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## PowerPoint Slideshow about 'Binary Numbers' - noreen

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Presentation Transcript

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 = 2102 + 0101 + 5100
- 11001101 = 127 + 126 + 025 + 024 + 123 + 122 + 021 + 120 = 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

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

- 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
- Step 2: add 1

Riddle

- Is it 214?
- Or is it – 42?
- Or is it …?
- It’s a matter of interpretation
- How was it declared?

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

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 HexBinary to Hex

- Break a binary string into groups of four bits (nibbles)
- Convert each nibble separately

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?

- 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

- 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

- Integer part

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

- Fractional part

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
- Ex. Avogadro’s number
- 6.0221367 1023 particles
- 602213670000000000000000

- Ex. Fundamental charge e
- 1.60217733 10-19 C
- 0.000000000000000000160217733 C

- Ex. Avogadro’s number

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

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