EC Mathematical Techniques A Revision Notes EC Mathematical Techniques A Revision Notes Mathematical Techniques A begins with two weeks of intensive revision of basic arithmetic and algebra, to the level of GCSE mathematics. This revision will be tested in the first onehour examination which will take place at the beginning of week and which carries a weight of.% in your overall mark for the module. This intensive revision programme has been introduced because past experience has shown that many students taking this module have forgotten much of the algebra they learned at school. This lack of basic proficiency leads to their struggling with the module throughout the year. Some even fail the examination at the end of the module. Therefore if your basic algebra is weak, it is essential that you revise thoroughly now. These revision notes are intended to help with you with your revision. You may also find it helpful to obtain a GCSE textbook, such as the Letts publication, Revise GCSE Mathematics (only the sections on arithmetic and algebra are relevant, though the statistics section is a useful preparation for Statistical Techniques A which you will take in term ). If you find yourself struggling it may also a good idea to ask more mathematically competent friends to coach you! None of the maths in this module is "maths for the sake of maths" - understanding it will make you better at economics!! Revision notes. Addition and subtraction with positive and negative numbers The key point to understand here is that the + and - signs serve two distinct purposes: (i) We place a + or - sign in front of a number to indicate whether that number is positive or negative. Thus (+) is a positive number and (-) is a negative number. The brackets are not strictly necessary but have been added to make it absolutely clear that the + and - signs are attached to the numbers. (ii) We place a + or - sign between two numbers to indicate whether the operation of addition or subtraction is to be performed. This results in four possibilities: (a) Adding a positive number, e.g. (+) + (+). The result is (+) (b) Subtracting a positive number, e.g. (+) - (+). The result is (+) (c) Adding a negative number, e.g. (+) + (-). The result is (+) (d) Subtracting a negative number, e.g. (+) - (-). The result is (+) To understand these rules, we could think of Anne's bank balance being (+) dollars and John's bank balance being (+) dollars. Then in case (a) Anne and John are perhaps planning an evening out together, so are adding their bank balances together to see how much they have to spend. They arrive at their combined bank balance, (+) dollars. In case (b) Anne and John wish to see which of them is the richer, by subtracting John's balance from Anne's D:\Temp\ECa.RevN.doc.doc G.T.Renshaw 00
EC Mathematical Techniques A Revision Notes in order to calculate the difference between them, which is (+) dollars. In other words, Anne has two dollars more than John. Cases (c) and (d) are a little more difficult. In case (c), Anne's bank balance is (+) dollars as before, but John's account is in the red; his balance is (-) dollars. When they add their balances, therefore, they find that their combined balance is only (+) dollars. In case (d), Anne's bank balance is (+) dollars and John's is (-), as in case (c). This time, though, they are subtracting John's balance from Anne's in order to calculate the difference between them, which is (+) dollars. In other words, Anne has eight dollars more than John. John would need to pay eight dollars into his account in order to have the same balance as Anne. In summary, adding a positive number [case (a)] and subtracting a negative number [case (d)] both result in addition of the two numbers. Subtracting a positive number [case (b)] and adding a negative number [case(c)] both result in subtraction of the second number from the first. In practice of course we don't usually bother to place a + sign in front of positive numbers, nor bother with the brackets. So cases (a) - (c) above would actually be written as: (a) (b) (c) (d) +. The result is (Anne and John's combined bank balance). -. The result is (the difference between Anne' balance and John's). + (-). The result is (their combined balances, when John is in the red). - (-). The result is (the difference between Anne's balance and John's, when John is in the red). It makes no difference to the application of the rules if the first number (, in our examples) is negative. Of course, it changes the answers, because we are now assuming that Anne's bank balance is (-) instead of. [Check for yourself that the answers to (a), (b), (c) and (d) in this case are -, -, - and - respectively.]. Multiplication and division with negative numbers One approach to understanding the rules for multiplication of positive and negative numbers is as follows: If a number is multiplied by +, the number is left unchanged. If a number is multiplied by -, the number is left unchanged in absolute magnitude but its sign is reversed; that is, if it was previously positive it becomes negative and vice versa. Thus: (+) x (+) (+) (+) x (-) (-) (-) x (+) (-) (-) x (-) (+) However, consistency clearly requires that these rules must hold for multiplication by any number, not just -. This implies that: D:\Temp\ECa.RevN.doc.doc G.T.Renshaw 00
