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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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Why binary
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
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
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
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
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
Iterate

  • 51/2 = 25, remainder 1

  • 25/2 = 12, remainder 1

  • 12/2 = 6, remainder 0


Iterate1
Iterate

  • 6/2 = 3, remainder 0

  • 3/2 = 1, remainder 1

  • 1/2 = 0, remainder 1


Recap
Recap

127 + 126 + 025 + 024

+ 123 + 122 + 021 + 120

205


Adding binary numbers
Adding Binary Numbers

  • 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
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
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 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
Two’s Complement

  • Step 1: exchange 1’s and 0’s

  • Step 2: add 1


Riddle
Riddle

  • Is it 214?

  • Or is it – 42?

  • Or is it …?

  • It’s a matter of interpretation

    • How was it declared?


Hexadecimal numbers
Hexadecimal Numbers

  • 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


Decimal binary hex

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
Binary to Hex

  • Break a binary string into groups of four bits (nibbles)

  • Convert each nibble separately


Addresses
Addresses

  • 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
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


Ip addresses
IP Addresses

  • 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
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

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

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


Ascii
ASCII

  • 0  00110000

  • 1  00110001

  • A  01000001

  • B  01000010

  • a  01100001

  • b  01100010


Booleans
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
Fractions

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


Converting decimal to binary ii
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
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
Converting decimal to binary IV

  • Put together the integral and fractional parts

  • 98.6  1100010.1001100110011001


Scientific notation
Scientific notation

  • Used to represent very large and very small numbers

    • Ex. Avogadro’s number

      •  6.0221367  1023 particles

      •  602213670000000000000000

    • Ex. Fundamental charge e

      •  1.60217733  10-19 C

      •  0.000000000000000000160217733 C


Floats
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
Mantissa and Exponent

  • .11000101001100110011001  200000111

  • Mantissa

  • .11000101001100110011001  200000111

  • Exponent


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