The Mathematics of Banking and Finance By Dennis Cox and Michael Cox Copyright 2006 John Wiley & Sons Ltd

Size: px

Start display at page:

Wilfrid Palmer
5 years ago
Views:

3 Less than ( ), less than or equal to ( ) Appendix 281 A symbol meaning smaller or less than, for example, 3 4 is read as 3 is less than 4. This may be extended to include equality, for example an account has no funds or is overdrawn if the balance 0, or, in words, the balance is less than or equal to zero. Not equal to ( ) A symbol meaning not equal to, for example, 3 4 is read as 3 is not equal to 4. It is used in mathematical notation to exclude a specific value. This occurs at any time when a known value cannot occur. For example, if the value of a security 10, then it can take any value other than 10. Parentheses (),[],{} Brackets or parentheses are used to enable the reader to know the order in which things need to be done effectively which calculation is performed first and on what is the calculation conducted. Conventionally round brackets ( ) are used, however when a series of brackets are required both square [ ] and curly brackets {}may be employed. Effectively it makes little difference as long as people understand what you are trying to explain. For example, the result of would be 23, since the multiplication is done before the addition. (3 4) 5is 35, because in this case the parentheses override the normal precedence, causing the addition to be done first. Brackets or parentheses are also used to set apart the arguments in mathematical functions. For example, f (x) is the function f applied to the variable x. When evaluating complex expressions it is important to follow the rules in Table A.3. Another important rule is that when you have several multiplication and divisions within expressions, always work left to right. Table A.3 Priority Order of operations rules Operation 1 Calculations contained within parentheses are evaluated 2 Squaring (or raising to another exponent) 3 Multiplying or dividing 4 Adding or subtracting Significant digits Significant digits are the number of integers appearing in a figure after the decimal point. For example, is to three significant digits, whereas is only to two significant digits. Rounding The procedure of rounding has the effect of deleting the least significant digits of a number, then applying some rule of correction to the part retained. If you round a number to a certain number of significant digits and the first number rejected is less than 5 the last significant figure

4 282 Appendix is rounded down. If the first number rejected is 5 or greater, then it is rounded up. What this means is that rounded to two significant places is 12.76, while rounded to two significant places is As a general rule 5 is always rounded upwards, so will be rounded to whereas would be rounded down to Exponent or power If a number, referred to as the base, is multiplied by itself a number of times, then another form of notation is generally used. Referred to as the exponent or power, this records the number of times the base is multiplied by itself. The multiplier is shown as a superscript after the base number itself. A series of examples are presented in Table A.4. Table A.4 Examples of exponent Base Exponent Notation Result ,024 In words, 3 2 is referred to as three squared, whereas 2 3 is referred to as two cubed. Certain simple rules follow. Any base with an exponent 1 gives itself, that is to say the base number is not multiplied by anything. If the base is negative, the result will be positive for exponents that are even and negative for exponents that are odd. For example ( 1) 2 1 on the same basis as before that the product of two negatives equals a positive. Some basic rules for combining exponents are summarised below for any base x. Rule for multiplication: x n x m x m n, for example Rule for division: x n x m x n m, for example Rule for raising a power to a power: (x n ) m x n m or (3 2 ) Negative exponents: A negative exponent indicates a reciprocal: x n 1/x n or 3 2 1/3 2 Identity rule: Any non-zero number raised to the power of zero is equal to 1, x 0 1(x not zero), so and 5, Square root ( ) The square root of a value, denoted, is the number that when multiplied by itself gives the original value. Thus 9 3, it corresponds to an exponent or power of 1 / 2. Absolute value ( x ) The absolute value of a number is a measure of how far that number is from zero. It is a measure of distance and is always a positive quantity. Put simply, the absolute value of a number is its size regardless of its sign. The absolute value of a number x is denoted by x, with the number being referred to inside the bars. For example, the absolute value of the number negative three is 3. Since 3 is three units away from zero, the absolute value of 3 is 3, or using the symbols, 3 3. Similarly, we know that the number 3 is also three units away from zero,

5 Appendix 283 so 3 3. Numerical opposites have the same absolute value, as they are the same distance away from zero. Factorial symbol (!) The factorial function is denoted by the exclamation mark!. The number 5!, is read as five factorial, or factorial five and is a shorthand notation for the expression The factorial of 1, or 1! 1, and by convention 0! is defined as 1. However, factorials only apply to positive integers. In general, if n is a positive whole number then n! n (n 1) (n 2) In this expression the dots are taken to replace all the terms between (n 2) and 5. Therefore 8! is the same as 8! Greek alphabet The Greek alphabet (Table A.5) is used throughout business mathematics to refer to different types of event. We saw a number of these characters throughout this text. Perhaps we use Greek (or collectively the Greeks) just to confuse the poor reader, or just to show how clever we all are. Nonetheless they are in common usage so will need to at least be recognised. If you mistake a gamma for a delta then you may find yourself taking the wrong action when buying or selling shares, for example. Table A.5 Greek alphabet alpha A iota I rho P beta B kappa K sigma gamma lambda tau T delta mu M upsilon epsilon E nu N phi zeta Z xi chi X eta H omicron O o psi theta pi omega Summary of operations In this appendix we have covered the following general notation: Addition ( ) Subtraction ( ) The plus or minus sign ( ) Multiplication ( ) Division ( ) The reciprocal of a number Integer Greater than ( ), greater than or equal to ( ) Less than ( ), less than or equal to ( ) Not equal to ( ) Parentheses ((), [], {})

6 284 Appendix Rounding Exponents Square root ( ) Absolute value ( x ) Factorial symbol (!) Greek alphabet. This notation is used throughout the book and throughout business mathematics. A range of other notation is also required to fully understand business mathematics and is considered specifically within the relevant chapter to which it relates.

The Legacy Continues GREEK LETTERS AND EMBLEMS

The Legacy Continues GREEK LETTERS AND EMBLEMS HOW TO BUY AN EMBLEM OR GREEK LETTER The BRICK legacy INSCRIPTION continues These instructions are specific for ordering a brick inscription where the donor