# 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.

## Example Candidate Response Booklet

Example Candidate Response (ECR) Booklets are a source of crucial information for Centres and Candidates as they use real candidate responses. We ask Senior Examiners to comment on five or more responses in terms of why the mark was awarded with commentary about how to improve the answer (if necessary).