1Number and Algebra. Surds

Transcription

1 Numer an Algera ffi ffi ( ), cue root ( ecimal or fraction value cannot e foun. When alying Pythagoras theorem, we have foun lengths that cannot e exresse as an exact rational numer. Pythagoras encountere this when calculating the iagonal of a square of sie length unit. A sur is a square root ), or any tye of root whose exact

2 NEW CENTURY MATHS ADVANCED for the Australian Curriculum0 þ0a Shutterstock.com/totojang9 n Chater outline Proficiency strans -0 an irrational numers* U F R C -0 Simlifying surs* U F R -0 Aing an sutracting surs* U F R -04 Multilying an iviing surs* U F R -0 Binomial roucts involving surs* U F R C -0 Rationalising the enominator* U F R C *STAGE. n Worank irrational numer A numer such as or that cannot e exresse as a fraction a rational numer Any numer that can e written in the form a ; where a an are integers an ¼ 0 rationalise the enominator To simlify a fraction involving a sur y making its enominator rational (that is, not a sur) real numer A numer that is either rational or irrational an whose value can e grahe on a numer line simlify a sur To write a sur x in its simlest form so that x has no factors that are erfect squares sur A square root (or other root) whose exact value cannot e foun

3 Chater n In this chater you will: (STAGE.) efine rational an irrational numers an erform oerations with surs (STAGE.) escrie real, rational an irrational numers an surs (STAGE.) a, sutract, multily an ivie surs (STAGE.) exan an simlify inomial roucts involving surs (STAGE.) rationalise the enominator of exressions of the form a c SkillCheck Worksheet StartU assignment MAT0NAWK009 Simlify each exression. a (y) (4m) c ( x) Exan each exression. a (x þ ) 4(y ) c ( þ w) ( y) e (a þ ) f k( þ k) Select the square numers from the following list of numers Exan an simlify each exression. a (m þ )(m þ ) (y þ )(y 4) c (n )(n ) ( þ )( þ ) e ( )(4 þ ) f (a þ f )(a þ f ) g (x þ 4) h (y ) i (k þ ) j (a )(a þ ) k (t þ )(t ) l (m þ 4)(m 4) -0 an irrational numers ffi ffi A sur is a square root ( ), cue root ( ), or any tye of root whose exact ecimal or fractional value cannot e foun. As a ecimal, its igits run enlessly without reeating (like ), so they are neither terminating nor recurring ecimals. is rea as the square root of or simly root. Rational numers such as fractions, terminating or recurring ecimals, an ercentages, can e exresse in the form a ; where a an are integers (an ¼ 0). are irrational numers ecause they cannot e exresse in this form. 4

4 NEW CENTURY MATHS ADVANCED for the Australian Curriculum0 þ0a Rational numers can e exresse in the form a Irrational numers cannot e exresse in the form a Integers 4 =, = = 4, Terminating ecimals 0., =., 8 % = 0.,. Recurring ecimals = = = 0....,,, 8 Transcenental numers Non-surs whose ecimal value also have no attern an are non-recurring, for examle, π =.49, cos 8 = , Examle Select the surs from this list of square roots: ¼ :48... ¼ : ¼ 9: ¼ 9 So the surs are, an ¼ Examle Is each numer rational or irrational? a.% 4 ffi c 0 0: _ e 48 a :% ¼ : 00 ¼ 8 [.% is a rational numer. 4 ffi ¼ 4 [ 4 ffi is a rational numer. c 0 ¼ [ 0 is an irrational numer. 0: _ ¼ 0:... ¼ 4 which is in the form of a fraction a which can e written as 4 The igits run enlessly without reeating. which is a recurring ecimal which is a fraction e [ 0: _ is a rational numer. 48 ¼ : The igits run enlessly without reeating. [ 48 is an irrational numer.

5 Chater Square roots ffi The symol stans for the ositive square root of a numer. For examle, 4 ¼ (not ). Furthermore, it is not ossile to fin the square root of a negative numer. It is only ossile to fin the square root of a ositive numer or zero, ecause the square of any real numer is ositive or zero. Summary For x > 0, x is the ositive square root of x. For x ¼ 0, x is 0. For x < 0, x is unefine. Your calculator will tell you that there is a mathematical error if you enter, for examle, : Worksheet on the numer line MAT0NAWK009 on a numer line The rational an irrational numers together make u the real numers. Any real numer can e reresente y a oint on the numer line. 0 0% π 0 :44::: irrational (sur) ¼ 0: rational (fraction) 0:::: rational (fraction) Examle 0 4 0% ¼. rational (ercentage) :0::: irrational (sur).4 Use a air of comasses an Pythagoras theorem to estimate the value of irrational (i) on a numer line. Ste Using a scale of unit to cm, raw a numer line as shown. Ste Construct a right-angle triangle on the numer line with ase length an height unit as shown. By Pythagoras theorem, show that XZ ¼ units. Ste With 0 as the ffi centre, use comasses with raius XZ to raw an arc to meet the numer line at A as shown. ffi The oint A reresents the value of an shoul e aroximately.44 0 Z X 0 Z X 0 A

