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Chapter 3: Data

3.1 Binary

• Binary: Information represented by only two values. There are no communication errors or misunderstandings because there are no small differences. It is needed to represent data and program instruction because of how computers work.
• A system with separate states is said to be digital and if there are two states it is binary.
• Digital: Information represented by certain fixed values.
• A system such as this, where there is a continuous range between two values, is said to be analogue.
• Analogue: Using signals or information represented by a continuously variable quantity.
• Representing information
  • There are only two digits, 1 and 0, but to represent the symbols in text there must be at least 52 separate items of information - 26 lower-case and 26 upper-case letters.
  • In a similar way, the two binary digits (bits) can be combined into groups.
  • Binary digits: The smallest unit of data that is represented in a computer. It has a single binary value, either 1 or 0.
• Number systems
  • Binary is a number system based on two digits, 0 and 1.
  • The difference is that the denary system works in powers of 10, whereas the binary system uses powers of 2.
• Denary system
  • Every digit has a value and the one to the left has a value 10 times higher than the one to the right.
• Binary system
  • The binary system has a similar place value but they increase by powers of 2.
  • Byte: The basic combination of bits used to represent an item of information. A byte typically consists of 8 bits.
• Binary arithmetic
  • Binary numbers can be manipulated in the same way as denary ones.
• Denary addition
  • When addition is performed n denary, a ‘carry-over’ is used if the result is greater than 9.
  • Binary addition works int he same way but a ‘carry-over’ is needed if the result is greater than 1.
• Binary addition
  • We are adding eight-bit numbers and this has caused a problem. All eight bits have been used and the 1 that was carried over in the last column has nowhere to go - it has been carried out.
  • Therefore the result of the calculation would be wrong. This is called an overflow error.
  • This condition occurs when a calculation produces a result that is greater than the computer can deal with or store.
  • The processor is informed that an error has occurred.
• Signed and unsigned numbers
  • When an integer is indicated as being positive or negative it is described as being signed.
  • Sign and magnitude
    • In a multiple-bit binary number, the left-most it, the one with the greatest value, is called the most significant bit (MSB). We can use this to represent signed integers.
• Two’s complement
  • The most common method used to represent signed integers in modern computers is two’s complement.
  • This method works on the principle above-the result of any number added to its negative equivalent should be zero.