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4.3 Converting with Base Systems

5 min read•june 18, 2024

Number base systems are the foundation of how we represent and manipulate numbers in different formats. They're crucial in computing, where and 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 . 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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  • 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: 7510=7×81+5×80=115875_{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: 11012=1×23+1×22+0×21+1×20=13101101_2 = 1 \times 2^3 + 1 \times 2^2 + 0 \times 2^1 + 1 \times 2^0 = 13_{10}
  • for converting from base 10 to another base
    • Divides the base 10 number by the new base repeatedly until the becomes 0
    • Collects the remainders in reverse order to form the digits of the new base representation
    • Example: 4210=(42÷2=2142_{10} = (42 \div 2 = 21 0,21÷2=100, 21 \div 2 = 10 remainder 1,10÷2=51, 10 \div 2 = 5 remainder 0,5÷2=20, 5 \div 2 = 2 remainder 1,2÷2=11, 2 \div 2 = 1 remainder 0,1÷2=00, 1 \div 2 = 0 remainder 1)=10101021) = 101010_2
  • 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: 11012=((1×2+1)×2+0)×2+1=13101101_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,…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,…,9,A,B,…,F,10,11,…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: 11012+1012=(1+1=01101_2 + 101_2 = (1+1=0 carry 1,0+0+1=1,1+1=01, 0+0+1=1, 1+1=0 carry 1,1+0+1=01, 1+0+1=0 carry 1)=1001021) = 10010_2
  • Addition in hexadecimal
    • Adds each column, starting from the rightmost, and carries over when the sum is 16 or greater
    • Example: 3E16+2A16=(14+10=24,2+3+1=6)=68163E_{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: 11012−1012=(1−1=0,0−0=0,1−1=0,1−0=1)=100021101_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: 3E16−2A16=(14−10=4,3−2=1)=14163E_{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: 11012×1012=(1101×1=1101,1101×0=0000,1101×1=1101000)=100000121101_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: 3E16×2A16=(3E×A=276,3E×2=7C0)=A7E163E_{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: 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 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 ( 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
  • 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 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

Key Terms to Review (24)

