Elementry Mthemticl Concepts nd Opertions After studying this chpter you should be ble to: dd, subtrct, multiply nd divide positive nd negtive numbers understnd the concept of squre root expnd nd evlute n lgebric expression plot points on grph The im of this chpter is to introduce the bsic opertions (ddition, substrction, multipliction nd division) used in lgebr. The min focus will be on the properties of rel numbers nd the bsic mnipultion of lgebric expressions, collecting lgebric expressions with like terms. Arithmetic is the ABC of Mth. Addition, subtrction, multipliction, nd division re the bsics of Mth nd every Mth opertion known to humnkind. In one wy or nother, every eqution, grph, nd mny other things cn be broken down into the ABCs of Mth: the four bsic opertions. As people sy, Mth is lnguge, nd ddition, subtrction, multipliction, nd division re its lphbet, long with the number line s well. The properties re bsiclly proven wys to pply the mechnics of rithmetic to certin situtions (source: www.writework.com).
Is Mthemtics only for few kids? Is Mthemtics only for few clever kids who re geneticlly disposed towrds the subject? Mny techers believe, either explicitly or implicitly, tht some children my be born with mthemticl ptitudes or mthemtics genes, nd others re not. Some techers even believe tht children from certin groups (bses on fctors such s gender, ethnicity nd rce) re blessed with superior mthemticl bility. Some techers feel there is not much tht cn be done to chnge or improve the innte bility of those unfortunte children who re inherently not good t mthemtics. The mthemticl interests nd knowledge young children bring to school my indeed differ, but the cuses re more likely to be their vrying experiences, rther thn their biologicl endowment. While techers should be wre of nd sensitive to these differences, they should never lose sight of the fct tht ll children, regrdless of their bckgrounds nd prior experiences, hve the potentil to lern Mthemtics. In fct, the gps in erly Mthemtics knowledge cn be nrrowed or even closed by good Mthemtics curricul nd teching. Techers should strive to hold high expecttions nd support for ll children, without ny ungrounded bises. When techer expects child to succeed (or fil), the child tends to live up to tht expecttion. Source: www.erlychildhoodustrli.org
ELEMENTARY MATHEMATICAL CONCEPTS AND OPERATIONS. Simple Opertions on Positive nd Negtive Numbers FIGURE. An Arbic public telephone keypd The digits,, etc. originte from Arbic numerls. Before dopting these symbols, Europens used Romn numerls I, II, III, IV, V, etc. Negtive numbers were not fully ccepted until round 800. In this book, we will use the deciml point nd the thousnd seprtor in our figures, like this:, 45.67 The bove mens twelve thousnd, three hundred forty-five nd sixty-seven hundredths. FIGURE. The number line illustrting the order of numbers 0 7 5 0
4 In Mth there re positive nd negtive numbers, nd the number 0, which is neither positive nor negtive. We cn think of positive number s credit nd negtive number s debt. The order of numbers is illustrted in figure., the picture of the number line, for exmple: >, <, nd >. Numbers cn be dded, subtrcted nd multiplied. All numbers cn lso be divided, but not by zero. An exmple of the ddition of positive nd negtive numbers goes like this: suppose you hve debt of 00. Tht mens the blnce of your bnk ccount equls 00. If you deposit 400 in your bnk ccount, your new blnce is 00 + 400 = 00 euros. Or suppose gin you hve debt of 00, nd your boyfriend hs debt of 00. So the blnces of both of your bnk ccounts re 00 nd 00 respectively. Now fter your wedding you decide to join both of your bnk ccounts. Together you hve debt of 500, nd the blnce of your joint ccount equls 00 + 00 = 500 euros. The multipliction nd division of positive nd negtive numbers re illustrted in Tble.. TABLE. Multipliction nd division of positive nd negtive numbers or + + + + Exmple. Multipliction nd Division of Positive nd Negtive Numbers Referring to Tble. we see tht: (+) (+) =, (+) ( ) =, ( ) (+) =, nd ( ) ( ) =. The sme holds for division, i.e. + + =+, + =, + =, nd =. N.B. When there is no sign in front of the number, it is lwys positive. I.e. =+.
