1. Binary Numbers

2. Outcome Familiar with the binary system
Binary to Decimal and decimal to binary
Adding two binary sequences
Logic gates
Hexadecimal system

3. Reading

4. The Decimal Number System
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 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

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 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

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’, …)
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. The Binary Number System (con’t)
The binary number system is also a positional numbering system.
Instead of using ten digits, 0 - 9, the binary system uses only two digits, 0 and 1.
Example of a binary number and the values of the positions:

11. 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

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

13. 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 =

14. Example 1210
We repeatedly divide the decimal number by 2 and keep remainders
12/2 = and R = 0
6/2 = 3 and R = 0
3/2 = 1 and R = 1
1/2 = 0 and R = 1
The binary number representing 12 is 1100

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

16. Exercises
Find the binary the decimal number represented by the following binary sequences:
110101
Represent the number 135 in base 2.

17. 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.

18. 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

19. Example
= =

20. 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 ’

21. AND A
Logic Gate:
A*B B
Truth Table: A B
A*B 1

22. OR A
Logic Gate:
A+B B
Truth Table: A B
A+B 1

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

24. 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

25. n-bit Inputs
For convenience, it is sometimes useful to think of the logic gates processing n-bits at a time. This really refers to n instances of the logic gate, not a single logic date with n-inputs.
110001
001110

26. Logic Circuits ≡ Boolean Expressions
All logic circuits are equivalent to Boolean expressions and any boolean expression can be rendered as a logic circuit.
AND-OR logic circuits are equivalent to sum-of-products form.
Consider the following circuits: A
y=aB+Bc
abc B C A B C Y
aBc y Ab
y=abc+aBc+Ab

27. 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

28. 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

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

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

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

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

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

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

35. 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

36. 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.

37. 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.

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

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

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

41. Summary Convert binary to decimal Decimal to binary Binary operation
Logic gates
Use of logic gates to perform binary operations
Hexadecimal