WJEC FP1. Identities Complex Numbers, Polynomials, Differentiation. Identities. About the Further Mathematics. Ideas came from.

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1 Furter Matematics Support Programme bout te Furter Matematics Support Programme Wales Wales WJEC FP Sofya Lyakova FMSP Wales Te Furter Matematics Support Programme (FMSP) Wales started in July 00 and follows on te success of te Furter Matematics Support Project in England. Te FMSP Wales is managed by WIMCS in partnersip wit MEI and funded by te Wels ssembly Government. -year pilot project. ll scools in Wales are invited to register wit FMSP Wales to obtain a free access to online database of FM resources. Tuition is available in Sout Wales only. WJEC FP topics Comple numbers Polynomials (quadratics cubics quartics) Series and Proof by Inductions Matrices Differentiation Integral FP WJEC resources Ideas came from Jean van Scaftingen (Louvain-la-Neuve) en Sparks Vitaly Moroz (Swansea) Peter Gordon (NJIT) Identities Comple Numbers Polynomials Differentiation ( a b)( a b) a b ( a b) a ab b ( a b) a ab b ( a b) a a b ab b ( a b) a a b ab b a b ( a b)( a ab b ) a b ( a b)( a ab b ) Identities Epressions equations formulae and identity 7 7 r is an epressions is an equation is a formulae Some equations owever are true for all value of and are called identical equations or identities ( ) ( )

2 WJEC FP Comple Numbers WJEC FP Comple Numbers Not all numbers ave been around a tousand years ago. Wic of tese do you tink older tan oters? Wic are te newest? Fractions Positive integers Zero Negative integers Roots and surds π Solve tese equations:. + 7 = 0. 7 =0. ² = = 7. ² + 7 = 0 6. ² + 0 = 0 Etc WJEC FP Comple Numbers WJEC FP Comple Numbers -7 or 0-0/7 0 (-0 )??? Etc Etc Solve tese equations:. + 7 = 0. 7 =0. ² = = 7. ² + 7 = 0 6. ² + 0 = 0. =. = 0/7. = ± 0 (= ±.6 ). = -. ( + 7) = 0 =0 or = (-0 )??? ² + 0 = 0 = (-0 )??? WJEC FP Comple Numbers WJEC FP Comple Numbers Can you solve tis equation

3 WJEC FP Comple Numbers WJEC FP Comple Numbers You will ave already come across te Quadratic Formula: If a b c d 0 a b c 0 b b ac a Tis will solve NY quadratic equation (and tis is only really a part of it ) WJEC FP Comple Numbers WJEC FP Comple Numbers Cardano (and some oters) in te 6 t century were trying to find a general formula for CUIC equations like te one already known for quadratics. So if it is a number wat is it like? let i i i (Imaginary bit) (square bot sides) i i Tis i as some funny properties but it does follow te normal laws of algebra. WJEC FP Comple Numbers So if it is a number wat is it like? let i i (Imaginary bit) (square bot sides) Since i³=i² i i i i i i Since i =i³ i nd off we go again Tis i as some funny properties but it does follow te normal laws of algebra. i i i i i i 0 -i -i -i -6i -7i Etc Tis point is +i Real Part () Imaginary Part () Te comple numbers ave actually turned out to be stunningly useful for very practical subjects suc as Engineering Pysics and Computing. Tey are now found to be at te eart of equations of Quantum Teory wic ave massively broadened our understanding of our universe. Etc

4 WJEC FP Comple Numbers comple number z is of te form + yi were and y are real numbers. Te real part of z is denoted Re(z) (= ) and te Imaginary part of z is denoted Im(z) (=y). Notice terefore tat Im(z) is actually REL! (Te set of Real Numbers is terefore a subset of te set of Comple Numbers.) How can we make sure tat te comple numbers beave like numbers? Do tey obey te normal rules of algebra? Can tey be added subtracted multiplied and divided? WJEC FP Comple Numbers How can we make sure tat te comple numbers beave like numbers? Do tey obey te normal rules of algebra? Can tey be added subtracted multiplied and divided? Comple Number ritmetic: - Eamples: ( + i)+(- + 7 i)= +iy????? ( + i) - (- + 7 i) = +iy????? ( i) (- + i) = +yi????? ( + i) ( i ) = +yi????? WJEC FP Comple Numbers ( + i) ( i) = +yi????? WJEC FP Comple Numbers Teacing and learning resources WJEC FP Polynomials c 0 ROOTS OF QUDRTICS Wic of te graps below correspond to D<0 >0 <0 D=0 D>0 two different real roots two equal real roots no real roots a b WJEC FP Polynomials ROOTS OF QUDRTICS c 0 True or False:. quadratic equation always as two roots c d. quadratic equation can ave two real roots C. quadratic equation can ave two comple roots D. quadratic equation always as two real roots E. quadratic equation can ave one real and one comple root F. If α is a root of te equation (comple or real) ten -α is a factor of te polynomial c

