I NUMERI COMPLESSI
attività introduttive ai numeri complessi INGITTI
Tartalomjegyzék
- LA FORMULA DI EULERO
- y= e^x approssimazione della funzione esponenziale con un polinomio in un intorno a zero
- i, proprietà dell'unità immaginaria
- Lezione 1: y= cosx approssimazione della funzione coseno con un polinomio intorno a zero
- Lezione 2: y= cosx forma sintetica dell'approssimazione della funzione coseno con un polinomio intorno a zero
- Lezione 1: y= sinx approssimazione della funzione sinx con un polinomio intorno a zero
- Lezione 2: y= sinx forma sintetica dell'approssimazione della funzione seno con un polinomio intorno a zero
- Scoperta guidata alla formula di Eulero
- DALLA FORMULA DI EULERO AI NUMERI COMPLESSI
- Dalla formula di Eulero ai numeri complessi per alunni
- PERCORSO STORICO-MATEMATICO PER INTRODURRE I COMPLESSI
- Lezione 1: percorso storico-matematico per introdurre i complessi
- Lezione 2 : percorso storico-matematico per introdurre i complessi
- POTENZE DEI NUMERI COMPLESSI E SPIRALI NEL PIANO DI ARGAND-GAUSS
- Lezione 1. La danza dei numeri complessi: da z a z^n. Spirali nel piano di Argand Gauss
- Lezione 2. La danza dei numeri complessi: da z a z^n. Spirali nel piano di Argand Gauss
- Lezione 3. La danza dei numeri complessi: da z a z^n. Spirali nel piano di Argand Gauss
- L'OPERATORE DI ROTAZIONE
- Dal vettore unitario alla rotazione
- OPERAZIONI TRA NUMERI COMPLESSI
- Attività di scoperta: cosa fa davvero la moltiplicazione complessa?
y= e^x approssimazione della funzione esponenziale con un polinomio in un intorno a zero
Approssimazione della funzione esponenziale in un intorno di zero
Vogliamo approssimare con un polinomio, in un intorno di x=0, la funzione [math]y=e^x[/math][br][br]Ripassiamo le proprietà della funzione esponenziale [math]y=e^x[/math][br][list][*]la funzione ha come dominio tutti i reali e come codominio l'insieme dei reali positivi[/*][*]la funzione [math]y=e^x[/math] è una funzione strettamente crescente[/*][*]la funzione [math]y=e^x[/math] interseca l'asse delle ordinate nel punto (0,1)[/*][*]la funzione [math]y=e^x[/math] presenta un asintoto orizzontale sinistro di equazione [math]y=0[/math] [/*][*]la funzione [math]y=e^x[/math] è l'inversa della funzione[math]y=lnx[/math][/*][/list]
Grafico della funzione esponenziale
Determinazione del polinomio approssimante
La funzione esponenziale in [math]x=0[/math] vale [math]1[/math] ([math]e^0=1[/math]). Quindi se vogliamo approssimare la funzione con un polinomio di grado 0, necessariamente anche il polinomio calcolato in x=0 deve valere 1 cioè [math]P_0\left(0\right)=1[/math]. Tale polinomio è una retta parallela all'asse delle ascisse. Se rappresenti entrambe le funzioni su un file di geogebra, ti accorgerai che l'approssimazione è buona solo in un intorno piccolissimo di [math]x=0[/math]. Infatti, fissa un valore di[math]x[/math], a tale valore corrisponderà un punto E sulla curva esponenziale e un punto P sulla funzione polinomiale. La distanza EP tra le ordinate di tali punti (|[math]\left|y_E-y_P\right|[/math]) può fornire una indicazione del grado di approssimazione tra le due curve: più la distanza EP tra le due curve è piccola più è buona l'approssimazione. Completa la seguente tabella inserendo opportune formule nel foglio di calcolo. Nella finestra grafica, invece, sposta il punto P (e quindi la sua ascissa) e visualizza come l'approssimazione indicata dalla distanza EP , tra la funzione esponenziale e il polinomio, migliori man mano che l'ascissa di P si avvicini a 0. [br][br][br][br]
Usando geogebra costruisci, la visualizzazione della distanza che c'e tra la funzione [math]y=e^x[/math] e il polinomio approssimante al variare delle [math]x[/math] prese in un intorno di [math]x=0[/math]. [br]N.B. devi:[list=1][*]traccia la funzione esponenziale[math]y=e^{^x}[/math];[/*][*]traccia la funzione polinomiale [math]y=1[/math];[/*][*]traccia la retta [math]x=a[/math];[/*][*]si introdurrà automaticamente uno slider che mi farà variare le[math]x[/math] scelte; [/*][*]visualizzare, con il comando intersezione, i punti E e P che appartengono rispettivamente alle due funzioni e corrispondenti alla [math]x[/math] scelta;[/*][*]visualizzare il segmento EP che rappresenta la distanza tra tali punti;[/*][*]visualizzare il valore numerico di tale distanza.[/*][/list]
E' assolutamente necessario migliorare l'approssimazione
Aggiungiamo al polinomio di grado 0 un termine di primo grado, ottenendo il polinomio [math]P_1\left(x\right)=1+ax[/math]. Ci chiediamo quale potrebbe essere il valore di [math]a[/math] che renda l'approssimazione migliore possibile.[br]Per determinare il valore del coefficiente [math]a[/math] del termine di 1° grado possiamo fare valutazioni di diversa natura:[br][list][*]Una valutazione intuitiva utilizzando Geogebra[/*][*]Una valutazione algebrica utilizzando un foglio di calcolo[/*][/list]Usa Geogebra: [br][list=1][*]inserisci la funzione esponenziale;[/*][*]inserisci il polinomio [math]P_1(x)=1+ax[/math][br][/*][*]si crea automaticamente uno slider [math]a[/math] [/*][/list]Per capire quale potrebbero essere il segno e il valore di [math]a[/math], fai dei tentativi, variando il valore dello slider . Ricordati che puoi anche "animare" lo slider , regolando la velocità del movimento, per osservare meglio la posizione reciproca tra il polinomio e la funzione.
Considerazioni che si possono fare su a:
[list=1][*]Quale è il segno di [math]a[/math]? [/*][*]Quale valore di [math]a[/math] ti sembra possa essere adeguato per una buona approssimazione? [/*][*]Ti sembra che il polinomio, che è una retta passante per il punto [math](0,1)[/math], sia una retta "particolare" per la funzione [math]y=e^x[/math]?[/*][/list]
Valutazione di tipo algebrico del coefficiente a del termine di 1° grado.
L'idea è la seguente: determiniamo il valore di [math]a[/math] ( coefficiente del termine di primo grado del polinomio) seguendo un ragionamento molto semplice : [br] [br][list][*]Prendiamo un valore di [math]x[/math] abbastanza vicino allo zero, per esempio [math]x=0,2[/math][/*][*] calcoliamo la funzione esponenziale in [math]x=0,2[/math]: [math]y=e^{0,2}[/math] [/*][/list][list][*] calcoliamo il valore del Polinomio di 1° grado in [math]x=0,2[/math] : [math]P_1\left(0,2\right)=1+a\left(0,2\right)[/math][/*][*]ricaviamo [math]a[/math] risolvendo la seguente equazione : [math]P_1\left(0,2\right)=e^{0,2}[/math][/*][/list]se risolviamo, rispetto ad [math]a[/math] l'equazione [math]1+a\left(0,2\right)=e^{0,2}[/math], otterremo [math]a=1,107014[/math] [br][br]Se ripeti la stessa procedura per un valor di [math]x[/math] che si avvicina ulteriormente a 0, otterrai un valore di [math]a[/math] sempre più vicino al valore che ti permette di approssimare meglio la funzione esponenziale con un polimonio [math]P_1(x)[/math]. Usa il foglio excel per ricavare [math]a[/math] (con un'approssimazione di sei cifre decimali) per i seguenti valori :[math]x=0,1[/math] ;[math]x=0,01;x=0,001;x=0,0001;x=0,00001[/math][br]
procedura per ricavare a
Valutazione del polinomio di primo grado approssimante
in seguito alle valutazioni fatte scrivi il polinomio di 1° grado, [math]P_1\left(x\right)[/math] che meglio approssima, in[math]x=0[/math], la funzione esponenziale:
Ulteriore miglioramento del polinomio approssimante
Proviamo a migliorare il polinomio approssimante aggiungendo un termine di secondo grado al polinomio [math]P_1\left(x\right):[/math] [math]P_2\left(x\right)=1+x+bx^2[/math].[br]Per determinare il valore del coefficiente [math]b[/math] del termine di secondo grado , possiamo fare anche qui valutazioni di diversa natura:[br][list][*]una valutazione intuitiva usando Geogebra[/*][*] una valutazione algebrica[/*][*]una valutazione di natura geometrica (una valutazione globale), ovvero, scegliendo un intorno di zero non troppo piccolo e minimizziamo l'area sottesa tra il polinomio e la funzione esponenziale usando Geogebra[/*][/list]
Valutazione intuitiva del coefficiente b
Usa Geogebra: [br][list=1][*]inserisci la funzione esponenziale;[/*][*]inserisci il polinomio [math]P_{2_{ }}(x)=1+x+bx^2[/math][br][/*][*]si crea automaticamente uno slider [math]b[/math] [/*][/list]Per capire quale potrebbero essere il segno e il valore di [math]b[/math], fai dei tentativi, variando il valore dello slider . Ricordati che puoi anche "animare" lo slider , regolando la velocità del movimento, per osservare meglio la posizione reciproca tra il polinomio e la funzione.
Valutazione intuitiva del coefficiente b
Considerazioni che si possono fare su b:
[list=1][*] Quale è il segno di [math]b[/math]? [/*][*]Quale valore di [math]b[/math] ti sembra possa essere adeguato per una buona approssimazione? [/*][/list]
Ti sarai reso conto che, pur variando il valore dello slider, sembrerebbe non semplice capire l'esatto valore del coefficiente [math]b[/math]. [br]Analizziamo un altro tipo di valutazione:[br]
Valutazione di tipo algebrico del coefficiente b del termine di 2° grado.
