4.3 Converting with Base Systems

Number base systems are the foundation of how we represent and manipulate numbers in different formats. They're crucial in computing, where binary and hexadecimal reign supreme. Understanding these systems helps us grasp how computers process data and perform calculations.

Converting between bases is a key skill in working with different number systems. Whether you're using place value methods or repeated division, mastering these techniques opens doors to understanding computer science and digital electronics on a deeper level.

Number Base Systems

Number base conversion methods

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• Place value method for converting from base 10 to another base
  • Expresses the base 10 number as a sum of powers of the new base multiplied by their respective coefficients
  • Coefficients are determined by the digits in the new base representation
  • Example: $75_{10} = 7 \times 8^1 + 5 \times 8^0 = 115_8$
• Place value method for converting from another base to base 10
  • Multiplies each digit in the number by the base raised to the power of its place value position (rightmost digit has a place value of 0, increasing by 1 for each position to the left)
  • Sums the resulting products to obtain the base 10 equivalent
  • Example: $1101_2 = 1 \times 2^3 + 1 \times 2^2 + 0 \times 2^1 + 1 \times 2^0 = 13_{10}$
• Repeated division method for converting from base 10 to another base
  • Divides the base 10 number by the new base repeatedly until the quotient becomes 0
  • Collects the remainders in reverse order to form the digits of the new base representation
  • Example: $42_{10} = (42 \div 2 = 21$ remainder $0, 21 \div 2 = 10$ remainder $1, 10 \div 2 = 5$ remainder $0, 5 \div 2 = 2$ remainder $1, 2 \div 2 = 1$ remainder $0, 1 \div 2 = 0$ remainder $1) = 101010_2$
• Repeated multiplication method for converting from another base to base 10
  • Starts with the leftmost digit and multiplies it by the base, then adds the next digit to the right
  • Repeats this process, moving right, until all digits have been processed
  • The final result is the base 10 equivalent of the original number
  • Example: $1101_2 = ((1 \times 2 + 1) \times 2 + 0) \times 2 + 1 = 13_{10}$

Error identification in base conversions

• Incorrect place values
  • Occurs when the wrong power of the base is used for a given digit's position
  • Fix by ensuring each digit is multiplied by the base raised to the correct power
• Incorrect base usage
  • Happens when the wrong base is used during the conversion process
  • Fix by double-checking the base being converted to or from
• Incorrect remainders or quotients in repeated division
  • Results from errors in the division process or misrecording remainders
  • Fix by carefully rechecking division operations and remainder values
• Incorrect ordering of digits
  • Occurs when digits are written in the wrong order, especially when using the repeated division method
  • Fix by ensuring digits are written in the correct order, from left to right, based on the conversion method used

Arithmetic in non-decimal systems

• Counting in binary
  • Uses only the digits 0 and 1
  • Increments by 1, carrying over to the next place value when necessary
  • Example: $0, 1, 10, 11, 100, 101, \ldots$
• Counting in hexadecimal
  • Uses digits 0-9 and letters A-F (representing 10-15)
  • Increments by 1, carrying over to the next place value when necessary
  • Example: $0, 1, 2, \ldots, 9, A, B, \ldots, F, 10, 11, \ldots$
• Addition in binary
  • Adds each column, starting from the rightmost, and carries over when the sum is 2 or greater
  • Example: $1101_2 + 101_2 = (1+1=0$ carry $1, 0+0+1=1, 1+1=0$ carry $1, 1+0+1=0$ carry $1) = 10010_2$
• Addition in hexadecimal
  • Adds each column, starting from the rightmost, and carries over when the sum is 16 or greater
  • Example: $3E_{16} + 2A_{16} = (14+10=24, 2+3+1=6) = 68_{16}$
• Subtraction in binary
  • Subtracts each column, starting from the rightmost, and borrows when necessary
  • Example: $1101_2 - 101_2 = (1-1=0, 0-0=0, 1-1=0, 1-0=1) = 1000_2$
• Subtraction in hexadecimal
  • Subtracts each column, starting from the rightmost, and borrows when necessary
  • Example: $3E_{16} - 2A_{16} = (14-10=4, 3-2=1) = 14_{16}$
• Multiplication in binary
  1. Creates a partial product for each binary digit in the multiplier
  2. Shifts each partial product left by the place value of the corresponding multiplier digit
  3. Sums the partial products to get the final result
  • Example: $1101_2 \times 101_2 = (1101 \times 1 = 1101, 1101 \times 0 = 0000, 1101 \times 1 = 1101000) = 1000001_2$
• Multiplication in hexadecimal
  • Follows the same process as binary multiplication, but uses the hexadecimal multiplication table for single-digit multiplications
  • Example: $3E_{16} \times 2A_{16} = (3E \times A = 276, 3E \times 2 = 7C0) = A7E_{16}$

Common Number Bases and Their Applications

Binary (base 2)

• Widely used in computer systems and digital electronics
• Represents data using only the digits 0 and 1
• Each digit is called a bit (binary digit)
• Example: ASCII character encoding, digital logic circuits

Octal (base 8)

• Used in some older computer systems and digital electronics
• Represents data using digits 0-7
• Often used as a compact representation of binary, with each octal digit corresponding to three binary digits
• Example: Unix file permissions, color codes in early computer graphics

Hexadecimal (base 16)

• Commonly used in computer systems and digital electronics
• Represents data using digits 0-9 and letters A-F (representing 10-15)
• Often used as a compact representation of binary, with each hexadecimal digit corresponding to four binary digits
• Frequently used to represent colors in web design and computer graphics (RGB color model)
• Example:

  #FF0000

  represents red, memory addresses in programming

Number Systems and Notation

Positional Notation

• A system for representing numbers where each digit's position determines its value
• The value of each digit is multiplied by the base raised to the power of its position
• Used in most modern number systems (e.g., decimal, binary, hexadecimal)

Base Conversion

• The process of changing a number's representation from one base to another
• Involves using methods like repeated division or multiplication
• Essential for working with different number systems in computing and mathematics

Radix Point

• The symbol used to separate the integer and fractional parts of a number in positional notation
• Equivalent to the decimal point in base 10, but applicable to all number systems
• Example: In binary, 101.01 represents 5.25 in decimal, with the radix point separating whole and fractional parts

Complement System

• A method for representing negative numbers in digital systems
• Commonly used in computer arithmetic to simplify subtraction operations
• Two's complement is widely used in binary systems for signed integer representation