1. What is a byte? What is it? How would you use it?
How does it apply to computers?

2. Binary, Decimal, Hexadecimal,& Octal Goofy video
Numbers Systems
Binary, Decimal, Hexadecimal,& Octal
Goofy video

3. Common Bases
Base 2
Base 8
735
Base 10
129
Base 16
BC94E

4. Positional Notation
So how can we express the numbers in different bases?
Positional notation is writing out each symbol and the calculation that represents that position. For example:
BASE TEN is our number system. Think of the number 56,423 broken down in base 10 positional notation. The base 10 number 56,423 becomes:
(5 x 104) + (6 x 103) + (4 x 102) + (2 x 101) + (3 x 100)
10000
1000
100 10 1
104
103
102
101

5. Base 2 - BINARY
IF… BASE 10 multiplies a number 0-9 by 10position and then adds the numbers together Then… Base 2 multiplies a number _______ by _____position and then adds the numbers together BASE 2 number system is called the binary number system. Computers use binary numbers to communicate because a 0 or 1 is simply and on or off. One binary digit is referred to as bit. A byte contains 8 bits.

6. Base 2 - BINARY
BASE 2 multiplies a number from 0-1 by 2position and then adds the numbers together
So let’s look at a typical base 2 number.
For example becomes:
0 x x x x x x x x 20 =
x x x x 1=
=39
128 64 32 16 8 4 2 1 27 26 25 24 23 22 21 20

7. Base 2 - BINARY
128 64 32 16 8 4 2 1 27 26 25 24 23 22 21 20
GO to the netschool page and click on the BINARY game.
Practice your binary numbers. See how high you can score…

8. Base 8 - Octal
The BASE 8 or OCTAL NUMBER SYSTEM is based on 8 numbers 0-7.
Let’s look at the octal number 362 converting to decimal form.
0 x x x x 80 =
0 x x x x 1 =
= 242
4096
512 64 8 1 84 83 82 81 80

9. Base 16 - Hexadecimal
The HEXADECIMAL NUMBER SYSTEM is based on 16 numbers 0-9 and A-F. Notice that the letters A - F are used in hex to represent the digits 10 – 15.
BASE 16 multiplies a number 0-9 & A-F by 16position and then adds the numbers together.
D3C216 = D x x C x x 160 =
= 54210
65536
4096
256 16 1
164
163
162
161
160

10. CONVERSIONS:
You can convert numbers between decimal and binary with two possible techniques: the ladder technique or positional notation technique. Start with a decimal number. For example 39 using the ladder technique: 39/2 = 19 remainder 1 19/2 = 9 remainder 1 9/2 = 4 remainder 1 4/2 = 2 remainder 0 2/2 = 1 remainder 0 1/2 = 0 remainder 1 You continue to divide the result until you get 0. The remainder numbers are what represent the binary numbers. So 39 in binary form = The first remainder you get becomes the first bit.

11. CONVERSIONS:
The other technique is called the positional notation technique. It looks at the base 2 values and pulls out the largest value from the decimal number and then finds the other combination of numbers going from largest to smallest.
For example 39 using the positional notation technique:
First you look for the largest base 2 position less than 39.
For 39 the largest number is 32 or 25.
Then the largest number less than = 7 which is 4 or 22.
Then look for the largest number less than 7-4 = 3 which is 2 or 21.
And finally the largest number less than 3-2 = 1 which is 1 or 20.
39= or
128 64 32 16 8 4 2 1 27 26 25 24 23 22 21 20

12. Data Representation
Now computers store, present, and help us modify several different types of data:
Numbers
Text
Images and graphics
Audio
Video
All of this data is stored in the computer as binary digits. 

13. Analog and Digital Data
Signal
Analog signal is a continuous signal which represents physical measurements.
Digital signals are discrete time signals generated by digital modulation.
Applications
Thermometer
PCs, PDAs
Bandwidth
Analog signal processing can be done in real time and consumes less bandwidth.
There is no guarantee that digital signal processing can be done in real time and consumes more bandwidth to carry out the same information.
Memory
Stored in the form of wave signal
Stored in the form of binary bit
Power
Analog instrument draws large power
Digital instrument draws only negligible power
Cost
Low cost and portable
Cost is high and not easily portable
While Analog is cheaper it has more errors and size limitations.

14. Numeric Representation - Integers
Remember
bit - binary digit, a 0 or 1
byte - a series of 8 bits,
The largest positive integer that can be represented with one byte is (25510). So we have to use multiple bytes to store larger integers.
What about negative integers? Most computers use two’s compliment.
(see bwagner’s presentation of two’s compliment)
Addition of different bases needed