L'idea è la seguente: determiniamo il valore di [math]b[/math] ( coefficiente del termine di 2° grado del polinomio) seguendo il seguente ragionamento: [br] [br][list][*]Prendiamo un valore di [math]x[/math] abbastanza vicino allo zero, per esempio [math]x=0,2[/math][/*][*] calcoliamo la funzione esponenziale in [math]x=0,2[/math]: [math]y=e^{0,2}[/math] [/*][/list][list][*] calcoliamo il valore del Polinomio di 2° grado in [math]x=0,2[/math] : [math]P_2\left(0,2\right)=1+\left(0,2\right)+b\left(0,2\right)^2[/math][/*][*]ricaviamo [math]b[/math] risolvendo la seguente equazione : [math]P_2\left(0,2\right)=e^{0,2}[/math][/*][/list]se risolviamo l'equazione [math]1+\left(0,2\right)+b\left(0,2\right)^2=e^{0,2}[/math], otterremo [math]b=[/math]0,53507[br][br]Se ripeti la stessa procedura per un valor di [math]x[/math] che si avvicina ulteriormente a 0, otterrai un valore di [math]b[/math] sempre più vicino al valore più adatto. Usa il foglio excel per ricavare [math]b[/math] (con un'approssimazione di sei cifre decimali) per i seguenti valori :[math]x=0,1[/math] ;[math]x=0,01;x=0,001;x=0,0001;x=0,00001[/math][br]
procedura algebrica per ricavare il coefficiente b del termine di 2° grado
Valutazione del polinomio di secondo grado approssimante
in seguito alle valutazioni fatte su [math]b[/math] indica il valore di b sotto forma di frazione algebrica e scrivi il polinomio di 2° grado, [math]P_2\left(x\right)[/math] che meglio approssima, in[math]x=0[/math], la funzione esponenziale:
Valutazione di natura geometrica per ricavare il coefficiente b del termine di 2° grado
[br]Passiamo ad un altro tipo di valutazione, basato sull'area sottesa tra le due funzioni in un intervallo fissato. L'idea sulla quale si basa questa valutazione è che "più le due funzioni sono vicine", più l'area "racchiusa" tra le due funzioni è piccola; man mano che le due funzioni si avvicinano l'area diminuisce.[br][br]Per questo tipo di valutazione , utilizziamo Geogebra: [br][list][*]Inserisci le due funzioni: la funzione esponenziale [math]y=e^x[/math]e il polinomio [math]P_2(x)=1+x+bx^2[/math]. [/*][*]Scegli un intorno di 0, per esempio [math]-1\le x\le1[/math] , [/*][*]usa il comando di Geogebra che si chiama [b]IntegraleTra, [math]IntegraleTra(f,P_2(x),-1,+1)[/math] [/b]er determinare l'area racchiusa tra le due funzioni nell'intervallo scelto. [br] [/*][/list]
Ti sembra che l'"idea" possa funzionare? ti sembra che la valutazione di b sia più accurata?
Miglioriamo il polinomio approssimante aggiungendo un termine di terzo grado.
Al fine di migliorare attraverso un polinomio l'approssimazione della funzione esponenziale , aggiungiamo un termine di terzo grado al precedente polinomio [math]$P_2=1+x+\frac{1}{2}x^2$[/math]. Chiamiamo [math]c[/math] il coefficiente del termine di terzo grado [math]P_3\left(x\right)=1+x+\frac{1}{2}x^2+cx^3[/math]. [br][br][br]Ci limitiamo in questo caso ad una valutazione di tipo algebrico, come già fatto precedentemente Ripetiamo la valutazione di tipo numerico.[br][br]Attribuisci alla [math]x[/math] gli stessi valori scelti precedentemente e compila la seguente tabella:[br][br][list][*]Prendiamo un valore di [math]x[/math] abbastanza vicino allo zero, per esempio [math]x=0,2[/math][/*][*] calcoliamo la funzione esponenziale in [math]x=0,2[/math]: [math]y=e^{0,2}[/math] [/*][/list][list][*] calcoliamo il valore del Polinomio di 3° grado in [math]x=0,2[/math] : [math]P3\left(0,2\right)=1+\left(0,2\right)+\frac{1}{2}\left(0,2\right)^2+c\left(0,2\right)^3[/math][/*][*]ricaviamo [math]b[/math] risolvendo la seguente equazione : [math]P_3\left(0,2\right)=e^{0,2}[/math][/*][/list]se risolviamo l'equazione [math]1+\left(0,2\right)+\frac{1}{2}\left(0,2\right)^2+c\left(0,2\right)^3=e^{0,2}[/math] rispetto a [math]c[/math], otterremo [math]c=...[/math]. [br][br]Se ripeti la stessa procedura per un valor di [math]x[/math] che si avvicina ulteriormente a 0, otterrai un valore di [math]c[/math] sempre più vicino al valore che permette al polinomio [math]P_3(x)[/math] di approssimare meglio la funzione esponenziale. Usa il foglio excel per ricavare [math]c[/math] (con un'approssimazione di sei cifre decimali) per i seguenti valori :[math]x=0,1[/math] ;[math]x=0,01;x=0,001;x=0,0001;x=0,00001[/math][br]
Procedura per ricavare c
Valore del coefficiente c del termine di 3° grado del polinomio approssimante.
Dalla valutazione algebrica puoi dedurre il valore di [math]c[/math]. Scrivi tale valore decimale anche in forma frazionaria:
Scrivi il polinomio di terzo grado ottenuto:
[math]P_3(x)=[/math]
Osservazioni sulle caratteristiche del polinomio approssimante
Confronta i polinomi usati per approssimare la funzione esponenziale intorno a [math]x=0[/math]:[br][math]P_0(x)=1[/math];[br][math]P_1(x)=1+x[/math];[br][math]P_2(x)=1+x+\frac{1}{2}x^2[/math];[br][math]P_3(x)=1+x+\frac{1}{2}x^2+\frac{1}{6}x^3[/math];[br] Come avrai notato, man mano che aumenta il grado del polinomio, l'approssimazione migliora. In altre parole: [br][b]1)[/b] [b]il grafico del polinomio “tocca” quello di [math]y=e^x[/math] in [math]x=0[/math] con un contatto sempre più accurato man mano che aumenta il grado. [br][b]2) l’approssimazione è particolarmente buona vicino a x=0[/b] e l’intervallo in cui il polinomio approssima bene la funzione [b]si allarga aumentando il grado[/b].[/b][br]Ma quale altra osservazione si potrebbe fare?
Osserva il segno dei termini
[*][b]Confronta i polinomi usati per approssimare la funzione esponenziale intorno a [math]x=0[/math]:[br][math]P_0(x)=1[/math];[br][math]P_1(x)=1+x[/math];[br][math]P_2(x)=1+x+\frac{1}{2}x^2[/math];[br][math]P_3(x)=1+x+\frac{1}{2}x^2+\frac{1}{6}x^3[/math];[br][/b][/*][*][b][br][/b][/*][*][b]1) Che segno hanno tutti i termini dei polinomi?[/b][/*][*][b]2) Vedi qualche termine con il segno meno?[/b][/*][*][b]3) Secondo te potrebbe comparire un termine negativo nei prossimi polinomi?[/b][br][/*]
Osserva le potenze di x
[b]Confronta i polinomi usati per approssimare la funzione esponenziale intorno a [math]x=0[/math]:[br][math]P_0(x)=1[/math];[br][math]P_1(x)=1+x[/math];[br][math]P_2(x)=1+x+\frac{1}{2}x^2[/math];[br][math]P_3(x)=1+x+\frac{1}{2}x^2+\frac{1}{6}x^3[/math];[/b][list=1][*][b]Qual è la potenza di [math]x[/math] più alta che compare in [math]P_0(x)[/math]?[/b][br][br][/*][*][b]Qual è la potenza più alta in [math]P_1(x)[/math]?[/b][br][br][/*][*][b]Qual è la potenza più alta ina [math]P_2(x)[/math]?[/b][br][br][/*][*][b]Che relazione c’è tra il numero del polinomio [math]P_n[/math] e la potenza più alta di [math]x[/math]?[/b][/*][*][b]Quali potenze di [math]x[/math] compaiono nel polinomio [math]P_3(x)[/math]?[/b][/*][*][b]Mancano potenze tra [math]x^0[/math] e [math]x^3[/math]?[/b][br][/*][/list]
Osserva i coefficienti
Confronta i polinomi usati per approssimare la funzione esponenziale intorno a [math]x=0[/math]:[br][math]P_0(x)=1[/math];[br][math]P_1(x)=1+x[/math];[br][math]P_2(x)=1+x+\frac{1}{2}x^2[/math];[br][math]P_3(x)=1+x+\frac{1}{2}x^2+\frac{1}{6}x^3[/math];[br]Scrivi solo i coefficienti dei vari polinomi e, in particolare, quelli del polinomio[math]P_3(x)[/math]:[br] [math]1[/math] , [math]1[/math] , [math]\quad\frac{1}{2}[/math] , [math]\frac{1}{6}[/math] , ....[br]Domande:[br][list=1][*][b]Cosa noti nei numeratori delle frazioni?[/b][br][br][/*][*][b]I denominatori come cambiano?[/b][br][br][/*][*][b]Che relazione c’è tra la potenza di [math]x[/math] e il numero nel denominatore?[/b][br][br][/*][*][b]Qual è il denominatore del termine con [math]x^2[/math]?[/b][br][br][/*][*][b]Qual è quello del termine con [math]x^3[/math]?[/b][br][br][/*][*][b]Secondo te quale sarà quello del termine con [math]x^4[/math]?[/b][br][/*][/list]
Ipotizza come potrebbe essere il polinomio di 4° grado P_4(x)
[list=1][*][b]Quale sarà la potenza più alta di[math]x[/math] nel polinomio[math]P_4(x)[/math]?[/b][br][br][/*][*][b]Quali potenze di [math]x[/math] dovrebbero comparire?[/b][br][br][/*][*][b]Che segno avranno i termini?[/b][br][br][/*][*][b]Come sarà il coefficiente del termine con [math]x^4[/math]?[/b][br][/*][*]Se il termine con [math]x^3[/math] è [math]\frac{1}{6}x^3=\frac{1}{2\cdot3}x^3=\frac{1}{3!}x^3[/math], quale pensi che sia il termine con [math]x^4[/math]?[/*][*]Scrivi il polinomio P_4(x)=[/*][/list]
La formula che esprime il polinomio approssimante
Dopo aver ricavato il polinomio approssimante di terzo grado e dedotto il polinomio di 4° grado , possiamo intuire quale sia il polinomio [math]Pn\left(x\right)[/math] che approssima la funzione [math]y=e^x[/math][br]Suggeriamo la formula che esprime il polinomio in parte già ricavato. Useremo il simbolo di Sommatoria[br]La formula è la seguente: [br][br][img width=79,height=55]data:image/png;base64,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[/img][br]La [b]funzione esponenziale[/b] [math]y=e^x[/math] può essere approssimata con una [b]serie infinita, [/b] usando il simbolo di [b]sommatoria[/b]. [br]Si legge: “Somma, per k che va da 0 a infinito, di [math]\frac{x^k}{k!}[/math]. Spieghiamo la formula:[br][list][*]0 è il valore iniziale dell’indice[br][b]∞ [/b]indica che[b] [/b] la somma continua all’infinito[/*][*]Quindi stiamo sommando:[list][*]termine con k=0[/*][*]termine con k=1[/*][*]termine con k=2[br][/*][*]termine con k=3, etc[/*][/list][br][/*][*]L’[b]indice k[/b] è la variabile che conta i termini della somma.[br]Funziona come un [b]contatore[/b].[/*][*]Ogni volta che k[b] aumenta di 1[/b], otteniamo un nuovo termine della serie.[/*][*]Il [b]fattoriale[/b] di un numero intero k si scrive [b]k![/b].[br]Definizione di fattoriale: k!=k⋅(k−1)⋅(k−2)⋯2⋅1[/*][*]Per convenzione: 0!=1[/*][/list]
Scrivi i primi 10 termini del polinomio approssimante
Vogliamo ora visualizzare con geogebra il polinomio approssimante
Utilizza geogebra per visualizzare il polinomio approssimante. [br]Segui le seguenti istruzioni:[br][list][*]inserisci [math]y=e^x[/math][br][/*][*]inserisci il polinomio [math]P(x)[/math] nel seguente modo: usa la formula [b]somma: [/b][/*][/list]La [b]formula somma (espressione, variabile, valore iniziale, valore finale) [/b]rappresenta una sommatoria matematica (spesso indicata con il simbolo [img]data:image/gif;base64,R0lGODlhAQABAIAAAP///wAAACH5BAEAAAAALAAAAAABAAEAAAICRAEAOw==[/img][math]\Sigma[/math] - sigma maiuscolo), utilizzata per calcolare la somma dei valori di un'espressione al variare di una variabile tra un limite inferiore e uno superiore.[br]inserisci :[br]=somma([math]\frac{x^k}{k!},k,0,a[/math]) , [math]a[/math] è uno slider che automaticamente comparirà, che ti permetterà di aumentare a tuo piacere i termini della somma e avere polinomi che sempre più si adagiano sulla funzione esponenziale intorno allo zero. Nelle impostazioni dello slider assegna ad [math]a[/math] come valore minimo lo zero e come valore massimo un valore alto, per esempio 50. Ti accorgerai che il polinomio con un grado alto , approssimerà molto bene la funzione esponenziale non solo in un intorno di [math]x=0[/math] [br] [br][br]
Visualizzazione del polinomio approssimante la funzione esponenziale al variare del grado attraverso Geogebra
Considerazioni finali.
