Number & Logic

Main Topics of Study:

A Number systems

  • Definition of a general number system. The common rules of decimal, binary, octal and hexadecimal.
    From these, formulate the general rules for a number system with base N.
  • Definition of bit, byte, character, word as basic storage units.
  • Methods to convert between decimal, binary, octal and hexadecimal integers.
  • Methods to convert between decimal, binary, octal and hexadecimal fractions.
  • The special relationship between binary and octal and hexadecimal.

B Arithmetic operations

  • Addition, subtraction, multiplication and division performed TOTALLY within with binary, octal or hexadecimal systems.
  • Given that 10 is approximately 2 to the power 3 1/3, work out how many binary places are equivalent to a given number of decimal places.

C Use of memory

  • Representing binary numbers in bytes and words. Normally only 8 or 16 bit-words will be used in numeric questions to reduce the working.
  • 2’s complement method of holding negative numbers. The ACTUAL NUMERIC VALUE of the sign bit in a 2’s complement number.
  • Representing fractional values in memory. Floating point numbers. Normalisation of floating point numbers. Determine how a decimal number would be held in binary in float point form including negative numbers. Determine the decimal value of a floating-point binary number.
  • Fixed point representation of mixed integer and fractional numbers – normally the mid-point will be the implied binary point.
  • How memory holds non-numeric data. ASCII. Binary Coded Decimal. Methods of holding variable length string data.
  • Format of an instruction in a binary word. A simplified form is normally tested which includes (i) operation code, (ii) register number and (iii) single address.

D Matrix notation

  • How a 2-D matrix is held in 1-D memory.
  • Basic rules for adding, subtraction and multiplying matrices.
  • Develop an algorithm for adding, subtracting or multiplying two matrices.
  • Determine the inverse of a 2×2 or 3×3 matrix.
  • Matrix method of solving simultaneous equations.

E Iterative methods

  • Iteration as the idea of “homing-in” to provide an accurate answer as required. Understand how far to go to determine an answer to the required number of figures/decimal places.
  • Understand that an iterative equation could converge to an answer or diverge away from it. Means of determining whether a particular iterative equation will converge for a given problem. e.g. An equation may have two solutions (near x=2 and x=4). To solve for the solution near x=2, one particular iterative equation might converge near x=2 but another might diverge or home in on the solution near x=4.
  • Practical applications of iteration such as:
    i Newton-Raphson method to determine the square root of a number.
    ii Determine the reciprocal of a number (1/N).
    iii Solve an equation up to degree four by an iterative method.
    iv Solve simultaneous equations by an iterative method.

F Other Numerical methods

  • Graphical method of finding the “best fit” (linear programming). Determine inequalities in a linear programming problem. Plot suitable lines graphically to represent the built-in restrictions. Plot a suitable line to maximise or minimise (e.g. minimum costs or maximum profit). Alternatively, candidates can use the Simplex method to solve a problem.
  • Venn diagrams. Application to real problems.
  • Apply a given process to determine the best fit. The method will be defined on the examination papers. e.g. best route to take between different points.

G Financial

  • Arithmetic and geometric series. Determine the nth term and sum of n terms for each.
  • Application to financial situations – discount and depreciation. Inflation.
  • Interest – simple and compound.
  • Economic order quantity (EOQ).

H Statistics

  • Averages – definitions of mean, median, mode, range, inter-quartile range, frequency.
  • Calculation of averages for a set of numbers including data held in a frequency table. Determine the best average to use in a given situation.
  • Calculation of averages for tabular data with class intervals.
  • Dispersion. Standard deviation. Skewness. Normal distribution. Correlation treated non-numerically.
  • Probability. Definition. Simple problems involving:
    i Mutually exclusive events – the probability of either occurring.
    ii Independent events. Probability of both occurring.
    iii Conditional probability – probability of x given y has occurred = P(x|y). Bayes theorem.
  • Expected value.
  • Permutations and combinations. Simple problems.

F