated multiplication by 2 and extract the integer part of the result
0.75 x 2 =1.50  extract 1
0.5 x 2 =  extract = 0.112
 stop
 Combine the results from integer and fractional part, =
How about choose some of
Examples: try 5.625
write in the same order 4 2 1
1/2
1/4
1/8
=0.5
=0.25
=0.125

18. Fractional Numbers (cont.)
Exercise 3: Convert (0.8125)10 to its binary form
Solution:
x 2 =  extract 1
0.625 x 2 =  extract 1
0.25 x 2 =  extract 0
0.5 x 2 =  extract 1
 stop
 (0.8125)10 = (0.1101)2

19. Representing fraction with error
Example: Convert (0.6)10 to its binary form
0.6 x 2 = 1.2  extract 1
0.2 x 2 = 0.4  extract 0
0.4 x 2 = 0.8  extract 0
0.8 x 2 = 1.6  extract 1
0.6 x 2 = 
(0.6)10 = ( …)2

20. Fractional Numbers (cont.)
Errors
One source of error in the computations is due to back and forth conversions between decimal and binary formats
Example: (0.6)10 + (0.6)10 = 1.210
Since (0.6)10 = ( …)2
Lets assume a 8-bit representation: (0.6)10 = ( )2 , therefore 
Lets reconvert to decimal system:
( )b= 1 x x x x x x x x x 2-8
= 1 + 1/8 + 1/16 + 1/128 =
 Error = 1.2 – =

21. Bits, Bytes, and Words A bit is a single binary digit (a 1 or 0).
A byte is 8 bits
A word is 32 bits or 4 bytes
Long word = 8 bytes = 64 bits
Quad word = 16 bytes = 128 bits
Programming languages use these standard number of bits when organizing data storage and access.

22. Adding Two Integers: Base 10
From right to left, we add each pair of digits
We write the sum, and add the carry to the next column
0 1 1
Sum
Carry
1 9 8
Sum
Carry 4 6 1 2 1 1 1 1

23. Example
= =

24. Binary subtraction

25. Binary subtraction (Cont)

26. Binary subtraction (Cont)

27. Exercise =?
= ?

28. Binary Multiplication
two = 8ten multiplicand
two = 9ten multiplier
____________
partial products
two = 72ten
Spring 2007, Jan. 17
ELEC 2200 (Agrawal)

29. Binary Division 1 3 Quotient 1 1 / 1 4 7 Divisor / Dividend 1 1
3 7 Partial remainder
3 3
4 Remainder /
1 0 0
Spring 2007, Jan. 17
ELEC 2200 (Agrawal)

30. Bitwise Operators: Shift Left/Right
Shift left (<<): Multiply by powers of 2
Shift some # of bits to the left, filling the blanks with 0
Shift right (>>): Divide by powers of 2
Shift some # of bits to the right
For unsigned integer, fill in blanks with 0
What about signed integers? Varies across machines…
Can vary from one machine to another! 1 53
53<<2 53 1 1 1
53>>2 1 1 1

31. Boolean Algebra to Logic Gates
Logic circuits are built from components called logic gates.
The logic gates correspond to Boolean operations +, *, ’.
Binary operations have two inputs, unary has one OR +
AND *
NOT ’

32. AND A
Logic Gate:
A*B
Truth Table: B A B
A*B 1 A B
Series Circuit:
A*B

33. OR A B A+B 1 A Logic Gate: A+B Truth Table: B A Parallel Circuit: B1 A
Parallel Circuit: B
A+B

34. NOT
Logic Gate:
(also called an inverter) A
A’ or A
Truth Table: a A 1

35. n-input Gates
Because + and * are binary operations, they can be cascaded together to OR or AND multiple inputs. A A B
A+B+C
ABC B C A A B
A+B+C
ABC B C C

36. NAND and NOR Gates
NAND and NOR gates can greatly simplify circuit diagrams. As we will see, can you use these gates wherever you could use AND, OR, and NOT. A B
AB 1
NAND A B
AB 1
NOR

37. XOR and XNOR Gates
XOR is used to choose between two mutually exclusive inputs. Unlike OR, XOR is true only when one input or the other is true, not both. A B
AB 1
XOR A B
A B 1
XNOR

38. Binary Sums and Carries
a b Sum a b Carry
XOR
AND 69
103
172

39. Design Hardware Bit by Bit
Adding two bits:
a b half_sum carry_out
Half-adder circuit a b
half_sum
carry_out
XOR
AND
Spring 2007, Jan. 17
ELEC 2200 (Agrawal)

40. Half Adder (1-bit) A B A B
S(um)
C(arry) 1 S
Half
Adder C

41. Half Adder (1-bit) A B
S(um)
C(arry) 1 A B
Sum
Carry

42. Full Adder A B
Cin A B
S(um)
Cout 1 S
Full
Adder
Carry In
(Cin)
Cout

43. Full Adder
H.A.
H.A. A B
Cin
Cout S

44. Full Adder
Cout S
Half
Adder C A B
Cin

45. 4-bit Ripple Adder using Full Adder
Cin
Cout S S3 A3 B3
Carry
Full
Adder A B
Cin
Cout S S2 A2 B2
Full
Adder A B
Cin
Cout S S1 A1 B1 A0 B0
Full
Adder A B
Cin
Cout S A B
Cin
Cout S
H.A.
Full Adder S0 A B S C
Half Adder

46. Working with Large Numbers
= ?
Humans can’t work well with binary numbers; there are too many digits to deal with.
Memory addresses and other data can be quite large. Therefore, we sometimes use the hexadecimal number system.

47. The Hexadecimal Number System
The hexadecimal number system is also known as base 16. The values of the positions are calculated by taking 16 to some power.
Why is the base 16 for hexadecimal numbers ?
Because we use 16 symbols, the digits 0 and 1 and the letters A through F.

48. The Hexadecimal Number System (con’t)
Binary Decimal Hexadecimal Binary Decimal Hexadecimal A B C D E F

49. The Hexadecimal Number System (con’t)
Example of a hexadecimal number and the values of the positions:
3 C B

50. Example of Equivalent Numbers
Binary:
Decimal:
Hexadecimal: 50A716
Notice how the number of digits gets smaller as the base increases.

51. Summary Convert binary to decimal Decimal to binary Binary operation
Logic gates
Use of logic gates to perform binary operations
Half adder
Full adder
The need of Hexadecimal Hexadecimal

52. Next lecture (Data representation)
Put this all together
negative and positive integer representation
unsigned notation
Signed notation
Excess notation
Tow’s complement notation
Floating point representation
Single and double precision
Character, colour and sound representation