# Bits, bytes, binary numbers, and the representation of

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Bits, bytes, binary numbers, and the representation of information

• computers represent, process, store, copy, and transmit everything as numbers

– hence "digital computer"

• the numbers can represent anything

– not just numbers that you might do arithmetic on

• the meaning depends on context

– as well as what the numbers ultimately represent – e.g., numbers coming to your computer or phone from your

wi-ﬁ connection could be email, movies, music, documents, apps, Skype call, ...

Encoding sound

• need to measure intensity/loudness often enough and accurately enough that we can reconstruct it well enough

• higher frequency = higher pitch • human ear can hear ~ 20 Hz to 20 KHz

– taking samples at twice the highest frequency is good enough (Nyquist)

• CD audio usually uses

– 44,100 samples / second – accuracy of 1 in 65,536 (= 2^16) distinct levels – two samples at each time for stereo – data rate is 44,100 x 2 x 16 bits/sample

= 1,411,200 bits/sec = 176,400 bytes/sec ~ 10.6 MB/minute

• MP3 audio compresses by clever encoding and removal of sounds that won't really be heard

– data rate is ~ 1 MB/minute

Binary numbers: use only digits 0 and 1 to represent numbers

• just like decimal except there are only two digits: 0 and 1

• everything is based on powers of 2 (1, 2, 4, 8, 16, 32, …)

– instead of powers of 10 (1, 10, 100, 1000, …)

• counting in binary or base 2:

0 1

1 binary digit represents 1 choice from 2; counts 2 things; 2 distinct values

00 01 10 11

2 binary digits represents 1 choice from 4; 4 distinct values

000 001 010 011 100 101 110 111

3 binary digits …

• binary numbers are shorthands for sums of powers of 2

11011 = 1 x 16 + 1 x 8 + 0 x 4 + 1 x 2 + 1 x 1 = 1 x 24 + 1 x 23 + 0 x 22 + 1 x 21 + 1 x 20

• counting in "base 2", using powers of 2

What's a bit?

• a bit represents one 2-way decision or a choice out of two possibilities

– yes / no, true / false, on / off, up / down, north / south, ...

• the abstraction of all of these is represented as 0 or 1

– enough to tell which of TWO possibilities has been chosen – a single digit with one of two values – hence "binary digit" – hence bit

• binary is used in computers because it's easy to make fast, reliable, small devices that have only two states

– high voltage/low voltage, current ﬂowing/not ﬂowing (chips) – electrical charge present/not present (RAM, ﬂash) – magnetized this way or that (disks) – light bounces off/doesn't bounce off (cd-rom, dvd)

• all information in a computer is stored and processed as bits

Powers of two, powers of ten

1 bit = 2 possibilities 2 bits = 4 possibilities 3 bits = 8 possibilities ... n bits = 2n

210 = 1,024 is about 1,000 or 1K or 103 220 = 1,048,576 is about 1,000,000 or 1M or 106 230 = 1,073,741,824 is about 1,000,000,000 or 1G or 109

the approximation is becoming less good but it's still good enough for estimation

• terminology is often imprecise:

– " 1K " might mean 1000 or 1024 (103 or 210) – " 1M " might mean 1000000 or 1048576 (106 or 220)

Bytes

• "byte" = a group of 8 bits treated as a unit

– on modern machines, the fundamental unit of processing and memory addressing

– can encode any of 28 = 256 different values, e.g., numbers 0 .. 255 or a single letter like A or digit like 7 or punctuation like $

ASCII character set deﬁnes values for letters, digits, punctuation, etc.

• group 2 bytes together to hold larger entities

– two bytes (16 bits) holds 216 = 65536 values – a bigger integer, a character in a larger character set

Unicode character set deﬁnes values for almost all characters anywhere

• group 4 bytes together to hold even larger entities

– four bytes (32 bits) holds 232 = 4,294,967,296 values – an even bigger integer, a number with a fractional part (ﬂoating point),

a memory address – current machines use 64-bit integers and addresses (8 bytes)

264 = 18,446,744,073,709,551,616

• no fractional bytes: the number of bytes is always an integer

Interpretation of bits depends on context

• meaning of a group of bits depends on how they are interpreted • 1 byte could be

– 1 bit in use, 7 wasted bits (e.g., M/F in a database) – 8 bits representing a number between 0 and 255 – an alphabetic character like W or + or 7 – part of a character in another alphabet or writing system (2 bytes) – part of a larger number (2 or 4 or 8 bytes, usually) – part of a picture or sound – part of an instruction for a computer to execute

instructions are just bits, stored in the same memory as data different kinds of computers use different bit patterns for their instructions

laptop, cellphone, game machine, etc., all potentially different

– part of the location or address of something in memory – ...

• one program's instructions are another program's data

– when you download a new program from the net, it's data – when you run it, it's instructions

Converting decimal to binary

repeat while the number is > 0: divide the number by 2 write the remainder (0 or 1) use the quotient as the number and repeat

the answer is the resulting sequence in reverse (right to left) order

divide 13 by 2, write "1", number is 6 divide 6 by 2, write "0", number is 3 divide 3 by 2, write "1", number is 1 divide 1 by 2, write "1", number is 0 answer is 1101

Hexadecimal notation

• binary numbers are bulky

• hexadecimal notation is a shorthand

• it combines 4 bits into a single digit, written in base 16

– a more compact representation of the same information

• hex uses the symbols A B C D E F for the digits 10 .. 15

0 1 2 3 4 5 6 7 8 9 A B C D E F

0 0000 4 0100 8 1000 C 1100

1 0001 5 0101 9 1001 D 1101

2 0010 6 0110 A 1010 E 1110

3 0011 7 0111 B 1011 F 1111

ASCII, using hexadecimal

• computers represent, process, store, copy, and transmit everything as numbers

– hence "digital computer"

• the numbers can represent anything

– not just numbers that you might do arithmetic on

• the meaning depends on context

– as well as what the numbers ultimately represent – e.g., numbers coming to your computer or phone from your

wi-ﬁ connection could be email, movies, music, documents, apps, Skype call, ...