5 WJEC FP Polynomials T. If are te roots of a quadratic ten α β equation a b c 0 b c - and. a a p. 7 Gaulter&Gaulter Furter Pure Matematics WJEC FP Polynomials ROOTS OF CUICS and QURTICS. How many roots? C D 0 C D E 0 T. If are te roots of a cubic equation a b c d 0 b c d ten α β - a a a. Wat combinations of comple and real roots are possible? p. 9 Gaulter&Gaulter Furter Pure Matematics Identities - Polynomials Eample a and b are unknown. a b ab Identities - Polynomials Eample a and b are unknown. a b ab Find a b a b a b Find a b a b a b a b WJEC FP Polynomials Teacing and learning resources WJEC FP Proof and Proof by Induction Go troug different types of proof in one lesson see lgebraic Proof powerpoint

6 WJEC FP Series ny even number can be defined by n n were n= ny odd number can be defined by n+ 7 n+ were n=0 ny odd number can be defined by n- 7 n- were n= ny square number can be defined as n² 9 n² were n= WJEC FP Series n n n r n n r r r ( n ) ( n ) r n r 0 n n r r ( r ) ( r ) WJEC FP Series = 0 0 WJEC FP Series = = = = (0 ) / = (n -) + n = n(n + ) / WJEC FP Series Conjecture:... ( n) n n r nn ( ) for n =. or nn ( ) r for n =. WJEC FP Matrices Let us talk about numbers again! Commutative law of addition: m + n = n + m. sum isn t canged at rearrangement of its addends. ssociative law of addition: ( m + n ) + k = m + ( n + k ) = m + n + k. sum doesn t depend on grouping of its addends. Commutative law of multiplication: m n = n m. product isn t canged at rearrangement of its factors. ssociative law of multiplication: ( m n ) k = m ( n k ) = m n k. product doesn t depend on grouping of its factors. Distributive law of multiplication over addition: ( m + n ) k = m k + n k. Tis law epands te rules of operations wit brackets (see te previous section). 6

7 7 WJEC FP Matrices Task. Solve =. Write your solution carefully step by step. Task. Solve =. Write your solution step by step avoid using division. Task. Solve =. Write your solution step by step avoid using division or fractions. Wat would cange if te commutative law of multiplication did not old?? WJEC FP Matrices Moving to simultaneous equations 7 WJEC FP Matrices Moving to simultaneous equations 7 7???? X X X X WJEC FP Matrices If we want to manipulate matrices like we manipulate number we must be able to: ) add matrices ) multiply matrices ) ave a zero matri ) ave an analogue of ) divide matrices?? If P and M two matrices is it always te case tat P+M=M+P? Is it always te case tat PM=MP? P0=0 P+0=P Eample a) Factorise b) Using te result in a) simplify Wat appens wen? 0 Identities - Differentiation Q. a) Simplify b) Using te result in a simplify Wat appens wen? 0 Identities Identities - Differentiation

8 LGERIC PROOF Furter Matematics Support Programme Wales Sofya Lyakova Furter Matematics Support Programme Wales aims to encourage more students to take Furter Matematics S/ level qualification Revision sessions (online face-toface video conferencing) Enricment activities Careers in Mats talks Starter Starter. ny even number can be defined by n a) lways b) Sometimes c) Never Starter. Wen you square a number te answer is positive a) lways b) Sometimes c) Never Starter. y ( y)( y) a) lways b) Sometimes c) Never

9 Starter. If two lines are eac perpendicular to a tird line tey must be parallel to eac oter a) lways b) Sometimes c) Never Starter. ( 7) a) lways b) Sometimes c) Never Starter 6. n odd number can be defined as n+ Starter 7. n odd number can be defined as n- a) lways b) Sometimes c) Never a) lways b) Sometimes c) Never proofs are cains of logical steps were every net step is based on a previous step and every step must be true! distinguis between practical demonstrations and proof

10 PROOF Part. lgebraic Proofs Part. Use of a counter-eample Part. lgebraic proof In tis section a number of general results about properties of numbers will be proved using algebra Part. Proof by contradiction Part. Proof by Induction Part. lgebraic proof Find te mistake in te proof below Part. lgebraic proof Proof? Let a = b Ten a² = ab (multiply by a) a² + a² = a² + ab (add a²) a² = a² + ab (simplify a² s) a² - ab = a² + ab - ab (subtract ab) a² - ab = a² - ab (simplify ab s) (a² - ab) = (a² - ab) (factorise) = (cancel (a² - ab)) Hmm We get nonsense because we ve actually divided by zero. We can t let tat appen. Part. lgebraic proof ertrand Russell matematician and pilosoper Part. lgebraic proof Eample. Prove tat te sum of squares of two consecutive integers is always odd. Eample. Prove tat te product of an even number and an odd number is always even

