2.1 Binary Numbers

1. Exam Points

• Represent data using different number bases/systems.
  • Decimal: 19 (0-9), Binary: 1111 (0-1), Hexadecimal: A9F (0-9 A-F)
  • Number of items that can be represented using n bits: 2^n
  • The largest number that can be represented using n bits: 2^n-1
  • Reresenting colors using different color bases.
• Convert between two number bases.
  • Decimal <-> Binary
  • Decimal <-> Hexadecimal
• Overflow errors and Round-off errors
• Analog signal and digital signal, and how analog signals are converted to digital signals.

2. Knowledge Points

(1) Representing Data

• Data are a collection of discrete or continuous values that convey information.

• Computing devices represent data digitally using bits (1 and 0).

• 1 byte = 8 bits.

• Commonly used number bases/systems:

• Converting numbers between different number bases.

  • Convert Binary/Hexadecimal to Decimal ( * and + )
  
  • Convert Decimal to Binary/Hexadecimal ( / and %, write reversely)
  

• Representing data using different number bases.

  

• ASCII

  • ASCII is used to convert text to binary form so computers can read it.
  • These standard character sets use 7 bits.
  • Therefore, ASCII can only represent 128 characters from 0 to 127.

(2) Representing Numbers

• Integer number

  • Computers use binary (base 2) notation. (Ex. 11 is 1011)
  • How many numbers can be represented using n bits? 2^n
  • The largest integer that can be represented using n bits? 2^n-1
  • Overflow Error: Integers are represented by a fixed number of bits, if an integer is out of the range, then overflow errors occur. (Ex. store 18 using 4 bits)

• Float number

  • Float numbers are imprecise in computers since computers use a fixed number of bits to store numbers. (Ex. 6.3 may be stored as 6.2999997).
  • Round-off Errors (舍入误差) : In programming languages, the fixed number of bits used to represent real numbers/float numbers limits the range and mathematical operations on values; this limitation can result in round-off and other errors.

(3) Representing Colors and Images

• The human eye primarily detects red, green, and blue. Other colors are a combination of these colors in different amounts.
• Examples
• Images are composed of pixels, each of which contains a specific color value.
• Resolution (分辨率): a measure of the sharpness of an image. (Ex. 1920 x 1080).
• Bit depth (位深) : The number of bits used to record the color of each pixel point. (记录每个像素点颜色时使用的位数。)

(4) Representing Videos

• Videos are made from a series of static images refreshed quickly.
• FPS(每秒帧数): refers to the number of frames per second. The higher the number of frames per second, the smoother.
• Ex. Movie: 24fps

(5) Representing Sound

• Signal : used to transmit information, normally electrical signals.
  • Analog signal: are continuous and vary smoothly over time. It can take on any value(infinite) within a certain range. (All signals that are natural or come naturally are analog signals).
  • Digital signal: are discrete and represented by a series of separate values.
  • Analog can be affected by noise during transmission while digital signals are more resistant to noise.
  • Analog signals can be converted to digital signals.
  • Convert analog signals to digital signals.
    • Sampling (采样)
    • Quantization (量化)
    • Encoding (编码)
      

3. Exercises

• Practice
• Solution