Revision Topic 1: Number and algebra
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1 Revisio Topic : Number ad algebra Chapter : Number Differet types of umbers You eed to kow that there are differet types of umbers ad recogise which group a particular umber belogs to: Type of umber Symbol Cotais Explaatio Natural umbers Ν 0,,, 3, Zero followed by the coutig umbers. Itegers Ζ, 4, 3,,, 0,,, 3, Ratioal umbers Q All fractios which are ratios of itegers Irratioal umbers Numbers that caot be expressed as ratios of itegers, icludig special umbers like π or e ad square roots of prime umbers Real umbers R All the ratioal ad irratioal umbers together The whole umbers with directio; cosist of all atural umbers with a positive or egative sig. Icludes all itegers as they ca be writte as fractios too. If a umber ca t be writte as a fractio usig itegers, it is irratioal. Ratioal ad irratioal umbers split the real umbers ito two groups. Approximatio ad estimatio Oly roud a aswer at the ed of a calculatio, uless you are estimatig. Uless specified otherwise, roud to 3 sigificat figures if your aswer is t exact. Estimate the aswer by roudig umbers to make a easy calculatio, ad use this estimate to check that your aswer makes sese. I summary, to roud to a give umber of decimal places or sigificat figures: Decimal places cout the umber of places to Sigificat figures cout the umber of figures the right startig from the decimal poit. startig with the leftmost o-zero figure. I both cases: Cout the umber of places/figures. The look at the ext digit to the right. If this digit is 5 or above, add to the previous digit; otherwise, leave the previous digit. Leave off the rest of the digits. Copyright Cambridge Uiversity Press 04. All rights reserved. Page of 0
2 Percetage errors va ve From your formula booklet, the percetage error ε is give by ε = v exact value ad v A is the approximate value of v. E 00%, where v E is the So, to calculate the percetage error, work out the differece betwee the exact value ad your estimated value, the divide this differece by the exact value ad multiply by 00. Expressig very large ad very small umbers i stadard form O your GDC, umbers are give i stadard form (also called scietific otatio) if they are too log to fit o the scree. You eed to rewrite the calculator otatio i the form a 0 k, where a < 0 ad k is a iteger. You should use your GDC for all questios ivolvig umbers expressed i stadard form. The followig table shows how umbers i stadard form are displayed o your calculator ad how you should iterpret them. Write this dow o paper as (do t forget to roud to 3 s.f.) To iput umbers writte i stadard form, use the followig key presses: How it will look writte o paper Copyright Cambridge Uiversity Press 04. All rights reserved. Page of 0
3 SI uits The followig are the SI base uits that you should be familiar with: Measurig Called You write Legth metre m Mass kilogram kg Time secod s Electric curret ampere A Temperature kelvi K Amout of substace mole mol Itesity of light cadela cd You eed to be able to covert betwee differet uits, ad to give your aswer i SI uits. Other uits are derived from these base uits, such as m for area or m 3 for volume; you might also use metres per secod, ms, for speed. Copyright Cambridge Uiversity Press 04. All rights reserved. Page 3 of 0
4 Chapter : Solvig equatios Whe solvig equatios, you first eed to kow how may solutios there are. You ca see this by lookig at the graph. You are expected to use your GDC to solve equatios. If you have a Texas calculator, you eed to rearrage the equatio so that it has just 0 o oe side. You would either draw a graph or use the equatio solver. I the followig examples, oly oe method is show, suggestig the quickest ad most efficiet way of fidig the solutio. Liear equatios A liear equatio has oe solutio. The easiest way to fid this solutio is to use the equatio solver. For example, to solve x 5 = 3: First, rearrage equatio to the form x 5 3 = 0. So the solutio is x = 4. Copyright Cambridge Uiversity Press 04. All rights reserved. Page 4 of 0
5 Pairs of liear equatios I this case, a solutio cosists of the values of two ukows. It is easiest to look for solutios by drawig the graphs of both equatios. The, use the itersect optio o the GDC to fid the values of x ad y where the lies cross. For example, to solve the pair of equatios y = 3x + 4 ad y = 6 x, draw these two lies ad look for their itersectio: So the solutio is x = 0.4 ad y = 5.. Quadratic equatios A quadratic equatio makes a parabola shape ad ca have up to two solutios. Drawig the graph ad lookig for the itersectio of the curve with the x-axis will show you how may solutios there are. The, to fid the solutio(s), use a calculator fuctio that picks out zeros or roots from the graph. There are three possible situatios: No solutio: the graph does ot cross the x-axis, e.g. x 3x+ 0 = 0 Drawig the graph should show this ad become your evidece as follows: Oe solutio: the graph touches the x-axis i oe place oly, e.g. x After drawig the graph, use the mi fuctio to fid the solutio as follows: 4x+ 4= 0 Copyright Cambridge Uiversity Press 04. All rights reserved. Page 5 of 0
6 Press the right arrow util the cursor has goe past the miimum poit the press twice. So the solutio is x =. Two solutios: the graph is partly above ad partly below the x-axis, makig two itersectio poits with the x-axis, e.g. x 4x 5= 0 After drawig the graph, fid the two itersectio poits usig the followig process o your GDC: Use the zero fuctio i calc meu to show the solutios oce you have draw the graph. Use the root fuctio o the G-Solv meu to show the solutios oce you have draw the graph by pressig Move left util the cursor is before the itersectio poit. Press, the move the cursor after the poit ad press ad agai to tell the GDC which poit to fid. Repeat the above steps for the secod itersectio poit. Press the key to get to the secod solutio. So the solutios are x = ad x = 5. Copyright Cambridge Uiversity Press 04. All rights reserved. Page 6 of 0