6 NEW CENTURY MATHS ADVANCED for the Australian Curriculum0 þ0a Exercise -0 an irrational numers Which one of the following is a sur? Select the correct answer A, B, C or D. A 4 B 00 C 0 D ffi 400 Which one of the following is NOT a sur? Select the correct answer A, B, C or D. A 84 B 9 C D ffi 0 Select the surs from the following list of square roots :0009 ffi 4: Is each numer rational (R) or irrational (I)? a : _ ffi ffi 8 c 4 e f :_ g 4 h % i ffi 0 j 0 k l 4 Arrange each set of numers in escening orer. a 4 ; ; 0 ; : _ ; 9 Exress each real numer correct to one ecimal lace an grah them on a numer line. a 4 4% c 4 e f g h 8% 9 Use the metho from Examle to estimate the value of on a numer line. ffi 8 a Use the metho from Examle to estimate the value of on a numer line y constructing a right-angle triangle with ase length units an height unit. Use a similar metho to estimate the following surs on a numer line. i 0 ii See Examle See Examle See Examle Investigation: Proof that ffi is irrational A metho of roof sometimes use in mathematics is to assume the oosite of what is eing rove, an show that it is false. This is calle a roof y contraiction. We will use this metho ffi to rove that is irrational. ffi Firstly, assume that is rational. This means we assume that can e written in the form a ; where ¼ 0, an a an are integers with no common factor. a ¼ ¼ a Squaring oth sies a ¼ is an even numer ecause it is ivisile y. [ a is even.

7 Chater If a is even, then a is also even ecause any o numer square gives another o numer. If a is even, then it is ivisile y an can e exresse in the form m, where m is an integer. ) a ¼ðmÞ ¼ 4m ¼ m ¼ ¼ m [ is even [ is even [ a an are oth even. This contraicts the assumtion that a an have no common factor. Therefore, the assumtion that is rational is false. [ must e irrational. Use the metho of roof just escrie to show that these surs are irrational. a Comare your roofs with those of other stuents. Puzzle sheet Simlifying surs MAT0NAPS009 Technology worksheet Excel worksheet: Simlifying surs quiz MAT0NACT0009 Technology worksheet Excel sreasheet: Simlifying surs MAT0NACT Simlifying surs Summary For x > 0 (ositive): x ¼ x x ¼ x x ¼ x Examle 4 Simlify each exression. a 4 c ¼ a 4 ¼ 4 4 ¼ 4 ¼ ¼ 4 ffi means 4 c ¼ð Þ ¼ ¼ 0 8

8 NEW CENTURY MATHS ADVANCED for the Australian Curriculum0 þ0a Summary The square root of a rouct For x > 0 an y > 0: xy ¼ x y A sur n can e simlifie if n can e ivie into two factors, where one of them is a square numer such as 4, 9,,,, 49, Examle Simlify each sur. ffi a 8 ffi ffi a 8 ¼ 4 ¼ ¼ 08 c is a square numer. 88 Metho Metho 08 ¼ ffi ¼ 08 ¼ 4 ¼ ¼ ffi 9 ¼ ¼ Metho involves simlifying surs twice ( 08 an ). Metho shows that when simlifying surs, you shoul look for the highest square factor ossile. c 4 ffi 4 ¼ 4 9 ¼ ¼ ¼ ¼ ffi ¼ 4 ¼ 4 Vieo tutorial Simlifying surs MAT0NAVT000 9

