# 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.