EC Mathematical Techniques A Revision Notes () (+) x (+) + () (+) x (-) - () (-) x (+) - () (-) x (-) + The way to remember these rules is that if the two numbers have the same sign (cases and ), the result is positive, while if they have different signs (cases and ), the result is negative. Division Since division simply reverses multiplication, consistency requires that division should obey the same sign rules as apply to multiplication. Thus: () (+) (+ ) + + + () (+) ( ) + - - () (-) ( + ) - + - () (-) ( ) - + Notice that the division operation can be written in two ways: either using the sign, or by writing a fraction in which the first number appears in the top (called the numerator) and the second in the bottom (the denominator). This second way is more convenient, hence is more common than the first. Again, the way to remember these rules is that if the two numbers have the same sign (cases and ), the result is positive, while if they have different signs (cases and ), the result is negative. From now on we shall follow the normal convention and omit the + sign and the brackets in front of positive numbers.. Brackets and when we need them When addition and subtraction are mixed with multiplication and division, you get different answers depending on which part of the calculation you do first. For example, 6 + D:\Temp\ECa.RevN.doc.doc G.T.Renshaw 00
EC Mathematical Techniques A Revision Notes is ambiguous. If you do the addition first you get 7, but if you do the division first you get 6 + 0. Mathematicians have set up certain conventions in an attempt to avoid ambiguity in such cases. Specifically, the convention is that you work from left to right in an expression, carrying out any division operations first, then any multiplication, then addition and finally subtraction. Applying these conventions to the expression above tells us to do the division first, so the correct answer is 0. But, these conventions are not followed universally, and so the only way to be sure of avoiding ambiguity is to use brackets to indicate which of any alternatives is correct in the particular case. The rule is that whatever is in the brackets must be done first. Thus but (6 + ) 7 6 + ( ) 6 + 0 However, because as noted earlier the division sign ( ) is not often used, these two expressions would not be written in this way. Instead we would write 6 + to denote (6 + ) 6 + When we write be divided by. That is why In contrast, we write it is understood that the whole of the numerator (that is, 6 + ) must 6 + 6 + to denote 6 + ( ). has the same meaning as (6 + ). 6 + To summarise, has the same meaning as (6 + ). In both expressions this means that the 6 and the must both be divided by. Overlooking this is one of the most common mistakes made by students taking Maths A. This rule will be considered further below, when we look at fractions.. Fractions In arithmetic the idea of fractions such as,, is familiar. As noted earlier, we often write, for example, to denote Thus a fractions (better described as a ratio or quotient) results whenever we divide one number by another. So we need to be clear about the rules for manipulating these fractions. Note that in any expression such as (also often written as /), is called the numerator and the denominator. D:\Temp\ECa.RevN.doc.doc G.T.Renshaw 00
EC Mathematical Techniques A Revision Notes "Cancelling" and Fractions It is easy to see that. 6 What we have done here (almost without thinking) is to divide both the numerator and the denominator by ; that is: 6 6 We divided numerator and denominator by because we saw that both numerator and denominator were divisible by without remainder; in other words we saw that was a factor of both the numerator and the denominator. This process is called cancelling and can be done whenever you can find something which is a factor of both the numerator and the denominator. Before cancelling, though, you must make sure that the whole of the numerator and the whole of the denominator are divisible by the factor, without remainder. The process of cancelling can also be reversed; that is, we can multiply numerator and denominator by any number. For example, x x Cancelling and its reverse are permissible because multiplying or dividing both numerator and denominator by the same thing leave a fraction unchanged in its value.. Addition and Subtraction of fractions Addition Consider: + The way we tackle this is to see, intuitively, that a half is the same thing as two quarters, so the answer is + + Another example is: + Again we see, almost without thinking, that 0 and 0, so + 0 + 0 9 0 What we need to do now is to make this intuitive process systematic, as follows. In the previous example, 0 is called the common denominator. A common denominator is any number which is divisible (without remainder) by the denominators of the two fractions to be added. Thus 0 is a common denominator because both and divide into it without D:\Temp\ECa.RevN.doc.doc G.T.Renshaw 00
