Binary Numbers

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# Binary Numbers - PowerPoint PPT Presentation

Binary Numbers. Why Binary?. Maximal distinction among values  m inimal corruption from noise Imagine taking the same physical attribute of a circuit, e.g. a voltage lying between 0 and 5 volts, to represent a number The overall range can be divided into any number of regions .

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## PowerPoint Slideshow about 'Binary Numbers' - noreen

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### Binary Numbers

Why Binary?
• Maximal distinction among values  minimal corruption from noise
• Imagine taking the same physical attribute of a circuit, e.g. a voltage lying between 0 and 5 volts, to represent a number
• The overall range can be divided into any number of regions
Don’t sweat the small stuff
• For decimal numbers, fluctuations must be less than 0.25 volts
• For binary numbers, fluctuations must be less than 1.25 volts

5 volts

0 volts

Binary

Decimal

It doesn’t matter ….
• Recall the power supply voltage measurements from lab 1
• Ideally they should be 5.00 volts and 12.00 volts
• Typically they were 5.14 volts or 12.22 volts
• So what, who cares
How to represent big integers
• Use positional weighting, same as with decimal numbers
• 205 = 2102 + 0101 + 5100
• 11001101 = 127 + 126 + 025 + 024 + 123 + 122 + 021 + 120 = 128 + 64 + 8 + 4 + 1 = 205
Converting 205 to Binary
• 205/2 = 102 with a remainder of 1, place the 1 in the least significant digit position
• Repeat 102/2 = 51, remainder 0
Iterate
• 51/2 = 25, remainder 1
• 25/2 = 12, remainder 1
• 12/2 = 6, remainder 0
Iterate
• 6/2 = 3, remainder 0
• 3/2 = 1, remainder 1
• 1/2 = 0, remainder 1
Recap

127 + 126 + 025 + 024

+ 123 + 122 + 021 + 120

205

• Same as decimal; if sum of digits in a given position exceeds the base (10 for decimal, 2 for binary) then there is a carry into the next higher position
Uh oh, overflow
• What if you use a byte (8 bits) to represent an integer
• A byte may not be enough to represent the sum of two such numbers
Bigger Numbers
• You can represent larger numbers by using more words
• You just have to keep track of the overflows to know how the lower numbers (less significant words) are affecting the larger numbers (more significant words)
Negative numbers
• Negative x is that number when added to x gives zero
• Ignoring overflow the two eight-bit numbers above sum to zero
Two’s Complement
• Step 1: exchange 1’s and 0’s
Riddle
• Is it 214?
• Or is it – 42?
• Or is it …?
• It’s a matter of interpretation
• How was it declared?
• Even moderately sized decimal numbers end up as long strings in binary
• Hexadecimal numbers (base 16) are often used because the strings are shorter and the conversion to binary is easier
• There are 16 digits: 0-9 and A-F
0  0000  0

1  0001  1

2  0010  2

3  0011  3

4  0100  4

5  0101  5

6  0110  6

7  0111  7

8  1000  8

9  1001  9

10  1010  A

11  1011  B

12  1100  C

13  1101  D

14  1110  E

15  1111  F

Decimal  Binary  Hex
Binary to Hex
• Break a binary string into groups of four bits (nibbles)
• Convert each nibble separately
• With user friendly computers, one rarely encounters binary, but we sometimes see hex, especially with addresses
• To enable the computer to distinguish various parts, each is assigned an address, a number
• Distinguish among computers on a network
• Distinguish keyboard and mouse
• Distinguish among files
• Distinguish among statements in a program
• Distinguish among characters in a string
How many?
• One bit can have two states and thus distinguish between two things
• Two bits can be in four states and …
• Three bits can be in eight states, …
• N bits can be in 2N states
• An IP address is used to identify a network and a host on the Internet
• It is 32 bits long
• How many distinct IP addresses are there?
Characters
• We need to represent characters using numbers
• ASCII (American Standard Code for Information Interchange) is a common way
• A string of eight bits (a byte) is used to correspond to a character
• Thus 28=256 possible characters can be represented
• Actually ASCII only uses 7 bits, which is 128 characters; the other 128 characters are not “standard”
Unicode
• Unicode uses 16 bits, how many characters can be represented?
• Enough for English, Chinese, Arabic and then some.
ASCII
• 0  00110000
• 1  00110001
• A  01000001
• B  01000010
• a  01100001
• b  01100010
Booleans
• A Boolean variable is something that is true or false
• Booleans have two states and could be represented by a single bit (1 for true and 0 for false)
• Booleans appearing in a program will take up a whole word in memory
Fractions
• Similar to what we’re used to with decimal numbers
Converting decimal to binary II
• 98.6
• Integer part
• 98 / 2 = 49 remainder 0
• 49 / 2 = 24 remainder 1
• 24 / 2 = 12 remainder 0
• 12 / 2 = 6 remainder 0
• 6 / 2 = 3 remainder 0
• 3 / 2 = 1 remainder 1
• 1 / 2 = 0 remainder 1
• 1100010
Converting decimal to binary III
• 98.6
• Fractional part
• 0.6  2 = 1.2
• 0.2  2 = 0.4
• 0.4  2 = 0.8
• 0.8  2 = 1.6
• 0.6  2 = 1.2
• 0.2  2 = 0.4
• REPEATS
• .100110
Converting decimal to binary IV
• Put together the integral and fractional parts
• 98.6  1100010.1001100110011001
Scientific notation
• Used to represent very large and very small numbers
•  6.0221367  1023 particles
•  602213670000000000000000
• Ex. Fundamental charge e
•  1.60217733  10-19 C
•  0.000000000000000000160217733 C
Floats
• SHIFT expression so it is just under 1 and keep track of the number of shifts
• 1100010.1001100110011001
• .11000101001100110011001  27
• Express the number of shifts in binary
• .11000101001100110011001  200000111
Mantissa and Exponent
• .11000101001100110011001  200000111
• Mantissa
• .11000101001100110011001  200000111
• Exponent