Abstraction – Number Systems and Data Representation

Memory• Computer memory is used

to store data • The smallest unit of

memory is a bit (Binary digIT)

• A bit can be off (no voltage) or on (has voltage) which we interpret to be 0 or 1

• Memory is organized into 8 bit contiguous groups called bytes. A megabyte is 1,048,576 bytes (over 1 million bytes). A gigabyte is over 1 billion bytes.

How does Memory Represent Values?

• The different patterns of the on and off bits in a byte determine the value stored

• Numbers are stored using binary numbers– 101 is 1 * 20 + 0 * 21 + 1 * 22 = 5

• Characters are internally represented as numbers– Different numbers represent different characters– There are several systems for assigning numbers to

characters: • ASCII, EBCDIC, and Unicode

Encodings Make Computers Powerful

• Voltages are interpreted as numbers• Numbers can be interpreted as characters• Characters can be interpreted to be part of a

link to Sun’s Java Site

0100 0001

off on off off off off off on

a

<a href=http://java.sun.com>Sun’s Java Site </a>

Notepad Exercise• Open notepad and type a

sentence in it• Save the file• Check the size in bytes by

leaving the cursor over the file name – Or right click and check

properties

• Now count the number of letters and spaces

Binary Numbers

• A bit is a binary digit with a value of 0 or 1– A group of 8 bits is a byte

• Computer memory is allocated in bytes• Numbers are stored using the binary number system

– With digits of 0 or 1 and powers of 2

• Other number systems– Decimal- digits of 0 to 9 and powers of 10– Octal - digits of 0 to 7 and powers of 8– Hexadecimal – digits of 0 to 9 and A, B, C, D, E, F and

powers of 16.

05-DrawingInJava 6

Converting from Binary to Decimal

• Multiply the digit value times the place value and add up the results to convert from binary to decimal

• The place values start with 20 on the right (which is 1) and increase to the left

7

Converting from Decimal to Binary

• Subtraction MethodKeep subtracting out

largest power of two until nothing remains

8

Converting from Decimal to Binary

• Division Method

Read result from top to bottom.

9

Binary Addition

• To add two decimal numbers you add the digits and if the total is greater than ten you carry the one into the next column

• To add two binary numbers – 0 + 0 = 0– 0 + 1 and 1 + 0 = 1– 1 + 1 = 0 with a carry of 1 into the next column to the left

00 10 111 01 01 001---- --- ------ 01 11 1000 00111001010 01010101101 -------------------10001110111

2’s Compliment Notation

• Computers actually only know how to add– So, how do they handle subtraction?

• Computers subtract by adding a negative number

• How do you represent a negative number in memory?– Positive numbers in 2’s compliment are just the

same as a binary number– For negative numbers reverse 0s and 1s and then

add 1

2’s Compliment Example

• To subtract 3 from 7• First represent both as a binary number

– 7 is 0000 0111– 3 is 0000 0011

• Reverse the 0s and 1s and then add 1 to get -3– 0000 0011 reversed is 1111 1100– add 1 0000 0001– The result is 1111 1101

Add the Negative Number

• To subtract 3 from 7• Add -3 to 7

– 7 is 0000 0111– -3 is 1111 1101– The result is 1 0000 0100

• Throw away the leftmost 1• The answer is 0000 0100 which is 4

Patterns Exercise

• How many different patterns of on and off bits are there in 3 bits? How many in 4 bits? How many in 8 bits?

• 000 is one pattern• 001 is another pattern• 010 is another pattern

Does the number of patterns matter?• Some garage door

openers in the 70s used 8 bits to set the code to use to open the door– Giving 256 different

patterns– Which is enough that

you won’t open your neighbors door

– But small enough that someone could try each one

Remote Entry Systems• With 8 bits for a code you

have a 1/256 chance of a random code working– You don’t want someone

opening your car in a place with lots of cars (like a mall)

• There are also radio scanners that can capture your code– So you want the code to

change each time • Modern remote entry

systems use a 40 bit rolling code