EC Mathematical Techniques A Revision Notes 6 remainder. (This follows from the fact that 0 x ). However, 0, 0, 0 etc are also common denominators because they too can be divided by both five and two without remainder. But ten is the lowest common denominator. It is the smallest number which is divisible by both and without remainder. (Using the lowest common denominator when adding fractions is not essential but simplifies the calculation.) How do we find a common denominator? Consider the example: + 7 We have to find a number which is divisible by both 7 and without remainder. It occurs to us that if we take the product of 7 and, that is 7 X or 77, this is certain to be divisible by both 7 and without remainder. So we can then write: and 7 x 7 x x 7 x 7 77 7 77 so + 7 77 + 7 77 77 The pattern here is that we took each fraction in turn and multiplied both its numerator and its denominator by the denominator of the other fraction. Thus in the example above where we took + becomes x x 0 and becomes x x 0 so + 0 + 0 9 0 Things are a little more tricky when there are three or more fractions to be added. The easiest method is to add them in pairs. For example, given + + we first add the / and the /, then add the result of this to /, as follows + + + + x x + + x x + 0 0 0 D:\Temp\ECa.RevN.doc.doc G.T.Renshaw 00
EC Mathematical Techniques A Revision Notes 7 Alternatively we can multiply numerator and denominator of each fraction by the product of the other two (or more) denominators. This gives: + + xx xx + xx xx + xx xx 0 60 0 + + 60 60 9 60 0 This second method does not always result in the lowest common denominator being found; in our example the common denominator is 60, whereas the lowest common denominator is 0. Consequently may lead to some unnecessary calculation, but this method does have the merit of being reliable and reasonably easy to remember. Subtraction The method is exactly the same as the method for addition. For example - 0-0 - 0 0. Multiplication and Division of fractions Multiplication The rule for multiplying one fraction by another is very simple. You simply multiply the two numerators together to obtain the numerator of the answer, and multiply the two denominators together to obtain the denominator of the answer. Thus in general: X x x 6 0 0 This generalises to any number of terms, e.g.: X X x x x x 60 9 0 Reciprocals From the multiplication rule, we can get X X X X D:\Temp\ECa.RevN.doc.doc G.T.Renshaw 00
EC Mathematical Techniques A Revision Notes Here, and are said to be reciprocals. By definition, their product is unity (in other words, when you multiply one by the other the result is ). "of" Sometimes we meet a question such as: what is half of? Obviously the answer is 9, but we need a procedure for answering this kind of question so that we don't have to rely on intuition (which can lead us astray) The key insight is to see that "half of " is found by multiplying by. That is, "half of " x 9. So "of" just means " multiplied by". For example, of x x x Division of Fractions From the multiplication rule above, we can get 0 X 0 X 0 X X But we also know that: 0 Combining these, we see that 0 0 X So dividing by is equivalent to multiplying by its reciprocal, This gives us the fundamental rule for dividing by a fraction: you simply invert the fraction and multiply instead of dividing. For example, Consider: The rule is that we first invert the fraction ( i.e. becomes ) it instead of dividing. Thus: and then multiply by X X D:\Temp\ECa.RevN.doc.doc G.T.Renshaw 00
EC Mathematical Techniques A Revision Notes 9 This answer may seem puzzling, but it may help to think in words. The question is: "How many two-thirds are there in?" And the answer is: "One and a half". This rule holds equally when one fraction is divided by another. Thus: X Here the question is: "How many quarters are there in an eighth?" And the answer is: "A half." (In other words, an eighth is only half as large as a quarter.) WARNING: A COMMON MISTAKE: Cancelling fractions when addition or subtraction is involved Students often go wrong when confronted with something like + If you try to cancel the with the, you get + +, which is wrong The way to understand why is correct while is wrong is to recall from our earlier discusssion that + is