Outcome
• Familiar with the binary system• Binary to Decimal and decimal to binary• Adding two binary sequences• Logic gates• Hexadecimal system
Reading
• http://www.math.grin.edu/~rebelsky/Courses/152/97F/Readings/student-binary
• http://en.wikipedia.org/wiki/Binary_numeral_system
• http://www.cut-the-knot.org/do_you_know/BinaryHistory.shtml
• http://www.binarymath.info/
The Decimal Number System
4
• 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?o Because we use 10 digits, the digits 0 through 9.
The Decimal Number System - base 10
5
• The decimal number system is a positional number system with a base 10.
• Example: 1011
• 10112 = 1000 + 000 + 10 + 1 = 1 x 23 + 0 x22 + 1 x 21 + 1 x 20 = 1110
1000 000 10 11 x 23 0x22 1 x 21 1x 20
The Binary Number System
6
• 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?o Because we use 2 digits, the digits 0 and 1.
The Binary Number System – base 2
7
• The decimal number system is a positional number system with a base 10.
• Example: 5623
• 5623 = 5000 + 600 + 20 + 3 = 5 x 103 + 6 x102 + 2 x 101 + 3 x 100
5000 600 20 35 x 103 6 x102 2 x 101 3 x 100
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– …
8
Base 10 and Base 2• Base 10
– Each digit represents a power of 10– 5417310 = 5 x 104 + 4 x 103 + 1 x 102 + 7 x 101 + 3 x 100
• Base 2
– Each bit represents a power of 2– 101012= 1 x 24 + 0 x 23 + 1 x 22 + 0 x 21 + 1 x 20 = 2110
9
The Binary Number System (con’t)
10
• 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:
1 0 0 1 1 0 1 26 25 24 23 22 21 20
Converting from Binary to Decimal
11
1 0 0 1 1 0 1 1 X 20 = 1 26 25 24 23 22 21 20 0 X 21 = 0 1 X 22 = 4 20 = 1 1 X 23 = 8 21 = 2 0 X 24 = 0 22 = 4 0 X 25 = 0 23 = 8 1 X 26 = 64 24 = 16 7710
25 = 32 26 = 64
Converting from Binary to Decimal (con’t)
Practice conversions:
Binary Decimal
11101 1010101 100111
12
Converting From Decimal to Binary (con’t)
13
• 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
25 24 23 22 21 20
32 16 8 4 2 1 1 0 1 0 1 0
42/2 = 21 and R = 021/2 = 10 and R = 110/2 = 5 and R = 05/2 = 2 and R = 12/2 = 1 and R = 01/2 = 0 and R = 1
4210 = 1010102
Example 1210
We repeatedly divide the decimal number by 2 and keep remainders
– 12/2 = 6 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
Exercises
• Find the binary the decimal number represented by the following binary sequences:– 110101– 10111010
• Represent the number 135 in base 2.
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.
17
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
18
1 98
+ 2 64
Sum
Carry
0 11
+ 0 01
Sum
Carry
2
1
6
1
4
0
0
1
0
1
1
0
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’
n-input Gates• Because + and * are binary operations, they can be
cascaded together to OR or AND multiple inputs.
AB
C
ABC
A+B+C
A+B+C
AB
ABC
ABC
ABC
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.
1101100101
01001101111101110111
10001111
0011110000001100
110001 001110
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
C
Babc
aBc
Ab
y=abc+aBc+Ab
y
A
B
C
Y
y=aB+Bc
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.
NAND
NOR
A B AB
0 0 1
0 1 1
1 0 1
1 1 0
A B AB
0 0 1
0 1 0
1 0 0
1 1 0
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.
XOR
XNOR
A B AB
0 0 0
0 1 1
1 0 1
1 1 0
A B A B
0 0 1
0 1 0
1 0 0
1 1 1
Binary Sums and Carries
a b Sum a b Carry0 0 0 0 0 00 1 1 0 1 01 0 1 1 0 01 1 0 1 1 1
29
XOR AND
0100 0101
+ 0110 0111
1010 1100
69
103
172
Full Adder
Cin A B S(um)
Cout
0 0 0 0 0
0 0 1 1 0
0 1 0 1 0
0 1 1 0 1
1 0 0 1 0
1 0 1 0 1
1 1 0 0 1
1 1 1 1 1
FullAdder
A B
S
Cout
Carry In(Cin)
4-bit Ripple Adder using Full Adder
FullAdder
A B
CinCout
S
S0
A0 B0
FullAdder
A B
CinCout
S
S1
A1 B1
FullAdder
A B
CinCout
S
S2
A2 B2
FullAdder
A B
CinCout
S
S3
A3 B3
Carry
A
BS
C
Half Adder
A
B
CinCout
SH.A. H.A.
Full Adder
Working with Large Numbers
36
0 1 0 1 0 0 0 0 1 0 1 0 0 1 1 1 = ?
• 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.
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.
37
The Hexadecimal Number System (con’t)
Binary Decimal Hexadecimal Binary Decimal Hexadecimal
0 0 0 1010 10 A 1 1 1 1011 11 B 10 2 2 1100 12 C 11 3 3 1101 13 D 100 4 4 1110 14 E 101 5 5 1111 15 F 110 6 6 111 7 7 1000 8 8 1001 9 9
38
The Hexadecimal Number System (con’t)
• Example of a hexadecimal number and the values of the positions:
3 C 8 B 0 5 1 166 165 164 163 162 161 160
39
Example of Equivalent Numbers
40
Binary: 1 0 1 0 0 0 0 1 0 1 0 0 1 1 12
Decimal: 2064710
Hexadecimal: 50A716
Notice how the number of digits gets smaller as the base increases.