PHY 201 (Blum) 1

Binary Numbers

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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

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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 volts5 volts

0 voltsDecimal Binary

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Range actually split in three

High

Low

Forbidden range

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It doesn’t matter …. Some of the standard voltages

coming from a computer’s power are ideally supposed to be 3.30 volts, 5.00 volts and 12.00 volts

Typically they are 3.28 volts, 5.14 volts or 12.22 volts or some such value

So what, who cares

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How to represent big integers

Use positional weighting, same as with decimal numbers

205 = 2102 + 0101 + 5100

Decimal – powers of ten 11001101 = 127 + 126 + 025 +

024 + 123 + 122 + 021

+ 120 = 128 + 64 + 8 + 4 + 1 = 205 Binary – powers of two

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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

1

0 1

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Iterate 51/2 = 25, remainder 1

25/2 = 12, remainder 1

12/2 = 6, remainder 0

1 0 1

1 1 0 1

0 1 1 0 1

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Iterate 6/2 = 3, remainder 0

3/2 = 1, remainder 1

1/2 = 0, remainder 1

0 0 1 1 0 1

1 0 0 1 1 0 1

1 1 0 0 1 1 0 1

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Recap

1 1 0 0 1 1 0 1

127 + 126 + 025 + 024

+ 123 + 122 + 021 + 120

205

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Finite representation Typically we just think computers do binary

math. But an important distinction between binary

math in the abstract and what computers do is that computers are finite.

There are only so many flip-flops or logic gates in the computer.

When we declare a variable, we set aside a certain number of flip-flops (bits of memory) to hold the value of the variable. And this limits the values the variable can have.

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Same number, different representation 5 using 8 bits 0000 0101 5 using 16 bits 0000 0000 0000 0101 5 using 32 bits 0000 0000 0000 0000 0000 0000

0000 0101

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Adding Binary Numbers Same as decimal; if the 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

1

3 9

+ 3 5

7 4

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Adding Binary Numbers

1 1 1 1

0 1 0 0 1 1 1

+ 0 1 0 0 0 1 1

1 0 0 1 0 1 0

carries

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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.

*The End of the World as We Know It

1 1

1 0 1 0 1 0 1 0

+ 1 1 0 0 1 1 0 0

1 0 1 1 1 0 1 1 0

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Biggest unsigned integers 4 bit: 1111 15 = 24 - 1 8 bit: 11111111 255 = 28 – 1 16 bit: 1111111111111111

65535= 216 – 1 32 bit:

11111111111111111111111111111111 4294967295= 232 – 1

Etc.

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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)

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Negative numbers Negative x is the number that when added

to x gives zero

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

1 1 1 1 1 1 1

0 0 1 0 1 0 1 0

1 1 0 1 0 1 1 0

1 0 0 0 0 0 0 0 0

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Two’s Complement

Step 1: exchange 1’s and 0’s

Step 2: add 1 (to the lowest bit only)

0 0 1 0 1 0 1 0

1 1 0 1 0 1 0 1

1 1 0 1 0 1 1 0

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Sign bit With the two’s complement approach,

all positive numbers start with a 0 in the left-most, most-significant bit and all negative numbers start with 1.

So the first bit is called the sign bit. But note you have to work harder than

just strip away the first bit. 10000001 IS NOT the 8-bit version of –

1

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Add 1’s to the left to get the same negative number using more bits

-5 using 8 bits 11111011 -5 using 16 bits 1111111111111011 -5 using 32 bits 11111111111111111111111111111011 When the numbers represented are whole

numbers (positive or negative), they are called integers.

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Biggest signed integers 4 bit: 0111 7 = 23 - 1 8 bit: 01111111 127 = 27 – 1 16 bit: 0111111111111111 32767=

215 – 1 32 bit:

01111111111111111111111111111111 2147483647= 231 – 1

Etc.

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Most negative signed integers 4 bit: 1000 -8 = - 23

8 bit: 10000000 - 128 = - 27

16 bit: 1000000000000000 -32768= - 215

32 bit: 10000000000000000000000000000000 -2147483648= - 231

Etc.

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Riddle

Is it 214? Or is it – 42? Or is it Ö? Or is it …? It’s a matter of interpretation

How was it declared?

1 1 0 1 0 1 1 0

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3-bit unsigned and signed

7 1 1 1

6 1 1 0

5 1 0 1

4 1 0 0

3 0 1 1

2 0 1 0

1 0 0 1

0 0 0 0

3 0 1 1

2 0 1 0

1 0 0 1

0 0 0 0

-1 1 1 1

-2 1 1 0

-3 1 0 1

-4 1 0 0

Think of an odometer reading 999999 and the car travels one more mile.