9 Chater Exercise -0 Simlifying surs See Examle 4 See Examle Simlify each exression. ffi ffi a c 0 ffi e 0:09 f g h Simlify each sur. a 0 f 4 ffi g 48 ffi k 88 ffi l 08 ffi 4 q Simlify each exression. a f g 4 k 48 l 0 ffi 0 4 c h m r 8 ffi 00 ffi 4 c 8 h 9 8 m i n s 0 9 ffi 4 ffi 40 i 0 n 8 e j o t e j o 00 ffi Which one of the following is equivalent to 4 0? Select A, B, C or D. A 8 B 0 C 8 D 0 0 Which one of the following is equivalent to? Select A, B, C or D. ffi 0 0 A B C 0 D 0 0 Decie whether each statement is true (T) or false (F). a ffi ¼ ffi ¼ 8 ¼ 9 c 9:4 9:4 ¼ ffi e : f The exact value of 0 is. 8 Just for the recor Unreal numers are imaginary! There exist numers ffi that are neither rational nor irrational, so they are also not real numers. For examle, is not a real numer, ffi ecause there is no real numer which, if square, equals. Numers such as ; 0 an 4 are calle unreal or imaginary numers an cannot e grahe on a numer line (that is, their values cannot e orere). Imaginary numers were first notice y Hero of Alexanria in the st century CE. In 4, the Italian mathematician Girolamo Carano wrote aout them, ut elieve negative numers i not have a square root. Imaginary numers were largely ignore until the 8th century when they were stuie y Leonhar Euler an the Carl Frierich Gauss. ffi ffi is efine to e the imaginary numer i, so ¼ i. ) ¼ ð Þ ¼ ffi ¼ i: Imaginary numers are useful for solving hysics an engineering rolems involving heat conuction, elasticity, hyroynamics an the flow of electric current. Simlify each imaginary numer. ffi a 00 c 4 4 0

10 NEW CENTURY MATHS ADVANCED for the Australian Curriculum0 þ0a -0 Aing an sutracting surs Just as you can only a or sutract like terms in algera, you can only a or sutract like surs. You may first nee to exress all the surs in their simlest forms. Puzzle sheet coe uzzle MAT0NAPS0094 Examle Simlify each exression. a þ 8 ffi 80 þ 0 e 8 þ 8 a þ ¼ c ffi ffi ffi 4 þ ¼ 8 4 ffi ffi 80 þ 0 ¼ þ 4 ¼ 4 þ ¼ ffi ffi e 8 þ 8 ¼ 4 9 þ 9 ¼ ffi þ ¼ c 4 þ f 0 8 ffi ffi ¼ Simlifying each sur. f 0 ¼ ffi 4 ¼ ¼ 0 ¼ Exercise -0 Aing an sutracting surs Simlify each exression. a 9 þ ffi þ g 4 þ j 4 þ Simlify each exression. a ffi 9 þ c 4 ffi þ ffi e 4 þ g 0 ffi ffi þ þ 4 i ffi ffi 8 e h ffi ffi 4 k þ 0 0 ffi 4 0 þ þ 4 þ f 4 ffi h þ 8 þ j For each exression, select the correct simlifie answer A, B, C or D. ffi a þ A B C D 4 A ffi B C 4 D 4 c ffi f 0 0 i ffi þ 4 l 0 ffi See Examle

11 Chater Simlify each exression. ffi a 8 þ 08 c e þ 4 f þ g h þ 99 i þ 8 j þ k 00 l 0 þ m þ n 0 4 o 48 4 þ 4 q 8 r 98 þ s þ 0 t 4 0 þ 8 u v ffi ffi w ffi þ 8 þ x þ 4 þ 4 y z 9 0 þ 4 Worksheet Multilying an iviing surs MAT0NAWK009 Puzzle sheet MAT0NAPS0004 Technology worksheet Excel worksheet: Simlifying surs quiz MAT0NACT0009 Technology worksheet Excel sreasheet: Simlifying surs MAT0NACT Multilying an iviing surs Summary The square root of roucts an quotients For x > 0 an y > 0: Examle Simlify each exression. a 90 0 a ¼ c 4 ffi 0 ¼ 4 0 e ¼ 40 ¼ ð Þ¼ ffi ¼ ¼ ffi 9 ¼ xy ¼ x y r x x ¼ y y e ¼ 84 ¼ 4 ¼ 90 ¼ ¼ 4 ¼ c 4 0 f f ffi ffi ¼ ¼ ffi ¼ ffi 8 ¼ 9 ¼

12 NEW CENTURY MATHS ADVANCED for the Australian Curriculum0 þ0a Examle 8 Simlify 4 0 : ¼ 8 0 ¼ 8 Exercise -04 Multilying an iviing surs Simlify each exression. a 0 8 g 0 ffi j ffi m ffi s 8 4 v ffi 0 8 ffi e h k 4 n 8 8 q t 8 w 8 4 ffi c 8 f 8 ffi i ffi 4 l 4 0 o 0 ffi 8 r u 90 x 48 Simlify each exression a c e f ffi g ffi h i j 4 k ffi l m ffi 4 4 n o q r s t ffi u Simlify the exressions elow. a y y e x x 4 Simlify : Select the correct answer A, B, C or D. A B C c f a a D See Examle