11 Part. lgebraic proof Question Part. lgebraic proof Solution (a) n (a) Write down an epression in terms of n for te nt multiple of. (b) Hence (i) prove tat te sum of two consecutive multiples of is always an odd number (ii) prove tat te product of two consecutive multiples of is always an even number. (b) (i) Let te first number be n so te second number is (n+) te sum is n + (n+) = n + n + = 0n + = (n + ) Wic is odd since n + is odd for all integer values of n. so we ave odd odd = odd Part. lgebraic proof Solution (continued) (b) (ii) Using n and (n + ) again product n (n + ) = n(n + ) if n is odd ten n + is even if n is even ten n + is odd as is odd we will always ave odd odd even wic is always even Part. lgebraic proof few important remarks: demonstration is not a proof unless you demonstrate all cases! Demonstration is useful to understand te nature of te result Part. Use of a counter-eamples Part. Use of a counter-eamples Eample. ll prime numbers are odd. Eample. Carlie says If is a positive integer ten is always prime. Sow tat Carlie is wrong. Sometimes you may met a conjecture tat is an unproven claim. If a conjecture turns out to be true it may be quite difficult to prove it for all possible cases. On te oter and if a conjecture is false you only need to find one case were it is fails in order to demonstrate its falseood. Suc a falilure is called a counter-eample

12 Part. Use of a counter-eamples Q. James says If you add two prime numbers togeter you always get anoter prime number. Sow tat James is wrong. Q. If is positive ten +0-² is also positive. Sow tat tis statement is false. Q. Petra says If n is a positive integer ten te value of n²+n+ is always prime. Sow tat Petra is wrong. Part. Proof by contradiction Sometimes to prove a conjecture one can start wit stating te opposite. ssume te opposite is true and call it our assumption. Start manipulating wit te assumption. You may end up wit a conclusion wic contradicts your assumptions. In tis case your assumption was wrong. Tis proves te original statement! Proof by contradiction Rational Numbers Proof by contradiction Rational Numbers Etc? Te followers of Pytagoras tougt tat every number could be written as a fraction. Te cult of te Pytagoreans was quite insistent on tis point. ut a man called Hippasus callenged Pytagoras Hippasus asked te question about te lengt of tis diagonal. Pytagoras own teorem said te lengt ad to be but tey couldn t find te fraction wic represented it. Tere s a good reason wy not Proof by contradiction Rational Numbers If we assume is rational ten it can be written as a fraction:? a = (were a and b ave no common factors) b = a b So a² is an even number so a is an even number b = a b = c so a = c b = c So b² is an even number and a = c so b is an even number ut if a and b are bot even tey ave a common factor of So we ave a contradiction so we can substitute c² for te a² in tis equation a b Proof by contradiction? Rational Numbers So wat appened to Hippasus wo first callenged tis idea tat every number was rational? Tey drowned im Or so te legend goes.

13 Part. Proof by induction Wy natural numbers are so special? Part. Proof by induction Wy natural numbers are so special? Every number as a successor is not a successor of any number No two numbers ave te same successor ny property wic belongs to and also to te successor of any number tat also as te same property belongs to all natural numbers. Part. Proof by induction Natural numbers are inductive Giuseppe Peano Italian matematician ny property wic belongs to and also to te successor if any number tat also as te same property belongs to all natural numbers. Suppose we want to prove tat someting is true for all numbers It would be enoug to sow tat )te statement is true for and )if it is true for an arbitrary number n ten it is true for its successor n+. Part. Proof by induction Suppose we want to prove tat someting is true for all numbers It would be enoug to sow tat )te statement is true for and )if it is true for an arbitrary number n ten it is true for its successor n+. Eample. is even. Does it mean tat every natural number multiplied by is even? If n is even ten (n+) = n+ = even + = even so true for every successor. So by induction is true for all natural numbers! ESY! Part. Proof by induction Eample. If tere are N pigeonoles and N+ object to be placed in tem ten one pigeonole must ave two or more objects in it. (Pigeonole Principle) It would be enoug to sow tat ) te statement is true for and If n= ten we ave one pigeonole and two objects. So tere are two objects in te same pigeonole. ) if it is true for an arbitrary number n ten it is true for its successor n+. n: tere are n pigeonoles and n+ objects and one of te pigeonoles as two or more objects in it. n+: Wat can we say about n+ pigeonoles and n+ objects? Can we find a pigeonole wic as two or more objects? Take one pigeonole at random 6

14 Part. Proof by induction Suppose we want to prove tat someting is true for all numbers It would be enoug to sow tat )te statement is true for and )if it is true for an arbitrary number n ten it is true for its successor n+. lgebraic Proofs PROOF Use of a counter-eample Proof by contradiction Proof by Induction Wen we do not agree wit a conjecture we try to find a countereample Use it wen you actually agree wit te conjecture Only used wen dealing wit positive integers Wat is net? - students. Your teacer will receive a list of eercises for you to try.. If you enjoyed tis session consider studying -level Mats and Furter Mats. Wat is net? - teacers. You will be ed list of eercises and solutions for your students to try on teir own.. Your feedback would be igly appreciated. Please me CFMSPWales@wimcs.ac.uk.If you enjoyed tis talk please mention it to oter scools in your area. More video talk organised by FMSP Wales:. Careers in Matematics. Infinity and eyond For more information CFMSPWales@wimcs.ac.uk Te History of Numbers Let Mats take you Furter 7

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