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Fractions Similar to what we’re used to with

decimal numbers

3.14159 =

3 · 100 + 1 · 10-1 + 4 · 10-2 + 1 · 10-3 + 5 · 10-4 + 9 · 10-5

11.001001 =

1 · 21 + 1 · 20 + 0 · 2-1 + 0 · 2-2 + 1 · 2-3 + 0 · 2-4 + 0 · 2-5

+ 1 · 2-6

(11.001001

3.140625)

Places 11.001001

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Two’s placeOne’s place

Half’s place

Fourth’s place Eighth’s

place Sixteenth’s place

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Decimal to binary 98.61

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

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Decimal to binary 98.61

Fractional part 0.61 2 = 1.22 0.22 2 = 0.44 0.44 2 = 0.88 0.88 2 = 1.76 0.76 2 = 1.52 0.52 2 = 1.04

.100111

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Decimal to binary Put together the integral and

fractional parts 98.61 1100010.100111

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Another Example (Whole number part)

123.456 Integer part

123 / 2 = 61 remainder 1 61 / 2 = 30 remainder 1 30 / 2 = 15 remainder 0 15 / 2 = 7 remainder 1 7 / 2 = 3 remainder 1 3 / 2 = 1 remainder 1 1 / 2 = 0 remainder 1

1111011

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Checking: Go to All Programs/Accessories/Calculator

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Put the calculator in Programmer view

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Enter number, put into binary mode

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Another Example (fractional part)

123.456 Fractional part

0.456 2 = 0.912 0.912 2 = 1.824 0.824 2 = 1.648 0.648 2 = 1.296 0.296 2 = 0.592 0.592 2 = 1.184 0.184 2 = 0.368 …

.0111010…

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Checking fractional part: Enter digits found in binary mode

Note that the leading zero does not display.

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Convert to decimal mode, then

Edit/Copy result. Switch to Scientific View. Edit/Paste

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Divide by 2 raised to the number of digits (in this case 7, including leading zero)

1 2

3 4

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Finally hit the equal sign. In most cases it will not be exact

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Other way around Multiply fraction by 2 raised to the desired

number of digits in the fractional part. For example .456 27 = 58.368

Throw away the fractional part and represent the whole number 58 111010

But note that we specified 7 digits and the result above uses only 6. Therefore we need to put in the leading 0. (Also the fraction is less than .5 so there’s a zero in the ½’s place.) 0111010

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Limits of the fixed point approach

Suppose you use 4 bits for the whole number part and 4 bits for the fractional part (ignoring sign for now).

The largest number would be 1111.1111 = 15.9375

The smallest, non-zero number would be 0000.0001 = .0625

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Floating point representation Floating point representation

allows one to represent a wider range of numbers using the same number of bits.

It is like scientific notation. We’ll do this later in the semester.

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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

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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

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Binary to Hex Break a binary string into groups of

four bits (nibbles) Convert each nibble separately

1 1 1 0 1 1 0 0 1 0 0 1

E C 9

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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

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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

0 0 0

0 0 1

0 1 0

0 1 1

1 0 0

1 0 1

1 1 0

1 1 1

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IP(v4) Addresses An IP(v4) 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?

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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”

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Unicode Unicode uses 16 bits, how many

characters can be represented? Enough for English, Chinese,

Arabic and then some.

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ASCII 0 00110000 (48) 1 00110001 (49) … A 01000001 (65) B 01000010 (66) … a 01100001 (97) b 01100010 (98) …

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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

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Boolean Operators A.k.a. logical operators Have Boolean input and Boolean output Standard:

AND OR NOT XOR (either or but not both) NOR = NOT(OR) NAND = NOT(AND)

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Truth Tables AND

INPUT OUTPUT

A B A AND B

0 0 0

0 1 0

1 0 0

1 1 1

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Truth Tables (Cont.) OR

INPUT OUTPUT

A B A OR B

0 0 0

0 1 1

1 0 1

1 1 1

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Truth Tables (Cont.) XOR (Excluded OR)

INPUT OUTPUT

A B A XOR B

0 0 0

0 1 1

1 0 1

1 1 0

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Numbers from Logic All of the numerical operations we have

talked about are really just combinations of logical operations

E.g. the adding operation is just a particular combination of logic operations

Possibilities for adding two bits 0+0=0 (with no carry) 0+1=1 (with no carry) 1+0=1 (with no carry) 1+1=0 (with a carry)

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Addition Truth TableINPUT OUTPUT

A BSum

A XOR B

CarryA AND

B

0 0 0 0

0 1 1 0

1 0 1 0

1 1 0 1

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All is NAND Actually you can use one logic gate

(the NAND) and a few tricks (like De Morgan’s theorem) to build all of the “combinatorial” circuitry (the circuitry that doesn’t involve memory)

NORs work too But we tend to think in ANDs, ORs

and NOTs

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Bit manipulation You can use an AND to select out

part of a word (where s is a 1 or 0, etc)s t u v w x y z

1 1 1 1 0 0 0 0

s t u v 0 0 0 0

AND

gives

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IP Addresses Revisted LaSalle’s IP address is what’s called a

Class B IP address Of the 32 bits the first two bits are 10

(this identifies us as Class B) The remaining 14 bits of the first two

bytes identify us as LaSalle The remaining 2 bytes are for our

internals use (to assign computers within LaSalle)

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In or Out To see if an address is local to

LaSalle, you would restrict your attention to the first two bytes.

HOW? AND it with FFFF0000

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Subnets A network (like LaSalle’s) can be

divided further into sub-networks Then subnet masks are used to

determine whether or not another computer is on the same subnet