13 Chater See Examle 8 Simlify : Select A, B, C or D. A 4 B C 0 D 0 Simlify each exression. a ffi 4 ffi c e 8 80 f ffi 8 Mental skills Maths without calculators Percentage of a quantity Learn these commonly-use ercentages an their fraction equivalents. Percentage 0% %.% % 0% 0% % % Fraction 4 Now we will use them to fin a ercentage of a quantity. Stuy each examle a 0% ¼ 0% 0 ¼ 0 c :% ¼ 8 ¼ ¼ 0 % ¼ 4 0 e ¼ % ¼ 4 0 ¼ 9 ¼ 4 f % 0 ¼ 0 ¼ 0 ¼ ¼ 4 ¼ 0 ¼ 40 Now simlify each exression. a % 44 % 0 c 0% % e 0% 0 f % 48 g 0% 8 h 0% 400 i % 4 j % 4 k % 0 l 0% 0 m.% 88 n % o 0% 0 % 80 4

14 NEW CENTURY MATHS ADVANCED for the Australian Curriculum0 þ0a -0 Binomial roucts involving surs Sur exressions involving rackets can e exane in the same way as algeraic exressions of the form a( þ c) an (a þ )(c þ ). Examle 9 Exan an simlify each exression. ffi a þ a + = = + + = = 0 = 0 Examle 0 Exan an simlify each exression. ffi ffi ffi ffi a þ 0 ffi ffi ffi a þ ¼ ffi ffi þ ¼ ffi ffi ffi ffi þ ¼ 4 þ 0 ffi ffi 0 ¼ ffi 0 ¼ ffi ffi 0 þ 0 ¼ ffi 9 0 þ 0 ¼ ffi 9 þ ¼ ffi ffi 9 0 þ ¼ 9 Summary (a þ ) ¼ a þ a þ (a ) ¼ a a þ (a þ )(a ) ¼ a

15 Chater Examle Exan an simlify each exression. a ffi þ c þ þ 4 4 ffi ffi ffi a ¼ þ Using (a ) ¼ a a þ ¼ þ ¼ ffi þ ffi þ ¼ þ Using (a þ ) ¼ a þ a þ ¼ ð4þþ þ ð9þ ¼ þ þ 4 ¼ þ ffi ffi c þ ¼ Using (a þ )(a ) ¼ a ¼ ¼ þ 4 4 ¼ 4 ¼ ð9 Þ ¼ 8 Note that ecause of the ifference of two squares, the answer is not a sur ut a rational numer. Using (a þ )(a ) ¼ a Exercise -0 Binomial roucts involving surs See Examle 9 See Examle 0 Exan an simlify each exression. ffi ffi ffi ffi ffi a þ c þ ffi ffi ffi e ffi ffi þ f 4 g 4 h ffi þ i ffi 4 þ ffi ffi ffi Which exression is equivalent to þ þ? Select the correct answer A, B, C or D. A 0 0 B þ C þ þ 0 0 þ D ffi þ þ þ 4 Exan an simlify each exression. ffi ffi ffi a þ c ffi ffi þ 4 þ ffi ffi e þ þ 4 g þ 0 0 ffi þ ffi ffi þ f 4 ffi ffi h þ

16 NEW CENTURY MATHS ADVANCED for the Australian Curriculum0 þ0a 4 Which exression is equivalent to þ? Select A, B, C or D. A B C þ 0 D þ Exan an simlify each exression. ffi ffi ffi a þ c þ 0 e þ f g þ h ffi þ Exan an simlify each exression. ffi ffi a þ þ c þ ffi þ e 0 þ 0 f þ g ffi ffi ffi þ h 4 ffi ffi 4 þ Which exression is equivalent to ffi ffi 4 þ 4? Select A, B, C or D. A ffi B 0 þ 0 C D 8 Exan an simlify each exression. a ffi 4 þ c ffi ffi þ þ 4 þ e 4 ffi ffi ffi þ 4 f 0 ffi See Examle Investigation: Making the enominator rational ffi If :44; what is the value of? Fractions containing surs in the enominator are ifficult to work with. When aroximating the value of ; it is ifficult to mentally ivie y.44. We can overcome this y making the enominator rational (that is, not a sur). What haens when we multily the numerator an enominator of a fraction y the same numer? a Simlify ffi : ffi Mentally aroximate the value of ; given that ffi :44: ffi c Check, using a calculator, that ¼ : Why is this true? a Is it true that ¼ ffi? Why? Simlify ffi : Comare your answer with those of other stuents. c Check, using a calculator, that ¼ : Exlain why ¼ ffi : e Show that ffi ¼ 0 :