ELEMENTARY MATHEMATICAL CONCEPTS AND OPERATIONS 5 EXTEND YOUR KNOWLEDGE Do you know why ( ) ( ) =? The xioms of rithmetic stte tht for ll numbers x, y, z we hve: x + x = 0, b x 0 = 0, c x (y + z) = x y + x z (distributivity ), nd d ( ) x = x. Now: + = 0 (by xiom ), ( ) ( + ) = ( ) 0 (by xiom b), ( ) + ( ) ( ) = 0 (by xiom c), + ( ) ( ) = 0 (by xiom d), Hence ( ) ( ) =.. Frctions nd Squre Roots A frction is the rtio of two whole numbers, i.e. 6 is the rtio of 6 nd. In this frction 6 is clled the numertor, nd the denomintor. The rtio of 6 nd equls : 6 =, for 6 is three times s big s : 6 =. Using nother exmple, we cn lso determine the rtio of nd 4 : 4 =, becuse is twice s big s 4 : = 4. Two times qurter equls hlf my be demonstrted by: hlf euro is twice s much s qurter of euro (50 cents is twice s much s 5 cents). For clcultions with frctions the following rules hold:
6 THEOREM. Clcultion Rules for Frctions Let, b, c, nd d be rel numbers, with b nd d unequl to zero. Then: The ddition of two frctions with the sme denomintor is defined by: d + c d = + c d (.) The ddition of two frctions with different denomintors is defined by: b + c d + bc = d bd (.) The multipliction of two frctions is defined by: b c d = c bd (.) The division of two frctions is defined by: b c = d b d c = d bc (.4) Eqution (.4) shows tht dividing by frction is equivlent to multiplying by its inverse (the inverse of number x 0 is ). Note tht we hve used the nottionl x convention b for b. Here is n exmple of the ppliction of eqution (.): Suppose you hve 5 of euro, 0 cents. You dd 5 of euro, 40 cents. The result is 60 cents, 5 of euro: 5 + 5 = + 5 = 5 Here is n exmple of the ppliction of eqution (.): Suppose you hve of euro, 50 cents. You dd of euro, 60 cents. The result is 0 cents, 5. euros: + 5
ELEMENTARY MATHEMATICAL CONCEPTS AND OPERATIONS 7 Since two frctions with different denomintor cnnot be dded immeditely, we use eqution (.): + 5 = 5 5 + 5 = 5 + 0 = 0 =. Here is n exmple of the ppliction of eqution (.): 5 = 5 = 5 Here is n exmple of the ppliction of eqution (.4): 4 = 4 = 4 = 4 6 = We will now turn to the definition of the squre nd the root of number. The squre of number is tht of number multiplied by itself, nd the squre root nswers the question Which number ws multiplied by itself in order to get this new number?. A squre root is useful for solving n eqution like x = 64 (see figure.). Definition. Squre Let x be ny number. The squre of x, x is defi ned by x = x x For exmple: 5 = 5 5 = 5 nd ( ) = ( ) ( ) = 9. FIGURE. The squre of 8 equls 64 becuse 8 8 = 64 8 7 6 5 4 b c d e f g h
8 Definition. Squre root Let x be non-negtive number, then the squre root of x, y = x, is tht number y 0 which, when squred, equls x, i.e.: y = AxB Exmple. The squre root of 9 equls, becuse = 9. Note lso tht ( ) = 9, so you might think tht would lso be the squre root of 9, however, the definition of squre root sttes tht only the non-negtive vlue pplies. Note tht negtive numbers do not hve squre root. For exmple: 9 does not exist becuse there is no number tht, when squred, equls 9. THEOREM. Clcultion Rules for the Squre Root Let, b 0, c > 0, nd d be ny number. Then consider the following rules: b = b (.5) B c = c (.6) d = d (.7) ( ) = (.8) Where d denotes the bsolute vlue of d, s defined by: d for d < 0 d = c 0 for d = 0 d for d > 0 (.9) For exmple: = nd =. Bsiclly, to rrive t the bsolute vlue of number expressed with minus sign, you just need to remove tht minus sign. Exmple. An exmple of eqution (.5) is: 8 = 4 = 4 = Exmple.4 An exmple of eqution (.7) nd the bsolute vlue (.9) is: ( 4) = 4 = 4
ELEMENTARY MATHEMATICAL CONCEPTS AND OPERATIONS 9. BEDMAS Just s it mtters in which order you put on your shoes nd socks, it mtters in which order you dd nd multiply. The order in which mthemticl opertions re executed is often referred to s BEDMAS (see figure.4). FIGURE.4 Which of the clcultors is right? BEDMAS is n cronym tht stnds for: B brckets E exponents (mens powers, squres nd roots) DM divide nd multiply, from left to right AS dd nd subtrct, from left to right Exmple.5 Let us clculte: ( + 6) 8 4 + 6 Following BEDMAS we need to strt with the Brckets: ( + 6) 8 4 + 6 = 9 8 4 + 6 There is n Exponent, 6, which hs to be done now: 9 8 4 + 6 = 9 8 4 + 6