another way of writing ( + ). This means that the addition must be performed first. So whenever addition or subtraction appear in the numerator or denominator of a fraction, the addition or subtraction must be done first before any cancelling takes place. Cancelling fractions when multiplication or division is involved In contrast to the above, it is quite OK to cancel between numerator and denominator when only multiplication or division signs appear in numerator and denominator. For example: x 6 (This is the same as the previous case, except that we now have a multiplication sign instead of a plus sign in the numerator.) In this case you can cancel the with the and get the right answer. For doing so gives: D:\Temp\ECa.RevN.doc.doc G.T.Renshaw 00
EC Mathematical Techniques A Revision Notes 0 x x Summarising, whenever you have something like + ; that is, something with addition or subtraction in the numerator or denominator, you must do the addition or subtraction first before doing any cancelling. But when you have something like x ; that is, something with only multiplication (or division) in the numerator and denominator, you can cancel individual elements in the numerator and denominator, if you wish. But you will also get the right answer if you choose instead to perform the multiplication or division in numerator or denominator first, and then do the cancelling.. Powers and roots Squares and square roots The product X is often written as. We read this as three squared and means threes multiplied together. Thus X 9. (It is called a square because is the area of a square with sides of length a.) Similarly X X, X X X X and so on. The process which is the reverse of squaring is called taking (or finding) the square root. Thus is the square root of 9 because 9. We write the square root of 9 as 9. Thus 9 because 9. Because squaring and taking the square root are mutually reversing processes, it follows that if you square a number and then take the square root, you arrive back where you started; that is: ( ). Similarly ( ). Negative square roots. Every positive number has square roots, one positive and the other negative. For example, 9 has two square roots. One, as we have just seen, is +. This is because (+) (+) X (+) 9. But (-) is also a square root of 9, because (-) (-) X (-) 9. (This is because when two negative numbers are multiplied together, the result is a positive number - see section. above). So the two square roots of 9 are + and - (sometimes written more compactly as ± ). In symbols, we write: 9 ±. We normally consider only the positive root unless the context of the problem requires us to consider both. D:\Temp\ECa.RevN.doc.doc G.T.Renshaw 00
EC Mathematical Techniques A Revision Notes Square roots of negative numbers. Suppose we are seeking the square root of (-9). Let us denote the number we are looking for by the symbol y (this is just a label). By definition, y is a number such that y (-9). Thus y is negative. But, from section XX above we know that if y is a positive number then y positive. So y cannot be positive if y is negative. y X y is Similarly we know from section XX above that if y is a negative number then y is positive. So y cannot be negative if y is negative. y X y Combining these two, we can say that y can be neither positive nor negative if y is negative. Therefore the square root of a negative number does not exist. (This is not quite true. In more advanced maths the concept of an imaginary number is introduced, and this number has the property that its square is negative. Thus to be completely precise we have to say that the square root of a negative number does exist, but is not a real number (but rather, is an imaginary number). This need not concern us greatly in this module.) 6. Decimals, fractions and percentages 6. Converting decimal numbers to fractions and vice versa To quickly refresh our memories as to the meaning of decimal numbers, recall that by definition 0. 0, 0.0, 00 0.00, 000 0. (which after cancelling between top and bottom) 000 and so on. The pattern is that the number of zeros in the denominator of the fraction is always equal to the number of digits after the decimal point. (Except that a zero or zeros at the right hand end don't count. E.g. 0.0 0. 0 ; 0.00 0. and so on. 00 Some conversions from fractions to decimals and vice versa are fairly obvious, e.g. 0. ; 0. ; 0. 0 The underlying rule for converting a fraction into a decimal is that you divide the numerator by the denominator, using decimal long division (being careful to get the decimal point in the right place!). For example, given we first divide into. This won t go, so we add a D:\Temp\ECa.RevN.doc.doc G.T.Renshaw 00
