B1-05.02 NUMBERING SYSTEMS

• Computers are used for repeated calculations and processing of huge amounts of data such as in military, science, commercial fields, and aviation.
• Digital computers offer speed, accuracy, and manpower savings, taking over routine jobs from personnel.
• Humans and computers require translation methods to communicate since they don't speak the same language.
• Humans use words and numbers in the decimal number system, while computers use coded electronic pulses.
• This topic will cover general number systems and binary/octal/hexadecimal number systems
• Methods for converting numbers between binary/octal/hexadecimal and the decimal system are explained
• These number systems translate easily into electronic signals for digital equipment.
• People primarily use the decimal system, and may know the Roman numeral system.
• Numbering systems have things in common like: unit, number, and base.

Common Numbering Systems

• Decimal: Base 10, using digits 0-9.
• Binary: Base 2, using digits 0 and 1.
• Octal: Base 8, using digits 0-7.
• Hexadecimal: Base 16, using digits 0-9 and letters A-F.
• A unit is defined as a single object such as apple or dollar.
• A number represents a unit or quantity.
• The Arabic numerals include 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.
• The base of a number tells how many symbols are used in that system.
• The base is always expressed in decimals.
• Base 10 has ten symbols (0-9).
• Base 3 has three symbols (0, 1, 2).
• A number system's base is indicated by a subscript following the number's value.
• 1010₂ is binary, 1010₁₀ is decimal and 1010₁₆ is Hexadecimal.
• The largest value symbol in a number system is one less than the base; for example, base 10 is 9, base 5 is 4 and base 3 is 2.
• Positional notation defines a number's value by not only it's symbol, but the symbol's position.
• Each position represents a power of the base, called the radix.
• A power is a number multiplied by itself that is written above and to the right of a number and is called an exponent.
• Zero is important in positional notation.

Digital Numbering Systems

• Decimal (Base 10) is the main method of counting and recording.
• Binary (Base 2) is used by computers for all calculations/processes depending on transistor states which are ON or OFF (1 or 0).
• Octal (Base 8) is used in computer application sectors and digital numbering systems.
• Octal numerals are easily converted from binary because grouping consecutive binary digits into groups of three equals an octal value.
• Hexadecimal (Base 16) is similar to octal because conversion from binary is easy.
• Each hexadecimal numeral represents four binary bits.
• Analogue = continuously changing values.
• Digital = distinct numerical values to represent waveforms.
• Computers use 0 and 1 (digital) to represent values.
• When entering a decimal number, calculators convert it to digital + calculate + convert the answer back to decimal for display.

Binary Terms

• Bit: A single digit that can be 0 or 1.
• Nibble: 4 bits.
• Byte: 8 bits.
• Word: 16 bits.
• Long word: 32 bits (2 x word).
• Very long word: 64 bits (4 x word).
• The binary numbering system, with base 2, uses only symbols 0 and 1.
• Numbering systems can be defined through a table of digit values, increasing from right to left.
• The decimal system (base 10) uses powers of 10 to determine the value of a position.
• The binary system uses powers of 2 to determine the value of a position.
• The binary numbering system is often converted to decimal for humans

Decimal Truth Table

• Each position is determined by lifting the base of the number to a power.
• The values are recorded across a page for the number to be converted.
• The values with "1" in their columns are recorded.
• It is essential to use the subscript to ensure the system being referred to is clear when working with number systems.
• Each octal numeral is represented by three binary digits and the two number systems are easily converted through substitution.
• 778 is followed by 1008, in sequence of 9910 which is followed by 10010.
• The advantage is binary can be displayed as octal, which is easily converted back to binary,
• Octal is more easily remembered
• A common method of interpreting data in aircraft is to memory inspect a computer memory location
• 08 18 28 38 78 108 118 128 → 168 178 208 218 → 268 278 308 318 → 768 778 → 1008
• Each octal represents three binary digits.

Hexadecimal Numbering systems

• Base 16 with symbols 0-9, A-F (radix).
• Useful in representing every byte (8 bits of binary) as two symbols.
• Hex uses the first 10 numbers of the decimal system: 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9.
• Number 10 in Hex uses the letter A, 11 uses B, 12 uses C, 13 uses D, 14 uses E, and 15 uses F.
• In the decimal number 16 in Hex, two digits have to be used - setting the first digit to 1 and increasing the second from 0 to F.
• To increment to 32, the first digit must be changed to 2 and the second digit from F to 0.
• Each hexadecimal digit represents four binary digits.
• Decimal to binary conversion: Repeatedly dividing by 2 to find the quotient and remainder digits.
• Two methods of doing a decimal to Binary conversion = division by 2, or repeated subtraction

Binary/Octal/Hexadecimal Conversions

• One method is converting the number to binary, then converting the number to decimal (using the binary/decimal truth table)
• Use conversion between binary as a base system + remember how to convert each system to/from digital.