Cosa puoi dire a conclusione del percorso effettuato? fai le tue considerazioni
Dalla formula di Eulero ai numeri complessi per alunni
INTERPRETAZIONE TRIGONIMETRICA di e^(ix)
La formula di Eulero:[br] [br][math]e^{ix}=cosx+i\cdot sinx[/math][br][br]a cui siamo giunti trovando i polinomi approssimanti le tre funzioni y=e^x , y=cosx e y=senx e introducendo l'unità immaginaria e le sue potenze è una relazione molto potente perché unisce importanti oggetti matematici: [list][*]la funzione esponenziale [/*][*]la funzione coseno [/*][*]la funzione senx[/*][*]l'unità immaginaria i[/*][/list]in un’unica formula scoperta da Leonhard Euler.[br]Le funzioni cosx e senx sono 'nascoste' in una particolare funzione esponenziale: [math]e^{ix}[/math].
Ma che tipo di oggetto è tipo di oggetto è [math]e^{ix}[/math]? Un numero reale o qualcosa di diverso?[br][br][br]
Possiamo disegnarlo come facevamo con [math]e^x[/math]?
[list][math]e^{ix}=cosx+i\cdot sinx[/math] suggerisce che [math]e^{ix}[/math] è formato da due parti le due parti del numero [math]e^{ix}[/math] ti ricordano qualcosa?[/list]dai almeno due indicazioni:
Separiamo le parti
Il numero [math]e^{ix}[/math] ha due parti, come possiamo descriverlo? [br][br][math]e^{ix}=cosx+i\cdot cosx[/math] [math]\longrightarrow[/math] ( ; )
Dove rappresentiamo normalmente coppie di 'numeri'?
Nascita del piano complesso
Nel nostro caso la coppia di numeri non può essere rappresentato su un normale piano cartesiano perché il primo numero è effettivamente un numero reale e quindi lo potrei inserire su un asse orizzontale in cui ci sono i 'semplici' numeri reali; il secondo numero della coppia invece, pur essendo un numero reale, è associato all'unità immaginaria i ([math]i\cdot senx[/math] ) pertanto ...è davvero la stessa cosa di un numero reale? Fai le tue considerazioni.
Per risolvere il 'problema' considera lo stesso piano cartesiano e prova a dare un nuovo significato agli assi; fai una proposta:[br][br]1) cosa metteresti sull'asse delle ascisse?[br][br]2) cosa metteresti sull'asse delle ordinate?
INTERPRETAZIONE ALGEBRICA
Torniamo alla formula di Eulero:[br][br][math]e^{ix}=cosx+isinx[/math][br][br]Se scriviamo questo numero nella forma [math]a+ib[/math], [br]1) quali sono [math]a[/math]e [math]b[/math]?[br]2) per un [math]x[/math] fissato, cosa sono [math]a[/math] e [math]b[/math]? [br]
z=e^(ix) come punto nel piano complesso
Nel piano complesso un numero [math]z=a+ib[/math] corrisponde al punto [math](a;b)[/math]Quindi il numero [math]z=e^{ix}[/math] corrisponde al punto [math]\left(a;b\right)[/math].
e^(ix) come punto che ruota
Dove si trovano nel piano complesso tutti i punti della forma [math](cosx;sinx)[/math]?
Dai una verifica formale di ciò che affermi
Il movimento del punto (cosx; sinx)
Immagina che [math]x[/math] varia assumendo i valori suggeriti nella tabella, scrivi per ogni [math]x[/math] il punto corrispondente:[br][br] [math]x[/math] [math]punto[/math][br]-----------|--------------[br] [math]0[/math] |[br] |[br] [math]\frac{\pi}{2}[/math] |[br] |[br] [math]\pi[/math] |[br] |[br] [math]\frac{3\pi}{2}[/math] |
[list]Che traiettoria descrivono questi punti che in qualche modo rappresentano [math]e^{ix}[/math]? Che tipo di movimento sembra?[/list]
Usando la finestra grafici di geogebra inserita sotto:[b][br][br]Crea uno slider [/b]per [math]x[/math]:[br] Seleziona lo strumento Slider (nella barra degli strumenti).[br]Clicca su un punto qualsiasi del piano. [br][list][*]Imposta:[br][/*][*][b]Nome[/b]: X[/*][*][b]Intervallo[/b]: da [code]0[/code] a [code]2π[/code][code][br][/code][/*][*][b]Incremento[/b]: ad esempio [code]0.01[br]Conferma.[br][/code][/*][/list] Ora hai un cursore che rappresenta un angolo in radianti.[br][b][br]Creare il punto dinamico[/b][br][list][*]Nella barra di inserimento scrivi:[/*][*]P=(cos(X), sin(X))[/*][*]Premi Invio.[/*][/list]Vedrai comparire il punto A che si muove al variare di [math]X[/math].[br][b][br]Attivare la traccia del punto[/b][br][list]Per vedere la “scia” del punto:Clic destro su P → seleziona [b]Attiva traccia[br][/b][/list]Muovendo lo slider, il punto disegnerà una figura.[br][b][br]Animare lo slider[/b][br][list]Clic destro sullo slider X Seleziona [b]Animazione attiva[br][/b][/list]Il punto inizierà a muoversi automaticamente.[br][br]Rispondi nella finestra risposta sotto:[br][list][*]Che figura disegna il punto A?[br][/*][br][*]Qual è il raggio di questa figura?[br][/*][br][*]Dove si trova il centro?[/*][/list]Verifica se le ipotesi che avevi fatto osservando i punti della tabella concordano con ciò che vedi al variare dello slider.[br][br]Continua nella stessa finestra grafici di geogebra inserita sotto:[br][b]Disegnare la circonferenza di riferimento[br][/b][list][*]Inserisci: c: x^2+y^2=1[/*][/list]In questo modo vedrai che il punto P si muove proprio su tale circonferenza.[br][b][br]Mostrare l’angolo[/b][br][list][*]Crea un punto:[br]B = (1, 0)[br][/*][*]Usa lo strumento [b]Angolo[/b] tra:[br][list][*]origine (0,0)[br][/*][*]B[br][/*][*]P[/*][/list][/*][/list]Si visualizza l’angolo x.[br]
Disegno dinamico
INTERPRETAZIONE GEOMETRICA
Dopo queste considerazioni, come descriveresti quindi [math]e^{ix}[/math] da un punto di vista geometrico?
INTERPRETAZIONE VETTORIALE
Se dovessi immaginare [math]e^{ix}[/math] come un vettore nel piano complesso:[br]1) che modulo gli assoceresti?[br]2) quale angolo (detto argomento) gli assoceresti per indicare la sua direzione? [br]3) cosa fa il vettore al variare dell'argomento?[br]
SINTESI FINALE
Grazie alla formula di Eulero abbiamo visto che possiamo leggere [math]e^{ix}[/math] in diversi modi. Scrivi sinteticamente le interpretazioni di [math]e^{ix}[/math] su cui hai avuto modo di riflettere durante questo percorso, esplicitando, in particolare, le seguenti interpretazioni:[list][*][b]trigonometrica[/b] (legame con [math]cosx[/math] e [math]sinx[/math])[/*][/list][math]e^{ix}[/math] rappresenta un numero complesso ...[br]le cui componenti sono:[br]parte reale: ...[br]parte immaginaria: ...[br]quindi è un modo compatto per esprimere...[list][*][b]algebrica o cartesiana[/b][/*][/list]il numero z rappresenta un numero complesso nella forma ...[br]le cui componenti sono:[br]parte reale: ...[br]parte immaginaria: ...[br]quindi è un modo algebrico [list][*][b]geometrica[/b] (legame con la circonferenza)[/*][/list]Nel piano complesso (piano di Piano di Argand-Gauss):[br] [math]e^{ix}[/math] è un punto sulla ... [br] di raggio ...[br] forma un angolo [math]x[/math] con l’asse...[br]quindi è un punto appartenente ad una figura geometrica [br][list][*][b]vettoriale[/b] (rotazioni)[/*][/list]Possiamo vedere (e^{ix}) come un vettore:[br]di lunghezza ...[br]direzione determinata dall’angolo...[br][br]Ripercorri sotto la sintesi introducendo le parti mancanti. [br]
Fai le tue considerazioni in relazione all'attività svolta:
[math]e^{ix}[/math] unifica: [list][*]trigonometria (seno e coseno)[/*][*]algebra (numeri complessi) [/*][*]geometria (circonferenza) [/*][*]vettori (rotazioni)[/*][/list]secondo te, quali tipi di fenomeni potrebbero essere descritti con l'uso di questo strumento 'potentissimo'?
Lezione 1: percorso storico-matematico per introdurre i complessi
Questo percorso , storico-matematico , porterà gli studenti a sentire la[b] necessità di introdurre i numeri complessi[/b], non solo ad accettarli come un’invenzione astratta. [br][br]Il percorso è strutturato in tappe logiche che legheranno la soluzione delle equazioni di terzo grado alla nascita di i, unità immaginaria , e ad una [b] trattazione sistematica [/b]dei numeri complessi.[br]
Facciamo un passo indietro: le equazioni di 2° grado
Si pensa erroneamente che l'introduzione dei numeri complessi sia legata alle equazioni di 2° grado. [br]Assegnata una equazione di 2° grado: [br][img width=119,height=22]data:image/png;base64,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[/img] [br]in R, le sue soluzioni sono:[br][br][img width=157,height=44]data:image/png;base64,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[/img][br][br]La quantità sotto radice quadrata è il [b]discriminante[/b] o DELTA. [br]Perché [math]b^2-4ac[/math] si chiama discriminante?
Siamo in R: spiega cosa succede quando:[br][br][list][*][img width=39,height=22]data:image/png;base64,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[/img],[/*][*][img width=39,height=22]data:image/png;base64,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[/img][/*][*][img width=39,height=22]data:image/png;base64,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[/img][/*][/list]
Quindi, quando [math][/math][img width=39,height=22]data:image/png;base64,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[/img] , cosa possiamo dire delle soluzioni delle equazioni di 2° grado in R?
Equazioni di terzo grado: cenni storici.
