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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

• 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

• 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

• 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

• 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

• 205/2 = 102 with a remainder of 1, place the 1 in the least significant digit position

• Repeat 102/2 = 51, remainder 0

• 51/2 = 25, remainder 1

• 25/2 = 12, remainder 1

• 12/2 = 6, remainder 0

• 6/2 = 3, remainder 0

• 3/2 = 1, remainder 1

• 1/2 = 0, remainder 1

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

• 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

• 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 x is that number when added to x gives zero

• Ignoring overflow the two eight-bit numbers above sum to zero

• Step 1: exchange 1’s and 0’s

• 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

• 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

• 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?

• 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 uses 16 bits, how many characters can be represented?

• Enough for English, Chinese, Arabic and then some.

• 0  00110000

• 1  00110001

• A  01000001

• B  01000010

• a  01100001

• b  01100010

• 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

• Similar to what we’re used to with decimal numbers

• 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

• 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

• Put together the integral and fractional parts

• 98.6  1100010.1001100110011001

• 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

• 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

• .11000101001100110011001  200000111

• Mantissa

• .11000101001100110011001  200000111

• Exponent