EC Mathematical Techniques A Revision Notes zero and simultaneously enter a decimal point (see figure ). We then divide into 0, which goes times, so the answer is 0.. Figure : 0 ) 0 Consider as another example. To convert this to a decimal number, we divide into. This won t go, so we add a zero and enter a decimal point (see figure ). We then divide into 0, which goes once with remainder. We add another zero and divide into 0, which goes twice with remainder. We add another zero and divide into 0, which goes times without remainder, giving us our answer: 0. Figure 0 )0 0 6 0 0 In practice you will probably use your calculator for this, but it's worth seeing how it is done without a calculator. To convert a decimal number into a fraction, simply apply the definitions above. For example, 0., which after cancelling, 0 Similarly 0., which after cancelling, 000 A strange feature of decimal numbers is that some fractions ahve no exact decimal equivalent. For example, (approximately) 0.6666 with the 6's recurring indefinitely. 6. Percentages (i) To convert a decimal fraction into a percentage, simply multiply by 00. For example in the case of 0. we get: D:\Temp\ECa.RevN.doc.doc G.T.Renshaw 00
EC Mathematical Techniques A Revision Notes 00 0. x 00 0 (per cent) 0 0 (ii) To convert any fraction into a percentage, simply multiply by 00. For example,. We get: 00 X 00 0 (per cent). So, 0%. Similarly 9 as a percentage is: 00 X 00.% (approx.) 9 9 (iii) To convert any percentage into a fraction, simply reverse the above, i.e. divide by 00. Thus 0% becomes 0 00 0 X 00 0 00 (iii) To find, say % of 7, recall that % simply means five-hundredths. So we have to divide 7 into 00 parts and then take of them. To do this, we divide 7 by 00, then multiply the result by. This gives: 7 X 00 00. (per cent). (With some practice, you ll find you can amaze your friends by calculating, say, % of 6 (.%) in your head). 7. Some Additional Symbols To conclude these notes, here are some symbols which we will be using later and which may be new to you: means "is NOT equal to" means "is approximately equal to" means "is identically equal to". This conveys a stronger meaning than the sign. The difference between and is important in Economics and we shall return to it later. means "infinity". (Note that infinity is NOT "a very large number". Any number, however large, is finite; while the essence of infinity is that it is not finite) - means "minus infinity" x means the absolute value of x, i.e. ignoring its sign. For example, > means "greater than" (for example, > ) < means "less than" (for example, < ) The symbols > and < are called inequalities and can be used in a variety of ways. For example if we want to say that some number called z is positive (i.e. greater than zero), we can simply write: z > 0. If on the other hand we want to say the some number k is negative, we can write: k < 0. A third possibility is that we might want to say that some D:\Temp\ECa.RevN.doc.doc G.T.Renshaw 00
EC Mathematical Techniques A Revision Notes number y has a value which lies between and. We can write this as: < y <, or as: > y >. Both of these algebraic statements tells us that y is greater than but less than. We shall return to inequality statements later. Finally, some symbols which you will find useful in setting out your work clearly are: means "therefore" means "implies" or "leads to" (For example: " x x + ") A note on use of calculators Many of you will already own calculators and may be in the habit of using them for routine calculations. My view on calculators is that they are fine for speeding up long and laborious calculations where a lot of data are involved, such as in an applied economics exercise. But if used routinely for trivial calculations (such as 00 ), they are positively harmful because they lead to mental laziness and, more important, they prevent you from developing any "feel" for numbers. (Incidentally, it is worth noting that a calculator cannot in fact calculate 00, because it runs out of space in its display! The correct answer is 6, but the calculator cannot supply this answer, only 6.6666666...which is only an approximation.) However, we shall find later in the module that a calculator is essential for certain kinds of mathematical computation (specifically, those involving logarithms and exponents); and also for statistics in the second term. So you will definitely need a calculator for this module, but it should be of the "scientific" type, with keys marked: " log ", " ln ", and " yx ", and also with some statistical function keys. It need not cost more than - 0. If in addition you already own, or wish to buy, one of the more expensive "graphics" calculators, with a larger display screen and which can draw graphs (priced from around up), this is perfectly OK as long as you don't become too dependent on it. University rules allow you to use a calculator in examinations, provided it can only display numbers (i.e. not text or graphics). D:\Temp\ECa.RevN.doc.doc G.T.Renshaw 00