La risposta dei matematici al caso del DELTA <0, nelle equazioni di secondo grado, fu semplicemente accettare che i[b]n taluni casi l'equazione di secondo grado non avesse soluzioni reali[/b]. [b]Scoprirai in questo percorso che l'introduzione dei numeri complessi è legata alle equazioni di 3° grado e non a quelle di 2° .[br][/b][br]Viene detta [b]equazione di terzo grado[/b] o [b]cubica[/b] un'[url=https://it.wikipedia.org/wiki/Equazione]equazione[/url] che in [url=https://it.wikipedia.org/wiki/Equazione_polinomiale]forma polinomiale[/url] è tale che il grado massimo dell'incognita è il terzo. Pertanto, la sua forma canonica è: [br][br][img width=239,height=22]data:image/png;base64,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[/img][br]in cui [math]x[/math] è la variabile incognita e i coefficienti numerici sono indicati da [math]a,b,c,d[/math], reali.[br]Essendo l'equazione di terzo grado, può avere al massimo 3 radici, ovvero 3 soluzioni.[br][br]Per risolvere queste equazioni è necessario scomporre il polinomio in modo da ottenere delle equazioni di primo e secondo grado che possono essere risolte tramite procedimenti e formule note.
Casi di equazioni di terzo grado risovibili con metodi noti.
Quali equazioni di terzo grado del tipo [math]ax^3+bx^2+cx+d=0[/math] sei in grado di risolvere? [br]1. [math]d=0[/math] : [math]ax^3+bx^2+cx=0[/math] [br]2. [math]b=0[/math] e [math]c=0[/math] : [math]ax^3+d=0[/math][br]3. equazione completa: raccoglimento parziale e/o applicazione di Ruffini.[br][br]Scrivi le suluzioni delle seguentti tre equazioni di 3° grado, dopo averle svolte sul tuo quaderno:[br]1. [math]x^3+3x^2+2x=0[/math][br]2. [math]x^3-8=0[/math][br]3. [i][math]3x^3+2x^2-6x-20=0[/math] [br][/i][br]
Equazioni di 3° grado complete e forma ridotta.
Purtroppo però non sempre è possibile ricondurre l’equazione di 3° grado ad uno dei casi visionati.[br][br]Le equazioni di 3° grado che non rientrano nei casi elencati precedentemente, possono essere risolte o con metodi approssimativi o con una importante formula, detta formula di Cardano, alla determinazione della quale collaborarono matematici illustri, tra il 1500 e il 1600: [b]Del Ferro[/b], [b]Tartaglia, Cardano e Ferrari[/b] [br][br]Facciamo una importante premessa: [br][br]Si dimostra che tutte le equazioni di terzo grado [img width=239,height=22]data:image/png;base64,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[/img] possono essere scritte in una [b]forma ridotta , o depressa: [br][/b][br][img width=111,height=22]data:image/png;base64,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[/img][br][br]Vediamo come arrivare alla forma ridotta delle equazioni di 3° grado , partendo da una equazione completa.
I METODO- Riduzione di una equazione di terzo grado in una equazione di terzo grado depressa: metodo della Traslazione
Data la seguente equazione di terzo grado, [img width=239,height=22]data:image/png;base64,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[/img], ricondurla alla forma [img width=111,height=22]data:image/png;base64,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[/img]. [br]Procedimento: [br]1. esegui un cambio di variabile (una traslazione dell'incognita) ovvero poni: [math]x=y-\frac{b}{3a}[/math]. Questa sostutzione corrisponde ad una traslazione dell funzione rappresentata dal polinomio [math]P\left(x\right)=ax^3+bx^2+cx+d[/math] di vettore [math]v\left(\frac{b}{3a},0\right)[/math][br]2. sostituisci nell'equazione di terzo grado completa la nuova variabile per renderti conto che i termini di secondo grado non saranno più presenti. [br][br]Procedi facendo i calcoli sul quaderno e scrivi l'equazione ridotta risultante nello spazio risposta [br]Dovrebbe risultare [br][img]data:image/png;base64,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[/img][br][br] Inserisci nello spazio risposta l'immagine copiata dei vostri calcoli che portano all'equazione ridotta
II METODO- Riduzione di una equazione di terzo grado in una equazione di terzo grado depressa: metodo del completamento del cubo
Un altro modo per ridurre una equazione completa di terzo grado in forma depressa è tramite il[b] metodo del completamento del cubo. [/b]Abbiamo incontrato in passato, in diversi contesti , il [b]metodo del completamento del quadrato[/b]. Ricordi in quale contesto?
Metodo del completamento del cubo
Assegnata l'equazione di terzo grado completa: [math]ax^3+bx^2+cx+d=0[/math], vogliamo operare per ricondurla alla forma ridotta [math]y^3+p\cdot y+q=0[/math][br][br]1° passo: dividi tutta l'equazione per [math]a[/math]; [br][br]otterrai: [math]x^3+\frac{b}{a}x^2+\frac{c}{a}x+\frac{d}{a}=0[/math][br]2° passo: completa il cubo, ovvero somma e sottrai due monomi così che insieme ai primi due termini del polinomio costituiscano il cubo di un binomio e trascrivi i termini rimanenti;[br][br]otterrai: [math]x^3+3\cdot\frac{b}{3a}\cdot x^2+3\cdot x\cdot\frac{b^2}{9a^2}+\frac{b^3}{27a^3}-3\cdot x\cdot\frac{b^2}{9a^2}-\frac{b^3}{27a^3}+\frac{c}{a}x+\frac{d}{a}=0[/math][br]3° passo: Riporta l'equazione scrivendo i primi 4 monomi come cubo di un binomio, e aggiungi i termini rimanenti raccogliendo l'incognita nei termini in cui compare.[br][br]4° passo: fai le opportune sostituzioni in modo da ottenere l'equazione nella forma: [math]y^3+p\cdot y+q=0[/math]
Riporta tutti i calcoli che ti ha portato alla forma ridotta [math]$y^3+p\cdot y+q=0$[/math] con un'immagine inserita nello spazio risposta.
Un po' di storia...
E' da questa equazione in forma ridotta (che possiamo vedere anche nella forma [math]x^3+p\cdot x+q=0[/math] ) che parte la storia delle soluzioni di equazioni di 3° grado, in particolare quelle che non possono essere risolte riconducendole ad equazioni di grado più basso con i metodi visti sopra.[br][br]La soluzione delle equazioni di 3° grado è una delle vicende [b]più affascinanti della storia della matematica rinascimentale italiana, [/b]e coinvolge figure come [b]Scipione del Ferro, Niccolò Tartaglia, Gerolamo Cardano e Rafael Bombelli.[/b][br][br][b]Scipione del Ferro, [/b] professore a Bologna, fu il primo a trovare una soluzione. [br]Riuscì a risolvere un caso particolare di equazione cubica, [math]x^3+px=q[/math]. Dopo la morte di Del Ferro, il metodo arrivò (indirettamente) a [b]Niccolò[/b] [b]Tartaglia[/b], che lo riscoprì e lo estese. Tartaglia era noto per partecipare a sfide pubbliche tra matematici, e grazie alla sua scoperta riuscì a vincere importanti competizioni. [br][br][b]Gerolamo Cardano[/b], medico e matematico, personaggio molto discutibile, convinse Tartaglia a rivelargli il metodo, promettendo di non pubblicarlo.[br][br]Nel 1539 Niccolò Tartaglia, inviò in versi a Cardano la forma risolutiva delle equazioni cubiche prive del termine di secondo grado, chiedendo però di non divulgarla. Ecco come Tartaglia svelava la soluzione della equazione di terzo grado[math]x^3+px=q[/math]al suo collega Cardano:[br][br][img]data:image/png;base64,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[/img][br]Cardano riusci a decodificare i versi e non mantenne la promessa fatta a Tartaglia; infatti, quando venne a sapere che [b]Del Ferro aveva già scoperto la soluzione prima di Tartaglia[/b], ritenne di poter pubblicare il risultato senza violare la promessa.[justify][br]Nel 1545 pubblicò l’opera [b]Ars Magna[/b], in cui presentò:[/justify][list][*][b]la soluzione generale delle equazioni cubiche[/b][/*][*][b][b]la soluzione delle quartiche [/b](grazie all'aiuto del suo allievo Ludovico Ferrari) [br][/b][/*][/list]Questo causò una[b] forte polemica con Tartaglia[/b].[br][br]Altro personaggio fondamentale nella vicenda fu [b]Rafael Bombelli,[/b] che intervenne qualche decennio dopo, nel 1572, con il libro [b]L'Algebra[/b]. Il suo contributo fu fondamentale perché:[br][list][*]chiarì i passaggi della formula di Cardano[br][/*][*]affrontò il problema dei numeri “impossibili” (oggi detti: numeri complessi)[br][/*][/list]In particolare, studiò i casi in cui [b]la formula della cubica porta a radici quadrate di numeri negativi (il cosiddetto [i]caso irreducibile[/i]), dando senso operativo ai numeri complessi.[/b][br][br]Quindi la formula risolutiva delle equazioni di terzo grado, detta di Cardano, in realtà dovrebbe essere chiamata la [b]formula di Del Ferro, Tartaglia e Cardano. [br][br][/b]Se l'equazioni di terzo grado, in forma ridotta, la guardiamo come [math]x^3+px+q=0[/math] si ha che la sua soluzione è:[br][br][math]x=\sqrt[3]{-\frac{q}{2}+\sqrt{\left(\frac{q}{2}\right)^2+\left(\frac{p}{3}\right)^3}}+\sqrt[3]{-\frac{q}{2}-\sqrt{\left(\frac{q}{2}\right)^2+\left(\frac{p}{3}\right)^3}}[/math][br][br]la quantità sotto radice quadrata è il discriminante, DELTA, delle equazioni di 3°grado:[br][br][math]DELTA=\left(\frac{q}{2}\right)^2+\left(\frac{p}{3}\right)^3[/math][br][br][br]OSSERVA: Il DELTA dell'equazione di 3° ha una forma diversa rispetto al DELTA dell'equazione di 2° ; come per le equazioni di 2° grado, il segno del DELTA , sarà discriminante per le soluzioni dell'equazione.
Applicazione della formula di Cardano ad una equazione di 3° grado
Proviamo ad applicare la formula di Cardano ad una equazione di terzo grado ridotta:[br][br][math]x^3-3x+2=0[/math][br][br][math]p=-3[/math][br][math]q=2[/math][br][br]Scrivi la soluzione di Cardano, specificando il valore del DELTA:
Troviamo le altre radici
Nota una radice, trovata applicando la formula di Cardano, possiamo scomporre il polinomio di 3° grado come prodotto di un polinomio di 1° grado e uno di 2° [br][br](Avendo una radice, puoi applicare la regola di Ruffini per determinare la scomposizione). [br]Dopo aver svolto i passaggi sul tuo quaderno , scrivi la scomposizione del polinomio dato e quindi le soluzioni dell'equazione di partenza di 3°. [br]Le soluzioni dell'equazione sono tutte reali?
Equazioni di 3° grado irriducibili
[b]La formula della equazione cubica suggerita da Cardano, a volte però, porta ad avere radici quadrate di numeri negativi (il cosiddetto [i]caso irreducibile[/i]). [br][/b]Facciamo un esempio: [br][math]x^3-15x-4=0[/math][br]Trova una radice con la formula di Cardano, e commenta il risultato, specificando il valore del DELTA. [br]Scrivi il valore di tale radice e commenta il risultato ottenuto.