7 Chapter 3: Arithmetic ad geometric sequeces ad series You eed to uderstad the differece betwee sequeces ad series: Vocabulary Defiitio Notatio sequece A ordered list of umbers, If the letter u deotes the terms i a sequece, e.g., 5, 8,, (arithmetic the u is the first term, u the secod term ad sequece) or, 4, 8, 6, so o. (geometric sequece) series The sum you get whe you add Usig S to deote the terms of a series, we have up the umbers i a sequece to a S = u particular poit (or idefiitely if S = u+ u a ifiite series) S = u + u + u ad so o. 3 3 Arithmetic sequeces I a arithmetic sequece, from oe term to the ext the same umber is added each time. This fixed differece betwee cosecutive terms is called the commo differece. You are give these formulas for arithmetic sequeces i the formula booklet: The th term of a arithmetic sequece: u = u + ( ) d Sum of the first terms of a arithmetic sequece: S = ( u + u) or S = [ u + ( ) d] You may be asked to work out ay of the followig values from iformatio give formally or i a word problem. Value How to fid it First term, u Give a term u ad the commo differece d, subtract ( ) d from u. Commo differece, d Subtract ay term u i the sequece from the ext term u +, i.e. u + u. th term Work out u ad d; the put these ito the formula u = u + ( ) d (if you kow the value of you ca calculate the actual value). Sum of terms If give the first ad last term to be summed, use S = ( u + u ). If give the first term ad commo differece, use S = u + ( ) d. Alteratively, use the sum seq fuctio o your GDC (see below). Copyright Cambridge Uiversity Press 04. All rights reserved. Page 7 of 0
8 Geometric sequeces I a geometric sequece, to get from oe term to the ext you multiply by the same umber each time. This fixed multiplier is called the commo ratio. You are give these formulas for geometric sequeces i the formula booklet: The th term of a geometric sequece: u = ur u Sum of the first terms of a geometric sequece: ( r S ) = r where r u or ( r S ) =, r You may be asked to work out ay of the followig values from iformatio give formally or i the cotext of a fiacial problem. Value How to fid it ( ) First term, u Give a term u ad the commo ratio r, divide u by r. Commo ratio, r Divide ay term i the sequece by the previous term: u u. th term Work out u ad r; the put these ito the formula u = ur (if you kow the value of you ca calculate the actual value). Sum of terms Oce you kow the first term u ad the commo ratio r (ot equal to ), you ca fid the sum usig ( u r S ) = r (if r <, you might fid it easier to use ( u r S ) = ). r Alteratively, use the sum seq fuctio o your GDC (see below). Usig the sum seq fuctio o your GDC This method works for both arithmetic ad geometric sequeces ad follows the same format for each. For example, to fid the sum of the first 8 terms of the geometric sequece with u = ad r = 5 : Method I this case = 8 (the umber of terms to be summed), ad X takes the place of i the th term formula u. Put the calculator i sum seq mode ad eter the kow iformatio i this order: u x, x,,, The press [ENTER/EXE]. So S 8 = 953. Copyright Cambridge Uiversity Press 04. All rights reserved. Page 8 of 0
9 Chapter 4: Fiacial mathematics Currecy coversios You eed to be able to covert betwee two give currecies usig the ratio betwee them, called the exchage rate. For example, CHF = EUR. To exchage i the directio of the rate as give, e.g. CHF EUR, you multiply by the exchage rate. To exchage agaist the directio of the rate, e.g. EUR CHF, you divide by the exchage rate. Currecies are bought ad sold at differet rates. All aswers should be give to decimal places, as this is appropriate for currecy uits. Commissio may be charged as a percetage of the trasactio cost. Compoud iterest You eed to be able to work out how much iterest a ivestmet will ear over a period of time. To do this, you have to uderstad the followig vocabulary ad be able to idetify the correspodig values from a problem or some give data. Vocabulary Notatio o the GDC Defiitio PV, preset value PV This is how much your ivestmet is worth at the begiig of the time period. FV, future value FV This is how much your ivestmet is worth at the ed of the time period. N or Number of years the ivestmet will last r% I% Aual iterest rate of the ivestmet k, umber of compoudig periods per year Number of paymets per year C/Y P/Y The umber of times iterest is calculated i a year, e.g. iterest compouded yearly gives k =, iterest compouded mothly gives k = The umber of iterest paymets per year. For compoud iterest this should be set to, but a differet value may be used i other applicatios. You ca work out the future value usig the followig formula, which is give i the formula booklet: FV k r = PV + 00 k Copyright Cambridge Uiversity Press 04. All rights reserved. Page 9 of 0
10 You should use the TVM app o your GDC for fiacial questios, as show here: Access the TVM app usig Access the TVM app usig The eter the data you have ito the appropriate row o the scree: The eter the data you have ito the appropriate row o the scree: Highlight the variable that you wish to work out by usig the arrow keys, ad the press Press the F butto uder the variable you wish to work out. Iflatio, depreciatio ad other uses of the compoud iterest formula Iflatio is a measure of how much the cost of goods icreases by; it is ormally give as a percetage per year. Depreciatio describes how much the value of somethig decreases over time. The TVM app o your GDC ca be used for other fiacial calculatios, but you will eed to adapt the variables as follows: For loa repaymets For iflatio calculatios For depreciatio calculatios FV = 0 so that the loa is paid off at the ed of the time period; C/Y = P/Y I% should be the rate of iflatio; PV < 0; k = C/Y = P/Y = I% < 0; PV < 0; k = C/Y = P/Y = Copyright Cambridge Uiversity Press 04. All rights reserved. Page 0 of 0
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