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1.3 Binary Coding and Decoding

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By So Hee Park 12 min read

✅ Positive Integer

  • Integer: 8bits
  • there are two ways to encode positive integers into binary

1️⃣ Use dividing by 2

  • divide the Integer by 2 several times
  • take the final quotient and the remainders
  • from down to up, write the quotient and remainder down
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33/2 = quotient: 16, remainder: 1
16/2 = quotient: 8, remainder: 0
8/2 = quotient: 4, remainder: 0
4/2 = quotient: 2, remainder: 0
2/2 = quotient: 1, remainder: 0
from down to up, 33 = 1(quotient) and 00001
👉🏻 33 = 100001

2️⃣ Use the weights rule of the power of 2

  • use the weights rule of the powers of 2
  • draw positions, and each position is double to the one at the right
  • then add 1 in positions needed and 0 in positions not needed
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33 =  __  __  __  __  __  __
      32  16   8   4   2   1
  👉🏻  1    0   0   0    0  1

❓ 0 = 00000000
❓ 255 = 11111111

✅ Negative Integer

🚩 Sign Flag

  • 1️⃣ Change the positive Integer into binary
  • 2️⃣ Add a 1 at the very left, this is a special 1
  • ⭐️ Sign flag: special 1 to indicate that this Integer is a negative Integer
  • This special 1 is stored in a different place

  • sign flag means this Integer is a negative 🚩 Flags are electric signals stored in specific places for special circumstances

  • If the flag is off, you add it as part of the number
  • If the flag in on, you turn it into a negative symbol
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❓ code -78 to binary
78 = 01001110
➕ add 1 on the left, why make 8 spaces? bc 8bits
👉🏻 101001110

❓ code -16 to binary
16 = 00010000
➕ add 1 on the left
👉🏻 100010000

✅ Decimal Integer

FLU(Floating Point Unit)

  • Decimal numbers are stored in a special place called FLU(Floating Point Unit)

  • The convention used for this is IEEE754 for international technique, IE3754. The steps are as following

  • 1️⃣ Take the Integer part, transform into binary but discord all the 0s at the left
  • no extra 0s

  • 2️⃣ Take the Decimal part such as 0.xxx, we** multiply by 2 five times**, but transforming the appearing 1s into 0s. You take the 0s and the 1s AFTER the = from up to down

  • add it to the first coded number
  • 3️⃣ The dot needs to be represented, the decimal part is always 5bits. We are going to shift the decimal symbol(the dot) to the left until you leave it after the FIRST 1. AND you count how many steps you had to take. The number of steps need to be counted somehow.
  • ⭐️ In computing, we want things to always occupy the same number, this is a rule. NO variable length, always fixed length.

  • move the dot to the left until you reach the first number

  • add the number of jumps to 127

  • add 1 to the left if negative

  • or add 0 to the left if positie

  • 4️⃣ The number of steps, add 127 and then you code the result.
  • 5️⃣ We place that at the END of the number, leaving extra space in between, the IEEE754 want the number to occupy 32bits
  • 6️⃣ In order to have 32bits, if the number is negative, add a 1 at the left and BESIDES you light the sign flag. If the number is positive, add a 0, so that it occupies the same number as a negative one.
  • 7️⃣ We complete with 0s in order for it to sum up to 32bits
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❓ code -17.33 to binary
1️⃣ Integer part into binary
17 = 00010001 ➡️ discord all the left 0s ➡️ 10001
2️⃣ Decimal part, multiply by 2, five times
0.33 * 2 = 0.66
0.66 * 2 = 1.32
0.32 * 2 = 0.64
0.64 * 2 = 1.28
0.28 * 2 = 0.56
👉🏻 01010
then add to 17
👉🏻 10001.01010

👉🏻 1.000101010
count the jumps = 4

4 + 127 = 131
change 131 to binary 10000011
add the number 10000011 to the end


3️⃣
4️⃣
5️⃣
6️⃣
7️⃣

1️⃣
2️⃣
3️⃣
4️⃣
5️⃣
6️⃣
7️⃣
  • all the numbers that use this protocals are called floats since they are stored in the FPU
  • If instead of 32bits, we made this in 64bits(The standard is totally different procedure), we call them double floats
  • Double floats have more precision, always more than 2 decimals.
  • ⚠️ It is considered a bad programming technique to use double floats with things that could be in a regular float, since it is more accerlerated, more work for the PC.