Cerchiamo di capire in quale situazione ci troviamo.
Interpretazione grafica delle soluzioni di una equazione.[br]Ricordiamo che per esempio le soluzioni reali di una equazione del tipo [math]x^2-3x+2=0[/math] saranno gli zeri della funzione [math]y=x^2-3x+2[/math][br]Cosa rappresentano allora, da un punto di vista grafico, le soluzioni di una equazione di grado qualsiasi?
Rappresentazione grafica della funzione di 3° grado associata all'equazione irriducubile
la rappresentazione grafica realizzata su geogebra della funzione [math]y=x^3-15x-4[/math], darà una idea delle soluzioni della equazione associata [math]x^3-15x-4=0[/math][br][br]Rappresenta su geogebra tale funzione e osserva gli zeri della funzione (indicali, sul grafico, con un piccolo cerchietto rosso)
Rappresentazione grafica della funzione
Osservazioni sul grafico della funzione e relazione con le soluzioni dell'equazione associata.
Osserva il grafico della funzione [math]y=x^3-15x-4[/math] : [br][list][*]Quanti sono i punti di intersezione della funzione con l'asse delle ascisse? [/*][*]In base a quanto detto precedentemente, il numero dei punti di intersezione tra la funzione e l'asse delle ascisse che indicazioni da sulle soluzioni dell'equazione associata alla funzione [math]x^3-15x-4=0[/math]? [/*][*]Sapresti indicare il valore numerico preciso di almeno uno degli zeri della funzione? [/*][*]Verifica algebricamente che lo zero da te individuato sia soluzione dell'equazione [math]x^3-15x-4=0[/math].[/*][/list]
Una evidente contraddizione. Come risolverla?
Ripercorriamo i passi fatti:[br][list][/list][list][*]applicando la formula di Cardano, abbiamo potuto verificare che nella soluzione dell'equazione compaiono [b]radici quadrate di numeri negativi[/b]. Questo fu un grande problema per Cardano, perché i numeri negativi sotto radice [b]non avevano ancora un significato. Quindi sembrerebbe che l'equazione [math]x^3-15x-4=0[/math] non abbia soluzioni reali. [/b][/*][/list][list][*]rappresentando graficamente la funzione [math]y=x^3-15x-4[/math], si deduce che essa ha tre punti di intersezione con l'asse delle ascisse, quindi che l'equazione [math]x^3-15x-4=0[/math] ha tre soluzioni reali, di cui una è [math]x=4[/math], come verificato anche algebricamente. [br][br][/*][*][b]i due risultati sono evidentemente in contrasto. Tornando alla nostra storia, Gerolamo Cardano si accorse della contraddizione , ovvero che , anche quando la soluzione finale era reale [math]x=4[/math], nell'applicazione della sua formula comparivano radici quadrate di numeri negativi. Dedusse che la formula in alcuni casi (equazione irriducibile con DELTA<0) non funzionasse. [/b][/*][/list][br]
Nasce la necessità di "allargare " l'insieme dei numeri reali!
Quella che a Cardano sembrava una contraddizione in realtà era un chiaro segnale che [b]la matematica dell’epoca era troppo “stretta”. [br][br]La svolta arrivò con Rafael Bombelli, che fu fondamentale per superare la contraddizione: fu il primo a trattare seriamente quei “numeri strani” che comparivano nella formula di Cardano, cioè le radici di numeri negativi.[br]Bombelli li prese sul serio, anche se “impossibili”! Decise di [b]usarli comunque[/b], come strumenti di calcolo, anche se non capiva ancora cosa fossero davvero. [br]Introdusse regole di calcolo chiare, in pratica costruì le basi operative dei [b]numeri complessi.[/b][br][/b]
Lezione 1. La danza dei numeri complessi: da z a z^n. Spirali nel piano di Argand Gauss
NUMERI COMPLESSI E FORME IN CUI POSSONO ESSERE SCRITTI
[b]1. Numero complesso: dalla forma cartesiana alla forma trigonometrica[/b][br][br]Un numero complesso [math]z=a+ib[/math] può essere scritto in [b]forma[br]polare o trigonometrica[/b]: [math]z=r\cdot\left(cos\left(\alpha\right)+i\cdot sin\left(\alpha\right)\right)[/math][br][br]dove: [br][list][*] [math]r=\left|z\right|=\sqrt{a^2+b^2}[/math] è il [b]modulo[/b] [/*][*][math]\alpha=arctan\left(\frac{b}{a}\right)[/math] è l'[b]argomento [/b] ([b]l’angolo orientato[/b] che il vettore (a,b) forma con il semiasse reale positivo) [/*][/list]Questa forma è molto comoda per svolgere moltiplicazioni e potenze di Numeri complessi, mentre la forma cartesiana più utile per somme e sottrazioni[br][br][b]2. Numero complesso: dalla forma trigonometrica alla forma esponenziale[br][/b]Allo stesso modo sarà comoda e utile, per le operazioni tra complessi, anche la [b]forma esponenziale[/b] di un numero complesso:[br][br][math]z=r\cdot e^{i\alpha}[/math][br][br]che si basa sulla [b]formula di Eulero: [/b][b][math]e^{i\alpha}=cos\left(\alpha\right)+i\cdot sin\left(\alpha\right)[/math] [br][br][/b]Basta infatti moltiplicare entrambi i membri della formula di Eulero per[math]r[/math]:[br][br] [math]re^{i\alpha}=r\cdot\left(cos\left(\alpha\right)+i\cdot sin\left(\alpha\right)\right)[/math] al fine di ottenere la forma esponenziale di un numero complesso. [br][br]La forma esponenziale è più compatta , utile per prodotti e potenze.[br]
PRODOTTO TRA NUMERI COMPLESSI SCRITTI IN FORMA CARTESIANA
[b]1. Prodotto di numeri complessi in forma cartesiana[/b][br][br]Dimostra che: dati i due numeri complessi [b]in forma cartesiana[/b][br][br][math]z_1=a+bi[/math] [math]z_2=c+di[/math] [br][br] il loro [b]prodotto è[/b]:[br][br][math]z_1z_2=(ac−bd)+(ad+bc)i[/math][br][br]Scrivi sul quaderno la dimostrazione della formula del prodotto tra due numeri complessi in forma cartesiana:
PRODOTTO TRA NUMERI COMPLESSI SCRITTI IN FORMA TRIGONOMETRICA
[b]2. Prodotto di numeri complessi in forma trigonometrica[/b][br][br]Dimostra che : dati i due numeri complessi [math]z_1=r_1\cdot\left(cos\left(\alpha_1\right)+i\cdot sin\left(\alpha_1\right)\right)[/math] e [math]z_2=r_2\cdot\left(cos\left(\alpha_2\right)+i\cdot sin\left(\alpha_2\right)\right)[/math] , [br]il prodotto tra i due numeri è: [br][br][math]z_1\cdot z_2=r_1\cdot r_2\left(cos\left(\alpha_1+\alpha_2\right)+i\cdot sin\left(\alpha_1+\alpha_2\right)\right)[/math][br][br]Notate come[b] i moduli si moltiplicano e gli argomenti si sommano. [/b]Questo fatto è di cruciale importanza per interpretare il significato geometrico della moltiplicazione tra due numeri complessi e, di conseguenza, della potenza di un numero complesso. [br][br]Scrivi sul quaderno la dimostrazione della formula del prodotto tra due numeri complessi scritta sopra:
PRODOTTO TRA NUMERI COMPLESSI SCRITTI IN FORMA ESPONENZIALE
[b]3. Prodotto di numeri complessi in forma esponenziale[/b][br][br]Dimostra che: dati i due numeri complessi in forma esponenziale, [br][math]z_1=r_1\cdot e^{i\alpha_1}[/math], [math]z_2=r_2\cdot e^{i\alpha_2}[/math][br][br]il prodotto tra i due numeri è: [br][br] [math]z_3=r_1r_2\cdot e^{i\left(\alpha_1+\alpha_2\right)}[/math][br][br]Scrivi la dimostrazione sul quaderno o nello spazio ripsosta. [br][br]
VISUALIZZAZIONE GEOMETRICA DEL PRODOTTO DI DUE NUMERI COMPLESSI
E' possibile [b]visualizzare in modo molto esplicativo il prodotto tra due numeri complessi con GeoGebra.[/b] [br][br]Nella barra di inserimento di Geogebra sono stati inseriti:[br][br][list][*]il punto [math]A=1+i[/math], fissato. [br][/*][*]lo slider[math]r[/math], numero che va da 0 a 20, che esprime il modulo di un numero complesso, scritto in forma trigonometrica[/*][*]lo slider [math]\alpha[/math], angolo, che va da 0° a 360°[/*][*]il punto[math]B[/math], espresso nel seguente modo: [math]B=r\cdot\left(cos\left(\alpha\right)+i\cdot sin\left(\alpha\right)\right)[/math][/*][/list][br][math]A[/math] rappresenta un[b] numero complesso fisso[/b][br][math]B[/math] rappresenta un [b]numero complesso variabile[/b], di cui possiamo controllare modulo e argomento tramite i due slider[math]r[/math] e [math]\alpha[/math][br][br][list][*][math]C=A\cdot B[/math], prodotto tra i due numeri complessi [br][br]Inseriamo anche i due vettori:[br][/*][/list][math]OA=vettore((0,0),A)[/math], il vettore originale[br][math]OB=vettore((0,0),B)[/math], il vettore variabile[br][br][math]OC=vettore((0,0),C)[/math], il vettore trasformato, ovvero il vettore risultato del prodotto tra A e B[br][br]
OSSERVAZIONE GUIDATA ALL'APPLET DI GEOGEBRA RELATIVA AL PRODOTTO DI DUE NUMERI COMPLESSI
Facciamo alcune osservazioni, al fine di comprendere il significato geometrico della moltiplicazione tra numeri complessi: [br][br][b]1. Fissa[/b][math]r[/math][b] e muovi lo slider [/b][math]\alpha[/math]:[br][list][*]Cosa succede alla lunghezza del vettore [math]OC?[/math][/*][*]Cosa cambia nel vettore[math]OC[/math]?[/*][*]Descrivi il movimento del punto [math]C[/math].[/*][/list]
[b]2. Ora fissa [/b][math]\alpha[/math][b] e cambia [/b][math]r[/math][b]:[/b][br][list][*]Cosa succede a [math]OC[/math]?[/*][*]Cosa puoi dire sulla lunghezza di [math]OC[/math] e di [math]OA[/math]? [/*][*]Che movimento descrive il punto [math]C[/math]?[/*][/list][br]
Rispondi alle seguenti domande:[br][list][*]cosa vuol dire [b]moltiplicare A per un numero complesso[/b] [math]B=r\cdot\left(cos\left(\alpha\right)+i\cdot sin\left(\alpha\right)\right)[/math], di modulo [math]r[/math] e argomento [math]\alpha[/math]? Rispondi osservando la posizione del punto C [/*][/list][list][*]esprimi il concetto in termini di[b] trasformazioni del piano. Ovvero, componendo quali trasformazioni otteniamo il punto C, prodotto tra A e B, partendo dal punto A?[/b][/*][*][b]cosa significa moltiplicare un numero complesso per [math]i[/math], unità immaginaria?[/b][/*][/list]
DAL PRODOTTO TRA NUMERI COMPLESSI ALLA POTENZA DI UN NUMERO COMPLESSO
Sappiamo che la potenza di un numero reale o complesso è una [b]notazione compatta [/b]per rappresentare [b]moltiplicazioni ripetute dello stesso numero[/b], ed è fondamentale perché semplifica molto i calcoli e la scrittura di numeri grandi.[br]
POTENZE DI UN NUMERO COMPLESSO
[b]1. Potenza di un numero complesso scritto in forma cartesiana. [br]Assegnato [math]z=a+ib[/math], la potenza n-esima del numero z, si calcola a partire dalla formula della potenza n-esima di un binomio:[br][br][math]z=\left(a+ib\right)^n[/math] [br][/b][br]Risolvi i seguenti esercizi sul quaderno o nello spazio risposta: [br][br][math]z=(1+i)^3[/math][br][br][math]z=(2+2i)^4[/math][br][br][math]z=(1-i)^5[/math][br][br]
POTENZE DI UN NUMERO COMPLESSO