Encoding sound

• need to measure intensity/loudness often enough and accurately enough that we can reconstruct it well enough

• higher frequency = higher pitch • human ear can hear ~ 20 Hz to 20 KHz

– taking samples at twice the highest frequency is good enough (Nyquist)

• CD audio usually uses

– 44,100 samples / second – accuracy of 1 in 65,536 (= 2^16) distinct levels – two samples at each time for stereo – data rate is 44,100 x 2 x 16 bits/sample

= 1,411,200 bits/sec = 176,400 bytes/sec ~ 10.6 MB/minute

• MP3 audio compresses by clever encoding and removal of sounds that won't really be heard

– data rate is ~ 1 MB/minute

Binary numbers: use only digits 0 and 1 to represent numbers

• just like decimal except there are only two digits: 0 and 1

• everything is based on powers of 2 (1, 2, 4, 8, 16, 32, …)

– instead of powers of 10 (1, 10, 100, 1000, …)

• counting in binary or base 2:

0 1

1 binary digit represents 1 choice from 2; counts 2 things; 2 distinct values

00 01 10 11

2 binary digits represents 1 choice from 4; 4 distinct values

000 001 010 011 100 101 110 111

3 binary digits …

• binary numbers are shorthands for sums of powers of 2

11011 = 1 x 16 + 1 x 8 + 0 x 4 + 1 x 2 + 1 x 1 = 1 x 24 + 1 x 23 + 0 x 22 + 1 x 21 + 1 x 20

• counting in "base 2", using powers of 2

What's a bit?

• a bit represents one 2-way decision or a choice out of two possibilities

– yes / no, true / false, on / off, up / down, north / south, ...

• the abstraction of all of these is represented as 0 or 1

– enough to tell which of TWO possibilities has been chosen – a single digit with one of two values – hence "binary digit" – hence bit

• binary is used in computers because it's easy to make fast, reliable, small devices that have only two states

– high voltage/low voltage, current ﬂowing/not ﬂowing (chips) – electrical charge present/not present (RAM, ﬂash) – magnetized this way or that (disks) – light bounces off/doesn't bounce off (cd-rom, dvd)

• all information in a computer is stored and processed as bits

Powers of two, powers of ten

1 bit = 2 possibilities 2 bits = 4 possibilities 3 bits = 8 possibilities ... n bits = 2n

210 = 1,024 is about 1,000 or 1K or 103 220 = 1,048,576 is about 1,000,000 or 1M or 106 230 = 1,073,741,824 is about 1,000,000,000 or 1G or 109

the approximation is becoming less good but it's still good enough for estimation

• terminology is often imprecise:

– " 1K " might mean 1000 or 1024 (103 or 210) – " 1M " might mean 1000000 or 1048576 (106 or 220)

Bytes

• "byte" = a group of 8 bits treated as a unit

– on modern machines, the fundamental unit of processing and memory addressing

– can encode any of 28 = 256 different values, e.g., numbers 0 .. 255 or a single letter like A or digit like 7 or punctuation like $

ASCII character set deﬁnes values for letters, digits, punctuation, etc.

• group 2 bytes together to hold larger entities

– two bytes (16 bits) holds 216 = 65536 values – a bigger integer, a character in a larger character set

Unicode character set deﬁnes values for almost all characters anywhere

• group 4 bytes together to hold even larger entities

– four bytes (32 bits) holds 232 = 4,294,967,296 values – an even bigger integer, a number with a fractional part (ﬂoating point),

a memory address – current machines use 64-bit integers and addresses (8 bytes)

264 = 18,446,744,073,709,551,616

• no fractional bytes: the number of bytes is always an integer

Interpretation of bits depends on context

• meaning of a group of bits depends on how they are interpreted • 1 byte could be

– 1 bit in use, 7 wasted bits (e.g., M/F in a database) – 8 bits representing a number between 0 and 255 – an alphabetic character like W or + or 7 – part of a character in another alphabet or writing system (2 bytes) – part of a larger number (2 or 4 or 8 bytes, usually) – part of a picture or sound – part of an instruction for a computer to execute

instructions are just bits, stored in the same memory as data different kinds of computers use different bit patterns for their instructions

laptop, cellphone, game machine, etc., all potentially different

– part of the location or address of something in memory – ...

• one program's instructions are another program's data

– when you download a new program from the net, it's data – when you run it, it's instructions

Converting decimal to binary

repeat while the number is > 0: divide the number by 2 write the remainder (0 or 1) use the quotient as the number and repeat

the answer is the resulting sequence in reverse (right to left) order

divide 13 by 2, write "1", number is 6 divide 6 by 2, write "0", number is 3 divide 3 by 2, write "1", number is 1 divide 1 by 2, write "1", number is 0 answer is 1101

Hexadecimal notation

• binary numbers are bulky

• hexadecimal notation is a shorthand

• it combines 4 bits into a single digit, written in base 16

– a more compact representation of the same information

• hex uses the symbols A B C D E F for the digits 10 .. 15

0 1 2 3 4 5 6 7 8 9 A B C D E F

0 0000 4 0100 8 1000 C 1100

1 0001 5 0101 9 1001 D 1101

2 0010 6 0110 A 1010 E 1110

3 0011 7 0111 B 1011 F 1111

ASCII, using hexadecimal

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