✅ Text

☑️ Standard Text, Extended ASCII

  • for Standard Text, we use Extended ASCII
  • Extended ASCII is a FLC-8, each character is 8bits
  • each character 🟰 Integer
  • so you can code the Integer with the weights rule
  • there are no spaces
  • If you need a space, code the number 32
  • 👎🏻 However, Extended ASCII uses only 256 characters(from 0 to 255)

Screenshot-2025-10-06-at-00-44-22.png

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❓ Transform I_miss_you with ASCII code.
73 32 109 105 115 115 32 121 111 117 46

01001001001000000110110101101001………00101100

☑️ Extension of ASCII, UTF-8

  • UTF-8 is an extension of the extended ASCII
  • also known as Unicode
  • bc it is to be used with every language int he world
  • UTF 8 does not mean 8bits per symbol
  • it means multiple of 8bits per symbol
  • Some symbols are 8bits(1byte) like ASCII
  • Some symbols are 16bits(2bytes) like Greek, Cyrilic, Hebew, Arab...
  • Some symbols are 24bits(3bytes) like Korean

  • Some symbols are 32bits(4bytes) like Persian, Fenician, old languages and ALSO math

  • Each language, if possible uses its own version of ASCII
  • but due to Internet, a code was invented to allow intercommunication and translation between different languages

✅ Image

☑️ Two types of pictures

1️⃣ Raster

  • divided into pixels, get pixelized when increase size
  • JPG, gig, png
  • HR: Horizontal Resolution: how many pixels in horizontal
  • VR: Vertical Resolution: how many pixels in vertical
  • Resolution = HR * VR

✔️ Color Depth in Raster

  • Color Depth: how many bits you need for your image
  • 1 or 8 or 24
  • black and white ➡️ need 2 combos, need 1 bit(2^1) ➡️ color depth: 1
  • gray-scaled ➡️ need 256 combos, need 8 bits(2^8) ➡️ color depth: 8

  • full-colored ➡️ every color is R(0~255) + G(0~255) + B(0~255) ➡️ each R, G, B need 8 bits ➡️ need 24 bits ➡️ color depth: 24

    • pure red would be 255.0.0
    • pure green would be 0.255.0
    • black would be 0.0.0
    • white would be 255.255.255
    • as soon as there is one colored pixel, even if all pixels are black and white, color depth will be 24
    • if full color, each pixel will have 24bits, graphic card will write the bits
  • color codes are stored in graphic cards, in the driver
  • Red = 111111110000000000000000
  • I can use up to 2^24 colors, more or less 16000000(16 million)
    • (2^4)*(2^10)*(2^10) = more or less 16 _ 1000 _ 1000

  • If a graphic card had 64bytes, it can have 2^64 colors

    • (2^4)*(2^10)*(2^10)*(2^10)*(2^10)*(2^10)*(2^10)

✔️ Picture size in Raster

  • Picture size = HR * VR * D(color depth)
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Q: How many bits would a black and white picture have of 100 * 50?
- Picture size = HR * VR * D(color depth)
- 100 * 50 * 1 = 5000

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Q: How many bits would a gray picture have of 100 50?
- Picture size = HR * VR * D(color depth)
- 100 *  50 * 8 = 40000

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Q. If all the sound that my computer can make are 8, how many bits do my computer have?
- 3
- because 2^3 = 8

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Q: How many bits would a full color picture have of 100 * 50?
- 100 * 50 * 24 = 120000

2️⃣ Vector

  • more professional, more efficient when picture is not so detailed

  • svg

  • do not use pixel, use area by area/ zone by zone
  • call the color just once, then call the length/size/measurements of the area that has that color
  • red(255.0.0)* 10 meters + white(255.255.255) * 2 meters
  • the picture tends to be more compressed
  • 👍🏻 more efficient
  • 👍🏻 when the area is uniform, no big transition of color, use vector
  • ⭐️ big areas of same color(like football field, all green)
  • ↔️ image with many color changes, use Raster(like a christimas tree with lights)
  • if I make the picture bigger, zoom in, then just have to change the length/size/measurements

  • do not get pixelized

  • 👎🏻 however, as measuring device is more expensive,
  • vector is only used for professional purposes

📌 Main Computing Principle

  • 0: 0.5v

  • 1: 5v

  • If you have n bits to store 0s and 1s, you can have total of 2^n combinations
  • 1 bit: 2 combinations 0, 1
  • 2 bits: 4 combinations 00, 01, 10, 11