[b]2. Potenza di un numero complesso scritto in forma trigonometrica[/b][br][br]Per calcolare la potenza [math]z^n[/math](con [math]n\in N[/math]), scriviamo [math]z[/math] in forma trigonometrica e lo moltiplichiamo per se stesso [math]n[/math] volte. Otteniamo:[br][br] [math]z^n=r^n\cdot\left(cos\alpha+isin\alpha\right)^n[/math][br][br]Prova a calcolare:[br][list][*][math]z^2[/math], [/*][*][math]z^3[/math], per questa potenza sono utili le due seguenti formule, facilmente dimostrabili:[/*][/list][br] 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[/img][list][*] [math]z^4[/math], .........[/*][/list]con un po' di calcoli, ti puoi rendere conto che la potenza di un numero complesso scritto in forma trigonometrica è data dalla seguente formula: [br][br][math]z^n=r^n\left(cos\left(n\alpha\right)+i\cdot sin\left(n\alpha\right)\right)[/math] [b] Formula di de Moivre. [/b][br][br]Questo significa che, elevare un numero complesso a potenza diventa molto semplice perché, invece di eseguire molte moltiplicazioni, basta:[br][list][*]elevare il [b]modulo[/b] [math]r[/math]alla [math]n[/math][/*][*]moltiplicare l’[b]angolo[/b] [math]\alpha[/math] per [math]n[/math][br][br][/*][/list]P.S. Al termine dell'attività troverai una bellissima dimostrazione [b]per induzione[/b] della formula di De Moivre[br][br] [math]z^n=r^n\left(cos\left(n\alpha\right)+i\cdot sin\left(n\alpha\right)\right)[/math][br][br]ESERCIZIO:[br][img]data:image/png;base64,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[/img][br][br][br]
POTENZE DI UN NUMERO COMPLESSO - forma esponenziale
[b]3. Potenza di un numero complesso.[br][br][/b]La potenza [math]z^n[/math] (con [math]n\in N[/math]) di un numero complesso scritto in forma esponenziale, [math]z=r\cdot e^{i\alpha}[/math][br]è data dalla formula [br][br] [math]z^n=\left(r\cdot e^{i\alpha}\right)^n=r^n\cdot e^{in\alpha}[/math], di immediata dimostrazione.[br][br]Questo significa che, elevare un numero complesso alla potenza [math]n[/math] diventa ancora più semplice perchè, invece di eseguire molte moltiplicazioni, basta:[br][list][*]elevare il [b]modulo[/b] [math]r[/math]alla [math]n[/math][/*][*]moltiplicare l’[b]angolo[/b] [math]\alpha[/math], presente nell'esponente di e, per [math]n[/math][br][/*][/list]ESERCIZIO:[br][br][img]data:image/png;base64,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[/img][br]
CONSIDERAZIONI SULLE POTENZE DI NUMERI COMPLESSI
Riassumendo, sulla base delle formule espresse sia in forma cartesiana, sia trigonometrica che in forma esponenziale della potenza di un numero complesso [math]z[/math], dopo aver svolto gli esercizi assegnati, fai le dovute considerazioni su quale sia la forma più adeguata di un numero complesso per determinare una sua potenza. [br]
APPROFONDIMENTO 1. DIMOSTRAZIONE DELLA FORMULA DI DE MOIVRE
[b] Dimostrazione della formula di de Moivre [math]z^n=r^n\left(cos\left(n\alpha\right)+i\cdot sin\left(n\alpha\right)\right)[/math][/b][br][br][b]Per induzione su [/b][img width=10,height=22]data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAoAAAAWCAMAAADO+P1vAAAAAXNSR0IArs4c6QAAAFRQTFRFAAAAAAAAAAA6AABmADpmADqQOgAAOjpmOpC2OpDbZgAAZrb/kDoAkGZmkLbbkNv/tmYAtmY6tmZmttv/tv/btv//27Zm2////7Zm/9uQ//+2///be0k4qAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAATUlEQVQYV7WOSRKAIBADE0VUXFgURfj/P535g+aU6vQhwE9pkSZflhOQfOC4F9tnoK1cUDjcwKMgcJYDCqrrvFQFspwbqhNVfHN8evEFS9AC2E8jFygAAAAASUVORK5CYII=[/img][br][list=1][br][*][b]Base[/b][br] ([img width=39,height=22]data:image/png;base64,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[/img]): [/*][br][/list][img 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induttivo[/b]: supponiamo che valga per [img width=9,height=22]data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAkAAAAWCAMAAAAlz0ZsAAAAAXNSR0IArs4c6QAAAFdQTFRFAAAAAAAAAABmADpmADqQAGa2OgAAOgA6OgBmOpCQOpDbZjoAZrb/kDoAkGY6kGaQkJC2kNv/tmYAtpBmttv/tv//25A62//b2////7Zm/9uQ/9u2///bjf122wAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAASUlEQVQYV62LDQ6AIBSCsbQy7c/SfOb9zxneQTY24BtAZ9Vd+c+pFbifYM+cdAbqZoBAo8wXm2eSIaIskakB0e9BwJtMYxs76Acw7QLctpahQgAAAABJRU5ErkJggg==[/img]: [/*][/list][br][img 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[/img]=[br][b][br]Ricordando il passo induttivo[/b] per il primo fattore del secondo membro[b]:[br][br][/b][img 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[/img][br]Ma per le [b]formule di addizione del coseno e seno[/b]:[br][br][br][img 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[/img][br][br]Induzione completata.[br][br][br]
Dal vettore unitario alla rotazione
Con questa attività vogliamo far emergere, in modo graduale, che [math]e^{ix}[/math] non è solo un numero complesso ma è un particolare operatore.
Grazie alla formula di Eulero abbiamo visto che possiamo leggere [math]e^{ix}[/math] come un numero complesso che si presta a varie interpretazioni :[br][br][list][*][b]trigonometrica[/b] (legame con [math]cosx[/math] e [math]sinx[/math])[/*][/list][math]e^{ix}[/math] rappresenta un numero complesso [math]z=cosx+i\cdot sinx[/math], le cui componenti sono:[br]parte reale: cosx[br]parte immaginaria: sinx[br]quindi: è un modo compatto per esprimere insieme le funzioni coseno e seno e[br][br] [math]\left(cosx;sinx\right)[/math] rappresenta [u]solo[/u] un punto nel piano complesso.[br][br][list][*][b]algebrica o cartesiana[/b][/*][/list]il numero [math]z[/math] rappresenta un numero complesso nella forma algebrica [math]a+i\cdot b[/math][br]le cui componenti sono numeri reali:[br]parte reale: [math]a[/math][br]parte immaginaria: [math]b[/math] [br]In particolare il numero complesso [math]z=a+bi[/math] si vede come un [b]vettore[/b]: [math]v^{\longrightarrow}=\left(a;b\right)[/math] con[br][list][*]componente orizzontale = parte reale[/*][*]componente verticale = parte immaginaria[br][/*][/list]che è una descrizione cartesiana attraverso coordinate. [br]Il modulo di tale vettore è [math]\left|z\right|=\sqrt{a^2+b^2}[/math], mentre la sua direzione è [math]\Theta=arctg\left(\frac{b}{a}\right)[/math][br]N.B. in questo modo il vettore è rappresentato in [b][i]modo vettoriale[/i][/b] (coordinate:ti dicono [i]quanto ci si muove in orizzontale/verticale[/i]); [br][br][list][*][b]geeometrica[/b] (legame con la circonferenza)[/*][/list]Nel piano complesso (piano di Piano di Argand-Gauss):[br] [math]e^{ix}[/math] è un punto sulla circonferenza di raggio 1 che forma un angolo [math]x[/math] con l’asse dei numeri reali, quindi è un punto appartenente ad una figura geometrica: la circonferenza unitaria. Quindi, possiamo vedere [math]e^{ix}[/math] come un vettore di lunghezza [math]1[/math] e direzione determinata dall’angolo [math]x[/math] [br]N.B. in questo modo il vettore è rappresentato in [b][i]modo polare[/i][/b] (modulo+angolo: ti dicono [i]quanto è lungo e in che direzione punta)[/i]
Rappresentazione di un numero reale nel piano complesso
Abbiamo anche visto che il piano complesso è un particolare piano cartesiano:[br] sull'asse delle ascisse abbiamo posto i numeri reali[br] sull'asse delle ordinate abbiamo posto la parte immaginaria di un numero complesso (asse immaginario).[br]Con questa premessa, cerchiamo di capire cosa rappresenta [math]z=1[/math] nel piano complesso, rispondendo alle seguenti domande:[*] [/*][*] 1) Dove si trova il punto nel piano?[br][/*][br][*] 2) Quali sono le sue coordinate?[br][/*][br][*] 3) Quanto vale la parte immaginaria?[/*]
Il numero reale 1 sul piano complesso
Usa la finestra di geogebra sotto.[br]Prova a mettere il numero z=1 sull'asse dei reali nel piano complesso. Per indicare a geogebra che stai mettendo tale punto sull'asse dei reali usa l'[b]icona punto:[/b][br] nella tendina scegli 'Numero complesso' [br] posiziona ora il punto corrispondente a z=1 sull'asse dei reali
Il vettore unitario
Disegna, usando ancora la finestra di geogebra sopra, il vettore corrispondente al numero [math]z=1[/math] .[br]Nella finestra algebra, inserisci:[br][math]vector((0,0),(1,0))[/math] e guardando il vettore graficato, rispondi alle domande indicate sotto.[br]
[*]1) Nella finestra algebrica come è indicato il punto che hai posizionato corrispondente a z=1? [/*][*]2) Quanto vale il modulo (la lunghezza) del vettore?[/*][*]3) Cosa significa “unitario”?[br][/*][*]4) Possiamo immaginarlo come punto e/o come un vettore?[/*]
Introduzione di e^(i*X)
Con la finestra sottostante di GeoGebra[br][list=1][b]1.creare slider:[/b][/list]seleziona lo strumento 'Slider'[br]clicca sul piano vista grafico[br]e nella finestra che si apre impostare:[br] Nome: X [br] Tipo: numero (non angolo)[br] variabilità: 0 → 2π [ Min: [code]0 [/code]Max: [code]6.28[/code] (oppure [code]2π[/code])][br] incremento: [code]0.05 [/code] [Velocità animazione: media][br][br]IMPORTANTE:[br][list][*]anche se [math]X[/math] rappresenta un angolo, deve essere espresso in GeoGebra come un [b]numero reale[/b][br][/*][*]GeoGebra lavora in radianti di default[/*][/list][br][b] 2. Inserisci il numero complesso[/b][br]inserire nella vista algebra:[br][math]z=exp(i*X)[/math][br][br]attenzione a questi dettagli:[br][list][*]scrivi [b]i[/b] minuscola (unità immaginaria)[br][/*][*]usa [code]*[/code] per la moltiplicazione[br][/*][*]lo slider [code]X[/code] deve esistere PRIMA[/*][/list][br]
Cosa si vede
1) Cosa vedi per ogni valore [math]X[/math] fissato dello slider?[br]2) che tipo di movimento ha il punto, se attivi lo slider?[br]3) il movimento del punto, con lo slider attivo, che figura descrive?[br]4) quanto è la distanza che ha il punto dall'origine del piano complesso, è sempre la stessa o cambia?[br]5) se ad ogni punto associamo il vettore corrispondente, cosa cambia del vettore quando cambia il valore di [math]X[/math] in [math]e^{iX}[/math]?[br]
Moltiplicare e^(ix) per z=1
Abbiamo visto che il numero [math]z=1[/math] corrisponde al numero complesso[math](1;0)[/math] mentre [math]e^{ix}[/math] può essere pensato come un vettore unitario la cui direzione è individuata da [math]x[/math].[br]Cosa succede al numero [math]z[/math], corrispondente al punto [math](1;0)[/math] nel piano complesso, se lo moltiplichiamo per [math]e^{ix}[/math]?[br][br][math]1\cdot e^{ix}=e^{ix}[/math] ricorda la formula di Eulero [math]e^{ix}=\left(cosx+i\cdot sinx\right)[/math][br]quindi [math]z=1[/math], che corrispondente al vettore unitario [math](1;0)[/math] , se sottoposto ad una moltiplicazione per [math]e^{ix}[/math], diviene un nuovo vettore ancora di modulo unitario e direzione [math]x[/math]; perciò: [math]\left(1;0\right)\longrightarrow[/math] [math]\left(cosx;sinx\right)[/math]
Nel piano complesso, che tipo di trasformazione ha subito il vettore corrispondente a numero [math]z=1[/math] quando viene moltiplicato per [math]e^{ix}[/math]?