0 We continue with Division nd Multipliction from left to right: 9 8 4 + 6 = 9 4 4 + 6 = 9 + 6 nd we finish with Addition nd Subtrction from left to right: 9 + 6 = 8 + 6 = 44.4 Algebric Expressions Definition. Algebric expression An lgebric expression is mde up of the signs, or numbers, nd letters, or symbols of lgebr. It is lso composed of terms which cn be vrible or constnt. 6 term term # " 5 " x " coefficient vrible constnt Definition.4 Vrible Vribles re unknown vlues tht my chnge within the scope of given problem or set of opertions. Vribles re represented by letters most of the time, nd the ones often used re: x, y, nd z. However ny other letter could be used. Definition.5 Constnt A constnt is fi xed vlue tht does not chnge, like,, etc. Exmple.6 Addition nd Subtrction of Algebric Expressions How cn we simplify n expression such s x + 4y 5x + 0y 45xy + 5 yx? First we collect like terms; there re different sorts, x, y, nd xy: x y xy x 5x 4y + 0y 45xy + 5yx = x = 4y = 0xy Note tht xy = yx. After dding ll the terms we get x + 4y 0xy.
ELEMENTARY MATHEMATICAL CONCEPTS AND OPERATIONS.5 Expnding Brckets Expnding brckets, otherwise known s evluting n expression, involves removing the brckets in order to simplify the expression down to single numericl vlue. We cn expnd the brckets of the following expression: (x + )(x ) FIGURE.5 The bnn method (x + )(x ) Step : x x = x Step : x = x Step : x = x Add like terms Step 4 : = 6 Step 5 : x x + x 6 = Gther like terms nd simplify x +x 6 (Answer) A convenient wy of expnding these brckets is by using the bnn method, see figure.5: (x + )(x ) (x + )(x ) = x x + x 6 = x + x 6 Does this mke ny sense? Let us check the following exmple. We lredy know tht: ( + )( + 4) = ()(7) = 7 = (.0) Expnding the brckets first yields: ( + )( + 4) = + 4 + 6 + 8 = Which is the sme nswer s in (.0)..6 Fctorizing Algebric Expressions Fctorizing n lgebric expression is the opposite of expnding. You strt with sum or difference of terms nd finish up with product.
For exmple, by fctorizing the following expression: b + c You get (b + c) We hve used the common fctor method in the exmple: ll terms hve s fctor, so is the common fctor. We should now use the brckets to write down the common fctor outside the brckets s follows: ( ) To find out wht goes inside the brckets, divide the originl terms by the common fctor(s). For the exmple, divide the originl terms by : b = b c = c The lst step is to write these new expressions inside the brckets, like this: (b + c) Exmple.7 Fctorize yx + 4x yx + 4x = xy + x = x(y + ).7 The Crtesin Plne The Crtesin plne ws nmed fter the fmous French philosopher nd mthemticin Rene Descrtes (figure.6). When two perpendiculr number lines intersect, Crtesin plne is formed. Rene Descrtes, French philosopher, viewed the world with cold nlyticl logic. He sw ll physicl bodies, including the humn body, s mchines operted by mechnicl principles. His philosophy ws derived from the ustere logic of cogito ergo sum mening: I think therefore I m. In Mthemtics, Descrtes s chief contribution ws in nlyticl geometry. Descrtes s portrit is qudrisected by the xes of his gret dvnce in nlyticl geometry: the Crtesin Plne. It enbled n lgebric representtion of geometry (source: http://mthemticinspictures.com).
ELEMENTARY MATHEMATICAL CONCEPTS AND OPERATIONS FIGURE.6 René Descrtes (596 650) FIGURE.7 The coordinte plne y xis Origin 4 x xis C 4
4 The Crtesin Plne is plne with rectngulr coordinte system tht ssocites ech point in the plne with pir of numbers. A coordinte plne hs two xes: the x-xis (the horizontl line), nd the y-xis (the verticl line). These two xes originte in the origin O with coordintes (0, 0). In figure.7 point C hs coordintes (, ). is the distnce of this point from the origin in the direction of the x-xis, nd is the distnce of this point from the origin in the direction of the y-xis, hence the line goes downwrds becuse is negtive.