17 Chater Worksheet Rationalising the enominator MAT0NAWK00-0 Rationalising the enominator of the form ffi ; ; ; ; have enominators that are irrational. These exressions may e rewritten with a rational enominator y multilying oth the numerator an enominator y the sur that aears in the enominator. This metho is calle rationalising the enominator. Examle Rationalise the enominator of each sur. a 4 c 8 þ a ¼ ffi ffi ecause ¼ ¼ Because ¼ ; it is easier to aroximate y mentally multilying y.44 than y iviing y ¼ ffi 4 ffi ¼ 4 ¼ c 8 ¼ 8 ffi ffi ¼ 8 0 ¼ 8 0 ffi ffi þ þ ¼ ffi ffi ffi ffi þ ¼ Exercise -0 Rationalising the enominator See Examle By rationalising the enominator, which sur is equivalent to? Select the correct answer A, B, C or D. A B C D Rationalise the enominator of each sur. a c e f g h 4 i j k l 4 Which sur is equivalent to? Select A, B, C or D. A B C D 0 8

18 NEW CENTURY MATHS ADVANCED for the Australian Curriculum0 þ0a 4 Which sur is equivalent to? Select A, B, C or D. 8 A ffi B C Rationalise the enominator of each exression. ffi ffi þ a ffi c Simlify each exression, giving the answer with a rational enominator. a þ ffi þ c D ffi ffi Power lus ffi a Is it true that þ ¼ þ? Exlain. ffi Simlify þ : Is the enominator rational? c Use a calculator to check that the value of your answer to art is equal to the value of þ : The conjugate of þ is : Fin the conjugate of: a þ ffi ffi c þ The rocess shown in question involves rationalising a sur with a inomial enominator. By first fining the conjugate of the enominator, rationalise the enominator of each exression elow. a þ c ffi 4 The largest cue that can fit into a shere must have its eight vertices touching the surface of the shere. Exress the sie length, s, of the cue in terms of the iameter, D, of the shere. þ Squares are forme insie squares y joining the mioints of the sies of the squares as shown. If AB ¼ 4 cm, fin the exact length of the sie of the shae square. D C A B Six stormwater ies, each mm in iameter, are stacke as shown in the iagram. Fin the exact height, h, of this stacking. h 9

19 Chater review n Language of maths Puzzle sheet crosswor MAT0NAPS00 aroximate inomial enominator ifference of two squares exan irrational numer erfect square rouct Pythagoras theorem quotient rational numer rationalise real numer root simlify square numer square root sur unefine Why o you think a rational numer has that name? What is the ifference etween a rational numer an a real numer? What is a sur? 4 How o you simlify a sur? Why is an examle of an irrational numer that is not a sur? How o you rationalise the enominator of a sur exression? n Toic overview Coy an comlete this min ma of the toic, aing etail to its ranches an using ictures, symols an colour where neee. Ask your teacher to check your work. an irrational numers Simlifying surs Aing an sutracting surs Multilying an iviing surs Rationalising the enominator Binomial roucts involving surs 0

20 Chater revision Which one of the following is a rational numer? Select the correct answer A, B, C or D. A B 9 C D Is each numer rational (R) or irrational (I)? ffi a 8 e 8 ffi f Simlify each sur. a e 0 ffi i 48 m f 8 j 4 n 4 Simlify each exression. a 00 þ 8 c 98 þ 4 e ffi þ Simlify each exression. a ffi ffi 8 ffi e 4 g h 4 j ffi 8 k ffi 4 c 0: _ g 8 h þ c g 4 88 ffi k o þ þ 80 f c f i l ffi h 4 Exan an simlify each exression. a ð Þ 0 ð Þ c ð ffi ffi Þð þ Þ ð 4Þ e ð þ Þ f ð ffi ffi Þð þ Þ g ð þ 4 ffi ffi Þð4 Þ h ð 0 Þð 0 þ Þ Rationalise the enominator of each sur. a 0 4 e c f 4 þ l See Exercise -0 See Exercise -0 See Exercise -0 See Exercise -0 See Exercise -04 See Exercise -0 See Exercise -0