0x: 0x is a prefix used to denote that a number is in hexadecimal (base-16) format. This notation is crucial when dealing with different base systems, as it helps to clearly identify that the number following it is not in the usual decimal (base-10) format. The hexadecimal system is commonly used in computing and digital electronics because it allows for a more compact representation of binary numbers.
ASCII: ASCII, or American Standard Code for Information Interchange, is a character encoding standard that represents text in computers and other devices that use text. Each character in the ASCII set is assigned a unique numerical value, allowing for consistent communication and data exchange between different systems. This coding scheme is vital in converting data between base systems, especially in how digital information is processed and represented.
Base conversion: Base conversion is the process of changing a number from one base (or numeral system) to another. This process is essential in understanding how numbers are represented in different systems, such as binary, decimal, and hexadecimal. It involves techniques that allow us to interpret numerical values across various contexts, enabling operations like addition, subtraction, multiplication, and division to be performed in any base system.
Base-10: Base-10, also known as the decimal system, is a positional numeral system that uses ten distinct digits (0-9) to represent numbers. This system is fundamental in mathematics and everyday counting, as it allows for easy representation of values and operations through place value, where the position of a digit affects its contribution to the overall value. Base-10 is integral to understanding more complex numerical systems and operations, enabling seamless conversions and calculations across various mathematical frameworks.
Base-16: Base-16, also known as hexadecimal, is a numeral system that uses sixteen distinct symbols to represent values. The symbols include the digits 0 to 9 and the letters A to F, where A represents 10, B represents 11, C represents 12, D represents 13, E represents 14, and F represents 15. This system is commonly used in computing and digital electronics because it can represent large binary numbers in a more compact form.
Base-2: Base-2, also known as binary, is a numeral system that uses only two symbols, typically 0 and 1, to represent values. This system is foundational in computer science and digital electronics, as it aligns perfectly with the on-off states of electronic circuits. Base-2 allows for straightforward conversions to and from other numeral systems, facilitates efficient addition and subtraction operations, and forms the basis for multiplication and division methods that computers employ.
Base-8: Base-8, also known as octal, is a numeral system that uses eight distinct digits, from 0 to 7. This system is significant because it represents numbers using fewer digits than the decimal system, making it useful in computer science and digital electronics where binary and octal systems are commonly used to simplify binary representations.
Binary: Binary is a base-2 numeral system that uses only two symbols, typically 0 and 1, to represent numbers. This system is fundamental in computer science and digital electronics, as it is the language of computers, enabling them to perform calculations and store data. Understanding binary is crucial for converting between different base systems, as well as performing mathematical operations like addition, subtraction, multiplication, and division within these systems.
Complement system: The complement system is a crucial part of the immune system that enhances the ability of antibodies and phagocytic cells to clear pathogens from an organism. It consists of a series of proteins that work together to promote inflammation, opsonization, and cell lysis, ultimately supporting the body's defense mechanisms against infections and other harmful agents.
Divisor: A divisor is a number by which another number, called the dividend, is divided. In base systems, divisors play a crucial role in conversion processes and modular arithmetic.
Hexadecimal: Hexadecimal is a base-16 number system that uses sixteen symbols: the numbers 0-9 and the letters A-F to represent values from zero to fifteen. This system is commonly used in computing and digital electronics because it is more compact than binary, allowing for easier representation of binary-coded values. Each hexadecimal digit corresponds to four binary digits (bits), making it particularly useful for simplifying binary representation and making conversions more manageable.
Least significant digit: The least significant digit (LSD) is the digit in a number that holds the smallest value and is farthest to the right. In base systems, it plays a crucial role in determining the outcome of operations such as addition and subtraction, as changes to this digit can affect the overall value of the number. Understanding the least significant digit is essential for converting numbers between bases and performing arithmetic operations accurately, particularly in non-decimal systems.
Most significant digit: The most significant digit (MSD) is the digit in a number that holds the greatest value and therefore contributes most to the overall magnitude of the number. In the context of various base systems, the MSD is critical when converting numbers between bases and when performing addition or subtraction, as it influences how we determine place values and carry operations.
Number systems: Number systems are organized ways to represent and manipulate numbers, forming the basis of arithmetic and mathematics. These systems can vary in base, such as binary (base 2), decimal (base 10), and hexadecimal (base 16), allowing for different representations of numerical values. Understanding number systems is crucial for converting numbers from one base to another, which is a common task in various mathematical and computing contexts.
Octal: Octal is a base-8 numeral system that uses digits from 0 to 7. It serves as a convenient method for representing binary numbers in a more compact form since each octal digit corresponds to exactly three binary digits. This system plays a crucial role in computer science and digital electronics, where grouping bits can simplify the representation and manipulation of data.
Place value method: The place value method is a mathematical technique used to determine the value of a digit in a number based on its position. This method allows for the conversion between different base systems by analyzing how each digit contributes to the overall value of the number, depending on its place in the sequence. Understanding place value is essential for performing arithmetic operations and for converting numbers from one base to another.
Positional notation: Positional notation is a method of representing numbers where the position of each digit in a number determines its value. This system is fundamental to modern numerical systems, allowing for efficient representation and manipulation of numbers across various base systems, which directly influences how addition, subtraction, multiplication, and division are performed in mathematics.
Quotient: The quotient is the result of dividing one number by another. It shows how many times the divisor fits into the dividend without considering the remainder.
Radix: Radix is the base of a number system that determines how many unique digits, including zero, are used to represent numbers. It plays a crucial role in how numbers are expressed and manipulated in various base systems, such as binary (base 2), decimal (base 10), and hexadecimal (base 16). Understanding radix is essential for converting between different bases, performing arithmetic operations, and comprehending the structure of numerical systems.
Radix point: The radix point is a symbol that separates the integer part from the fractional part of a number in positional numeral systems. It serves as the equivalent of the decimal point in base-10, but can also appear in other bases such as binary, octal, or hexadecimal. The placement of the radix point determines the value of digits that follow it, affecting how numbers are represented and converted across different base systems.
Remainder: The remainder is the amount left over after performing a division when one number does not evenly divide another. It is an integer less than the divisor in any base system.
Repeated division method: The repeated division method is a technique used for converting numbers from one base to another by dividing the number by the base repeatedly and recording the remainders. This method systematically breaks down the number into its base components, making it particularly useful for converting decimal numbers to other bases like binary, octal, or hexadecimal. The order of the remainders collected during the division process is then reversed to obtain the final representation in the target base.
Repeated multiplication method: The repeated multiplication method is a technique used to convert numbers from one base system to another by multiplying the digits of the number by their corresponding place values and summing the results. This method simplifies the process of converting numbers between base systems by emphasizing the role of positional value, allowing for easier calculations and understanding of how different bases operate.
RGB: RGB stands for Red, Green, and Blue, which are the primary colors of light used in digital imaging and displays. By combining these colors in various intensities, a wide spectrum of colors can be created, making RGB essential for color representation in devices like monitors and televisions. Understanding RGB is crucial for converting color values across different base systems, particularly in digital graphics.
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4.4 Addition and Subtraction in Base Systems