Moltiplicare e^(ix) per z=1 Visualizzazione con geogebra
Usa geogebra sottostante e inserisci nella finestra algebrica le opportune indicazioni (già viste in precedenza o secondo le equivalenti indicazioni date) per rappresentare contemporaneamente: [br][b]1) il numero z=1 sull'asse dei reali del piano complesso e il suo vettore corrispondente. [/b][br] Per indicare a geogebra che stai mettendo tale punto sull'asse dei reali usa l'[b]icona punto[/b]; [br] nella tendina scegli 'Numero complesso' [br] posiziona ora il punto corrispondente a [math]Z=1[/math] sull'asse dei reali[br] Z=1 (attenzione metti Z come lettera maiuscola, per Geogebra z è una variabile) [br] P=(Z,0)[br] O=(0,0)[br] [math]v[/math]=Vector(O,P)[br]dovresti vedere:[list][*]punto [math]P=(1,0)[/math][br][/*][*]vettore [math]v[/math] sull’asse reale[/*][/list][b]2) la sua trasformazione dopo che è stato moltiplicato per [/b][math]e^{ix}[/math][b], cioè il vettore corrispondente al punto [/b][math](cosx;sinx)[/math] [b]sul cerchio unitario.[br][/b][br]-creare slider: [br] seleziona lo strumento 'slider'[br] clicca sul piano vista grafico[br] e nella finestra che si apre impostare:[br] Nome: X (non usare la lettera [math]x[/math] perchè geogebra associa a tale valore il valore di una variabile) [br] Tipo: numero (non angolo)[br] Variabilità: 0 → 2π [ Min: [code]0 [/code]Max: [code]6.28[/code] (oppure [code]2π[/code])][br] Step: [code]0.05 [/code] [Velocità animazione: media][br][br]- rappresenta [math]e^{iX}[/math]:[br] nella barra di input scrivi:[br] A = (cos(X), sin(X)) (questo punto rappresenta proprio [math]e^{iX}[/math]).[br][br]-Disegna il vettore dall’origine[br] Scrivi:[br] w = Vector((0,0), A) oppure: w= Vector(A) (Otterrai il vettore che parte dall’origine e arriva al punto A)[br][br][br]Rispondi sotto alle domande, riferendoti alla rappresentazione fatta nella finestra di geogebra sottostante.[br]1. Che relazione c'è tra il vettore [math]v[/math] di partenza (corrispondente a [math]z=1[/math]) e il vettore [math]w[/math] (corrispondente a [math]e^{ix}[/math])? [br]Confronta i loro moduli e le loro direzioni[br]2. Che tipo di trasformazione ha subito il vettore corrispondente a numero z=1 quando viene moltiplicato per [math]e^{ix}[/math]? (esplicita in modo chiaro le caratteristiche della trasformazione)[br]
Moltiplicare e^(ix) per z=2
Abbiamo visto che il numero [math]z=1[/math] corrisponde al numero complesso (1;0) mentre [math]e^{ix}[/math] può essere pensato come un vettore unitario la cui direzione è individuata da x.[br]Cosa succede al numero [math]z=2[/math] corrispondente al punto (2;0) nel piano complesso se lo moltiplichiamo per [math]e^{ix}[/math]?[br][br][math]2\cdot e^{ix}=2e^{ix}[/math] (ricordando la formula di Eulero [math]e^{ix}=\left(cosx+i\cdot sinx\right)[/math]), [math]2\left(cosx+isinx\right)=\left(2cosx+i2sinx\right)[/math][br]quindi [math]z=2[/math], che corrispondente al vettore [math](2;0)[/math] di modulo 2, se sottoposto ad una moltiplicazione per [math]e^{ix}[/math], diviene un nuovo vettore, non più di modulo unitario, ma di modulo 2 ([math]|z|=2[/math]) e direzione [math]x[/math]; perciò: [br] [math]\left(2;0\right)\longrightarrow[/math] [math]\left(2cosx;2sinx\right)[/math][br]
Moltiplicare e^(ix) per z=2 Visualizzazione con geogebra
Usa geogebra sottostante e inserisci nella finestra algebrica le opportune indicazioni (già viste in precedenza o secondo le equivalenti indicazioni date) per rappresentare contemporaneamente: [br][b]1) il numero z=2 sull'asse dei reali del piano complesso e il suo vettore corrispondente. [/b][br] Per indicare a geogebra che stai mettendo tale punto sull'asse dei reali usa l'[b]icona punto[/b]; [br] nella tendina scegli 'Numero complesso' [br] posiziona ora il punto corrispondente a [math]Z=2[/math] sull'asse dei reali[br] Z=2 (attenzione metti Z come lettera maiuscola, per Geogebra z è una variabile) [br] P'=(Z,0)[br] O=(0,0)[br] [math]v'[/math]=Vector(O,P')[br]dovresti vedere:[list][*]punto [math]P'=(2,0)[/math][br][/*][*]vettore [math]v'[/math] sull’asse reale[/*][/list][b]2) la sua trasformazione dopo che è stato moltiplicato per [/b][math]e^{ix}[/math][b], cioè il vettore corrispondente al punto [/b][math](2cosx;2sinx)[/math] [b]sul cerchio di raggio 2.[br][/b][br]-creare slider: [br] seleziona lo strumento 'slider'[br] clicca sul piano vista grafico[br] e nella finestra che si apre impostare:[br] Nome: X (non usare la lettera [math]x[/math] perchè geogebra associa a tale valore il valore di una variabile) [br] Tipo: numero (non angolo)[br] Variabilità: 0 → 2π [ Min: [code]0 [/code]Max: [code]6.28[/code] (oppure [code]2π[/code])][br] Step: [code]0.05 [/code] [Velocità animazione: media][br][br]- rappresenta [math]2e^{iX}[/math]:[br] nella barra di input scrivi:[br] A' = (2cos(X), 2sin(X)) (questo punto rappresenta proprio [math]2e^{iX}[/math]).[br][br]-Disegna il vettore dall’origine[br] Scrivi:[br] w' = Vector((0,0), A') oppure: w= Vector(A') (Otterrai il vettore che parte dall’origine e arriva al punto A')[br][br][br]Rispondi sotto alle domande, riferendoti alla rappresentazione fatta nella finestra di geogebra sottostante.[br]1. Che relazione c'è tra il vettore [math]v'[/math] di partenza (corrispondente a [math]z=2[/math]) e il vettore [math]w'[/math] (corrispondente a [math]2e^{ix}[/math])? Confronta i loro moduli e le loro direzioni[br]2. Che tipo di trasformazione ha subito il vettore corrispondente a numero z=2 quando viene moltiplicato per [math]e^{ix}[/math]? (esplicita in modo chiaro le caratteristiche della trasformazione)[br]3. Confronta i due casi, qui riassunti, ottenuti moltiplicando il numero [math]z[/math] per [math]e^{ix}[/math] :[br][math]\left(1,0\right)\longrightarrow\left(cosx,senx\right)[/math][br][br][math]\left(2,0\right)\longrightarrow\left(2cosx,2senx\right)[/math][br] la moltiplicazione per [math]e^{ix}[/math]come agisce sul vettore di partenza?
Domande ulteriori[br]Il numero preso in considerazione z=2 è numero con la parte immaginaria pari a....[list=1] e con modulo pari a .... [br][*]Attiva lo slider nella finestra geogebra sopra; che forma ha il moto del punto A' (corrispondente al vettore [math]2e^{ix}[/math])?[br][/*][br][*]Il raggio della circonferenza è cambiato rispetto al moto del punto A?[br][/*][br][*]Quanto vale il nuovo modulo?[/*][/list] 4. Cosa fa il numero 2 al vettore [math]e^{ix}[/math] (specificare se il vettore [math]2e^{ix}[/math], rispetto al vettore [math]e^{ix}[/math] , rimane lo stesso, oppure si dilata di..., oppure si contrae di ...[br][br]
Rendiamo dinamica la rappresentazione con GeoGebra
Nella finestra geogebra sottostante crea una rappresentazione dinamica della situazione: al variare del punto P che rappresenta [math]z[/math] sull'asse dei reali, (dovrai perciò inserire uno slider) possa variare di conseguenza [math]ze^{ix}[/math] ( legare il punto corrispondente a [math]z[/math], alle coordinate del punto trasformato dalla sua moltiplicazione con l'operatore [math]e^{ix}[/math] cioè[br][math](z,0)[/math] si trasforma in [math]\left(zcosx+izsinx\right)[/math]. (Dovresti essere in grado di mettere i comandi di geogebra necessari alla rappresentazione richiesta).