5 Summry Multipliction nd division of positive nd negtive numbers re illustrted in this tble: or + + + + Rules for clcultions with frctions re: d + c d = + c d b + c d = d bd + bc d + bc = bd bd d c d = c bd b c = d b d c = d bc b = b A c = c () = The squre root of negtive number does not exist. d = d, where d denotes the bsolute vlue of d, s defined by d for d < 0 d = c 0 for d = 0 d for d > 0 BEDMAS tells you the order to follow: brckets, exponents, division nd multipliction (from left to right), ddition nd subtrction (from left to right) x = x x Expnding brckets: y = x shows tht non negtive number y stisfies y = x Rules for clcultions with squre roots re: ( + b)(c + d) = c + d + bc + bd Wht is the difference between vrible, constnt nd coefficient?
6 Exercises Complete these exercises without using clcultor:. You borrowed,000 from the bnk lst month to buy your books, nd this month you pid bck 800. But you relize tht you need notebook, so you borrow nother 500 to buy one. Wht is the totl mount tht you owe the bnk now?. Lst week the blnce of your bnk ccount ws 600 in debit nd this week you withdrew 00. How much is your ccount now in debit by?. Determine the vlidity of the following sttements. Are they true or flse? 4 b 8< c 8 < d 0.7 4 e 5. > 7 f 4 < 6 8 g 4 6 8 h + = + i =.4 Clculte + b + c + 6 d 4 e + f g + 5 h 7 7 5
ELEMENTARY MATHEMATICAL CONCEPTS AND OPERATIONS 7.5 Clculte b c d e f g h 8 7 5 7 8 5 5 5 5 7 7 5.6 Clculte + (6 ) b 5 (8 7) c 5 (4 ) d (7 ) 4 e (4 ) f 5+ (4 7) g 4+ (7 ) + h 6 (8 7) 5.7 Clculte 5 4 7 8 8 b 7 + 8 c 5 7 + 7 5 + d 7 5 + 5 9 + 9 + 4 5 e 0 5 5 8 9 5 f g h i, hint: = 4
8 j k l.8 Clculte ( ) b ( ) c + 4 d ( + 4) e 0 5 + 8 f 0 (5 + 8) g 0 + h 0 ( + ).9 Evlute nd simplify 9 b 7 c 4 9 d ( 9) e 9 A 6 7 f A 8 g () h.0 Clculte (7 + 6) + b ( + ) + 4 5 6 7 c ( + 5 7) 6 + d ( + 5) 7 6 + e f g + 4 + h + 4 i + + 4 j + 4. Referring to figure.4, which clcultor is right ccording to BEDMAS?
ELEMENTARY MATHEMATICAL CONCEPTS AND OPERATIONS 9. Clculte ( )( + ) b 6 4 c 4 d A e 4 + 5 f 4 + 5 g + 0 8 h + 0 8 y 4 E A 4 4 D x B 4 C. Use this coordinte plne to nswer the following questions: Which point hs coordintes (, 4)? b Wht re the coordintes of the other points A, B,, E? c Is there point plotted t coordintes (, )?.4 Plot the following points in this coordinte plne. A = (4, 0) B = (, ) C = (, ) D = (, ) E = ( 4, 0) F = (, ) G = (, ) H = (0; 0.7) I = ( 0.8; ) J =, K = 5, 7 8 L = 7, 7
0 4 y 4 4 x 4.5 Simplify the following expressions by collecting like terms: 8x 4x + 7y 5x + x + y 0y b 5u 7v + 6uv 5vu + 8v 4u.6 Expnd the following expressions nd simplify the result. 7(x + y) + 8(x y) b 0(p + q) (p q) c 4(x + 6) + 5(x + 6) d 4( x + 5) + 5( x + 6).7 Simplify the following expressions by expnding the brckets nd dding like terms: [see fig.4] (x + 7)(x 4) b (x y)(x + y) c ( + b)( b) d (x + y) e (b + 6)(b 7) f (x + 5y) + (x y) g (7.5x x 4) + x x + 4 x.8 Show tht + = 5 + 6.9 Fctorize the following expressions: x + b (x + )(x + 5) + 7(x + ) c (x + )(x + 4) + (x + )(x ) d (x + )(4x + 9) 5(x + )