✅ Audio

  • Audio is vibration, need air to transmit
  • microphone has a small surface to capture vibration
  • microphone transforms the vibration to binary code
  • extension for audio is .wav
  • when compressed/summarized .mp3
  • if la, la, la, la, .mp3 summarizes sound and save la * 4
  • ⭐️ .mp3 does not lose sound, but just summarized
  • .mp3 is very compressed when there are uniform, repetitive sounds in the track
  • if many different sounds, .mp3 will not be so efficient

☑️ Sampling Frequency(Hertz)

  • how many times per second the computer pays attention to the microphone/vibration
  • sampling frequency ⬆️ better sound quality, capture more details of the sound wave ⬆️
  • sampling frequency ⬇️ cannot capture all the details of the sound ⬇️
  • if frequency is 1Hz, then listening 1 time/sec
  • if frequency is 1KHz, then listening 1000times/sec
  • if frequency is 1MHz, then listening 1000000 million times/sec
  • if frequency is 1GHz, then listening 1000000000 thousand million times/sec
  • ⭐️ If you capture a sound with a high sampling frequency, will get more details of the sound

☑️ Amplitudes(bits)

  • number of all the different sounds you want to capture
  • in an orchestra, you want to capture more instruments, need higher amplitude

  • higher amplitude ⬇️ more variety of sounds ⬇️

  • Q: If orchestra used 16 different sounds, how many bits would we need?
  • need 4 bits

  • Amplitude = 4

  • Q: If orchestra used 80 different sounds, how many bits would we need?
  • need 7 bits (2^7 = 128)
  • Amplitude = 7

☑️ Time(seconds)

  • how long the music lasts, measured in seconds
  • more time ⬆️ can save longer song ⬆️

☑️ Stereo

  • full stereo: use two sequences of bits(2bits) per ear, duplicate the sound
  • left and right ear listen to the whole song

  • need to multiply formula by 2

  • mono stereo: one sequence of bit, divide sound in two
  • left ear: background music, right ear: singer voice

✔️ Audio Size

  • size: Hertz * A * t
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Q: How many bits do you need for an orchestra for a 4 minute song, with 300 amplitude, sampled at 2GHz?
- size: Hertz * A * t
- 2*10^9 * 9 * (4*60)
 = 2000000000 * 9 * 240` bits

✅ Video

  • sequence of pictures + sound
  • .avi, .mov
  • compressed into .mp4
  • however, .mp4 suffered from forking
  • forking: people started using another form of video .wmv, not compatible with .mp4

☑️ FPS, Frames per second

  • how many images video shows per second
  • fps ⬆️ better quality of video ⬆️

☑️ Prediction in video

  • in video use prediction
  • as some pictures are important,
  • but other pictures are evident, so not as important
  • use picture I, P, B

✔️ Types of picture in video

  • Picture type I: Important
  • independent pictures, important pictures

  • mandatory to record them independently

  • Picture type P: Predicted
  • predicted pictures

  • can predict them, as it is very similar to the previous picture

  • Picture type B: Bidirectional
  • Bidirectional
  • extra movement to make video transition more natural
  • pictures used to improve transitions
  • make video look softer

✔️ Video size

  • picture size * fps * t + Audio * 2(stereo)
  • HR * VR * D * fps * t + Hertz * A * t * 2
  • ⭐️ audio is added
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Q: How many bits a 5 minute video have that has resolution of 1920\*1090 full color at 1000fps with stereo sound of 100 amplitudes sampled at a frequency of 25MHz?
- {(1920*1920 * 24) * 1000 * (5*60)} + {( 25 * 10^6)  * 7 * (5*60) * 2 bits}
  • ⭐️ only color depth and amplitude has to be converted into bits

✅ File

  • all the bits are stored in a unit called file

  • file: set of bits with information

  • metadata: extra information about the file
  • ⭐️ metadata is stored in the Operating system

  • Timestamp: created_date, last_modified_date

    • owner of the file
    • size of the file;lkj
    • extension of the file
    • permissions over the file, read/write/execute permissions, what you can do with the file
    • access control, who can access the file
  • Linux keeps metadata very far from data itself, 👍🏻 safe from haking
  • However, Windows keep metadata close to the data itself, not very safe

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CC BY Izzie [CC] Integers, Text part of the notes were completed thanks to Izzie, by referencing her notes. Thank you Izzie!

DAW bilingual, Computer System
This post is licensed under CC BY 4.0 by the author.