Generalizzazione: moltiplicare e^(ix) per z = a + ib
Moltiplichiamo ora [math]e^{ix}[/math] per un numero complesso generico del tipo [math]z=a+ib[/math]. [br]Un numero di questo tipo cosa rappresenta sul piano complesso?[br]qual è il suo modulo [math]|z|[/math]?[br]qual è la sua direzione [math]\Theta[/math]?[br] Come abbiamo visto precedentemente, per il numero z=1 e z=2, eseguiamo tutti i passaggi algebrici per capire come si trasforma [math]z=a+ib[/math] (che può essere visto come [math]\left(a,b\right)[/math][math][/math]) quando viene moltiplicato per [math]e^{ix}[/math][br][br][math]\left(a+ib\right)\cdot e^{ix}=\left(a+ib\right)\cdot\left(cosx+isinx\right)[/math] (avendo ricordato la formula di Eulero [math]e^{ix}=\left(cosx+i\cdot sinx\right)[/math]); svolgi i calcoli sul tuo quaderno e confronta il risultato con qui riportato:[br][math]\left(a+ib\right)\cdot\left(cosx+isinx\right)=acosx-bsinx+i\cdot\left(asinx+bcosx\right)[/math][br] Come vedi il numero complesso [math](a,b)[/math] dopo la moltiplicazione per l'operatore [math]e^{ix}[/math] diviene un nuovo numero complesso: [math](a,b)\longrightarrow\left(a',b'\right)[/math].[br]1) Scrivi sotto la parte reale e poi la parte immaginaria di questo vettore che è la trasformazione del vettore [math](a,b)[/math] causata dal fattore moltiplicativo [math]e^{ix}[/math]
Determinazione algebrica di (a+ib)e^(ix)
Sintesi concettuale
“Se moltiplico un numero complesso per [math]e^{ix}[/math], cosa sto facendo geometricamente?”
Conclusioni
Finora, hai visto come agisce [math]e^{ix}[/math] quando viene applicato ad un numero complesso [math]z[/math]; [br]1) riassumi che caratteristiche ha questo operatore?[br]2) cosa cambia dopo che l'operatore [math]e^{ix}[/math] agisce su un numero complesso (rispetto alle sue caratteristiche: modulo e direzione)? [br][list][*][br][/*][/list]3)quindi: [math]e^{ix}[/math] è un operatore di ...
Considerazioni sull'attività svolta:
Attività di scoperta: cosa fa davvero la moltiplicazione complessa?
Moltiplicazione tra complessi
Obiettivo[br]Capire che moltiplicare per un numero complesso A equivale a:[br][list][*]una [b]rotazione[/b][br][/*][*]e una [b]dilatazione[/b][/*][/list]Al fine di far capire meglio cosa succede quando si moltiplica un numero complesso [math]z[/math] per un complesso dato A si fa la seguendo procedura:[br]1. scelgo tre numeri complessi [math]z_1[/math], [math]z_2[/math] e [math]z_3[/math];[br]2. ne calcolo modulo e direzione;[br]3. visualizzo i tre vettori corrispondenti;[br]4. trovo modulo e direzione del fattore moltiplicativo [math]A[/math];[br]5. visualizzo il triangolo che ha come vertici i tre numeri complessi [math]z_1[/math], [math]z_2[/math] e [math]z_3[/math][br]6. moltiplico i tre numeri complessi [math]z_1[/math], [math]z_2[/math] e [math]z_3[/math] per il fattore moltiplicativo [math]A[/math] (numero complesso) ottenendo i vettori[math]z'_1[/math], [math]z'_2[/math] e [math]z'_3[/math];[br]7. visualizzo il triangolo che ha come vertici i tre numeri complessi [math]z'_1[/math], [math]z'_2[/math] e [math]z'_3[/math]. [br]Vogliamo scoprire che relazione c'è tra i due triangoli.[br]
Scelta dei numeri complessi e loro caratterizzazione
Siano dati i seguenti numeri complessi:[br][math]z_1=1+i[/math][br][math]z_2=2[/math][br][math]z_3=-1+i[/math][br]Calcola e riporta sotto il loro modulo e il loro argomento ricordando che [math]\left|z\right|=\sqrt{a^2+b^2};....\longrightarrow\Theta=arctg\left(\frac{b}{a}\right)[/math] ;[br][math]\left|z_1\right|=....\longrightarrow\Theta1=...[/math][br][math]\left|z_2\right|=....\longrightarrow\Theta_2=...[/math][br][math]\left|z_3\right|=....\longrightarrow\Theta_3=...[/math][br][br]I tre vettori hanno lunghezze diverse?[br]Che direzione hanno?
Visualizzazione nel piano
Nella barra di inserimento di GeoGebra [b]inserisci i numeri complessi[/b]:[br][math]z_1=1+i[/math][br][math]z_2=2[/math][br][math]z_3=-1+i[/math][br]GeoGebra li rappresenta automaticamente come punti.[br][b]Traccia i vettori[/b] corrispondenti a tali punti con i comandi:[br][math]vettore\left(\left(0,0\right),z_1\right)[/math][br][math]vettore\left(\left(0,0\right),z_2\right)[/math][br][math]vettore\left(\left(0,0\right),z_3\right)[/math][br]Determina [b]modulo[/b] e [b]argomento[/b] (angolo) con i seguenti comandi:[br][math]Length\left(z_1\right)[/math][br][math]Angle\left(z_1\right)[/math][br](ripetere per [math]z_2[/math]e per [math]z_3[/math])[br]Traccia il [b]triangolo iniziale[/b] con il comando:[br][math]Polygon\left(z_1,z_2,z_3\right)[/math]
Visualizzazione nel piano complesso
[*]Che tipo di triangolo è?[/*][*]Quanto sono lunghi i lati? (si può usare [math]Distance(z_1,z_2)[/math])[/*]
Introduzione del moltiplicatore
Scegliamo come numero complesso moltiplicatore un numero semplice del tipo [br][math]A=1+i[/math][br]stabilisci il suo modulo e il suo argomento (angolo) analogamente a quanto fatto prima:[br][math]\left|A\right|=...[/math][br][math]\Theta_A=...[/math]
Moltiplicazione
Svolgi le moltiplicazioni suggerite tra i vettori espressi in forma algebrica:[br][math]z^'_1=A\cdot z_1[/math];[br][math]z^'_2=A\cdot z_2[/math][br][math]z^'_3=A\cdot z_3[/math][br]Calcola e riporta sotto il loro modulo e il loro argomento ricordando che [math]\left|z\right|=\sqrt{a^2+b^2};....\longrightarrow\Theta=arctg\left(\frac{b}{a}\right)[/math] ;[br][math]\left|z^'_1\right|=....\longrightarrow\Theta^'1=...[/math][br][math]\left|z^'_2\right|=....\longrightarrow\Theta^'_2=...[/math][br][math]\left|z^'_3\right|=....\longrightarrow\Theta^'_3=...[/math][br][br]I tre vettori hanno lunghezze diverse rispetto ai vettori corrispondenti ai numeri complessi [math]z_1[/math], [math]z_2[/math] e [math]z_3[/math]?[br]Che direzione hanno rispetto a quelle dei vettori corrispondenti ai numeri complessi [math]z_1[/math], [math]z_2[/math] e [math]z_3[/math]?
Visualizziamo nella finestra di GeoGebra precedente:
[math][/math]Poni nella barra di inserimento anche il numero complesso che corrisponde al [b]moltiplicatore[/b] scelto:[br][math]A=1+i[/math][br]Determina [b]modulo[/b] e [b]argomento[/b] (angolo) con i seguenti comandi:[br][math]Lengthz\left(A\right)[/math][br][math]Angle\left(\Theta_A\right)[/math][br][br]Inserisci nella barra inserimento le moltiplicazioni suggerite tra i vettori:[br][math]z'_1=A\cdot z_1[/math][br][math]z'_2=A\cdot z_2[/math][br][math]z'_3=A\cdot z_3[/math][br][br][b]Traccia i vettori[/b] corrispondenti a tali punti con i comandi:[br][math]vettore\left(\left(0,0\right),z'_1\right)[/math][br][math]vettore\left(\left(0,0\right),z'_2\right)[/math][br][math]vettore\left(\left(0,0\right),z'_3\right)[/math][br]Determina [b]modulo[/b] e [b]argomento[/b] (angolo) con i seguenti comandi:[br][math]Length\left(z'_1\right)[/math][br][math]Angle\left(z'_1\right)[/math][br](ripetere per [math]z'_2[/math] e per [math]z'_3[/math])[br]Traccia il [b]triangolo nuovo[/b] con il comando:[br][math]Polygon\left(z'_1,z'_2,z'_3\right)[/math][br][br]
Confronto tra le lunghezze dei lati dei triangoli
Sempre nella stessa finestra di GeoGebra di prima, con il comando:[br][math]Distance(z_1,z_2)[/math][br][math]Distance(z'_1,z'2)[/math][br]individua e scrivi sotto tali distanze che rappresentano un lato prima della trasformazione e dopo la trasformazione.[br]Ripeti tali comandi per determinare altri lati corrispondenti dei due triangoli.[br]1) I lati sono cambiati? [br]2) Di quanto?[br]3) ) I lati hanno subito tutti la stessa dilatazione?[br]4) Le loro lunghezze in che relazione stanno con il modulo del moltiplicatore [math]A[/math]?
Confronto tra gli angoli dei triangoli
Sempre nella stessa finestra di GeoGebra di prima, con gli opportuni comandi, puoi visualizzare gli angoli tra due lati corrispondenti dei due triangoli:[br][math]Angle(z_1,z_2)[/math][br][math]Angle(z'_1,z'2)[/math][br]individua e scrivi sotto tali angoli che rappresentano un angolo tra due lati prima della trasformazione e dopo la trasformazione.[br]1)Gli angoli sono cambiati?[br]2) come si definiscono triangoli (prima della trasformazione e dopo) che godono di tali proprietà?
Rotazione
1) Il triangolo sembra ruotato?[br]2) Di quanto?[br][br] Suggerimento:[br]usando sempre la stessa finestra di Geogebra, confrontare gli angoli dei vettori prima di moltiplicarli per il fattore moltiplicativo [math]A[/math] e dopo; a tal proposito puoi inserire, per esempio, il seguente comando:[br][math]Angle(z'_1)-Angle(z_1)[/math][br]3) in che relazione è l'angolo tra i due vettori trasformati dal fattore moltiplicativo rispetto all'argomento(angolo) di quest'ultimo?[br]
Trasformazione geometrica di una figura legata al fattore moltiplicativo
Caratterizza sotto, i tipi di trasformazione geometrica che ha subito il triangolo trasformato dal fattore moltiplicativo [math]A[/math]
Trasformazione geometrica di un numero complesso (visualizzabile con un punto-vettore) legata al fattore moltiplicativo
Caratterizza sotto, i tipi di trasformazione geometrica che subisce un numero complesso trasformato dal fattore moltiplicativo [math]A[/math][br](specifica la trasformazione del suo modulo e del suo argomento in relazione alle caratteristiche del modulo e argomento del fattore moltiplicativo [math]A[/math])
Estensione dinamica
In una nuova finestra di Geogebra inserita sotto, crea un fattore moltiplicativo [math]A[/math] variabile inserendo due opportuni slider: [br][math]a=Slider(-3,3)[/math][br][math]b=Slider(-3,3)[/math][br][math]A=a+b\cdot i[/math][br] Ripeti tutte le operazioni fatte precedentemente e descrivi cosa succede al triangolo e al suo trasformato quando il fattore moltiplicativo [math]A[/math] cambia (crea triangoli dinamici).[br]
Considerazioni finali
Fai sotto le tue considerazioni sull'attività svolta.