tag:blogger.com,1999:blog-16373306369930859952024-03-04T23:12:44.568-08:00ALLENAMENTI DI MATEMATICA matematica per le gare di matematica
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gareslm.blogspot.comprofsergiolamalfahttp://www.blogger.com/profile/17034912698828182534noreply@blogger.comBlogger43125tag:blogger.com,1999:blog-1637330636993085995.post-75526683709123864002018-09-07T09:41:00.000-07:002018-09-07T09:41:00.033-07:00RIPASSO FONDAMENTI DI GEOMETRIA EUCLIDEA CON KAHOOT!<div style="text-align: center;">
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<span style="font-size: large;">RIPASSO : <a href="https://play.kahoot.it/#/k/47d49811-ef85-4ae5-a545-0ba56bf485dc" target="_blank">CLICCA QUI</a></span><br /><div style="text-align: center;">
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profsergiolamalfahttp://www.blogger.com/profile/17034912698828182534noreply@blogger.com0tag:blogger.com,1999:blog-1637330636993085995.post-35576423495870003202018-09-06T14:17:00.001-07:002018-09-07T06:20:45.479-07:00EQUAZIONI DIOFANTEE<div dir="ltr">
<span style="font-family: Arial, Helvetica, sans-serif;"><span style="font-size: large;">Sono equazioni a coefficienti intere e delle quali si cercano soluzioni intere.<br />
Consideriamo quelle lineari in x e y del tipo ax+by=c.<br />
Posto d=MCD(a, b) queste equazioni hanno soluzioni solo se d divide c.<br />
Se l'equazione ha soluzioni allora ne ha necessariamente infinite e sono date dalla somma di una soluzione particolare (x', y') e di una generica dell'equazione omogenea ax+by=0.<br />
Ad esempio 3x+4y=5 ha come soluzione particolare (3,-1). La soluzione dell'equazione dell'omogenea 3x+4y=0 è (4t,-3t). Quindi la soluzione è (4t+3,-3t-1).</span></span></div>
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<span style="font-family: Arial, Helvetica, sans-serif;"><span style="font-size: large;">Graficamente le soluzioni dell'equazione Diofantea sono i punti a coordinate intere che appartengono alla retta. Ad esempio le soluzioni dell'equazione 2x+3y=5 sono del tipo (1,1) soluzione particolare + (3t, -2t) soluzione omogenea. Quindi sono soluzioni tutte le coppie (1+3t,1-2t) al variare di t nei numeri interi. Ad esempio per t=1 ottengo C(4,-1) (vedi figura)</span></span></div>
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<span style="font-family: Arial, Helvetica, sans-serif;"><span style="font-size: large;"><span style="font-size: small;">video lezione </span></span></span></div>
profsergiolamalfahttp://www.blogger.com/profile/17034912698828182534noreply@blogger.com1tag:blogger.com,1999:blog-1637330636993085995.post-29178458646277778002018-09-04T10:55:00.003-07:002018-09-04T10:55:48.134-07:00QUADRILATERI CICLICI<div class="separator" style="clear: both; text-align: center;">
<span style="font-family: Arial, Helvetica, sans-serif;"><span style="font-size: large;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjf1zDp-FtQwb3lTzKXoMkzgo8hq8V2btl_oEogPzOTVIAExfOp4j5yGVxNFlye6cYDe-HAYCM7dAq42OcykhxD7-8hDFDz3nS0FibYvJ1ghnJC7p_6XmAVqKMx7yDMFgCW9ltE7rZtzQ/s1600/SS1.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="205" data-original-width="200" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjf1zDp-FtQwb3lTzKXoMkzgo8hq8V2btl_oEogPzOTVIAExfOp4j5yGVxNFlye6cYDe-HAYCM7dAq42OcykhxD7-8hDFDz3nS0FibYvJ1ghnJC7p_6XmAVqKMx7yDMFgCW9ltE7rZtzQ/s1600/SS1.png" /></a></span></span></div>
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<span style="font-family: Arial, Helvetica, sans-serif;"><span style="font-size: large;">Un quadrilatero si dice ciclico se si può iscrivere in una circonferenza. </span></span></div>
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<span style="font-family: Arial, Helvetica, sans-serif;"><span style="font-size: large;"><span style="font-family: Arial, Helvetica, sans-serif;"><span style="font-size: large;">1)Un quadrilatero è ciclico </span></span>SE E SOLO SE gli angoli opposti sono supplementari.</span></span></div>
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<span style="font-family: Arial, Helvetica, sans-serif;"><span style="font-size: large;">DIM: è conseguenza del fatto che l'angolo alla circonferenza è la metà dell'angolo al centro.</span></span></div>
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<span style="font-family: Arial, Helvetica, sans-serif;"><span style="font-size: large;"> </span></span></div>
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<span style="font-family: Arial, Helvetica, sans-serif;"><span style="font-size: large;">2) </span></span><span style="font-family: Arial, Helvetica, sans-serif;"><span style="font-size: large;"><span style="font-family: Arial, Helvetica, sans-serif;"><span style="font-size: large;"><span style="font-family: Arial, Helvetica, sans-serif;"><span style="font-size: large;">Un quadrilatero è ciclico </span></span>SE E SOLO SE sono uguali gli angoli che "insistono sullo stesso lato"</span></span></span></span></div>
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<span style="font-family: Arial, Helvetica, sans-serif;"><span style="font-size: large;">Un quadrilatero è ciclico </span></span><span style="font-family: Arial, Helvetica, sans-serif;"><span style="font-size: large;">SE E SOLO SE vale il teorema di TOLOMEO</span></span></div>
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<span style="font-family: Arial, Helvetica, sans-serif;"><span style="font-size: large;">TEOREMA DI TOLOMEO: AB<span style="font-size: small;">X</span>DC+AD<span style="font-size: small;">X</span>BC=AC<span style="font-size: small;">X</span>BD</span></span></div>
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<span style="font-family: Arial, Helvetica, sans-serif;"><span style="font-size: large;"><br /></span></span></div>
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<span style="font-family: Arial, Helvetica, sans-serif;"><span style="font-size: large;">Un quadrilatero è ciclico se e solo se (vale T. corde)</span></span></div>
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<span style="font-family: Arial, Helvetica, sans-serif;"><span style="font-size: large;"> AOxOC=BOxOD</span></span></div>
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profsergiolamalfahttp://www.blogger.com/profile/17034912698828182534noreply@blogger.com0tag:blogger.com,1999:blog-1637330636993085995.post-73482323485848878892018-09-04T07:38:00.002-07:002018-09-04T07:38:59.844-07:00ALLENAMENTO DI MATEMATICA BASE CON KAHOOT! #2<div class="separator" style="clear: both; text-align: center;">
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ALLENAMENTO BASE <a href="https://play.kahoot.it/#/k/6830bbc4-4f14-4927-8d89-bfa4e2dc4257" target="_blank">CLICCA QUI</a></div>
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profsergiolamalfahttp://www.blogger.com/profile/17034912698828182534noreply@blogger.com0tag:blogger.com,1999:blog-1637330636993085995.post-68287291181966925952018-09-02T09:59:00.001-07:002018-09-04T07:35:40.657-07:00ALLENAMENTO DI MATEMATICA BASE CON KAHOOT! #1<div class="separator" style="clear: both; text-align: center;">
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<a href="https://play.kahoot.it/#/k/d600733f-faec-4b28-af82-751ee6f79af6" target="_blank">1° INCONTRO</a></div>
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profsergiolamalfahttp://www.blogger.com/profile/17034912698828182534noreply@blogger.com0tag:blogger.com,1999:blog-1637330636993085995.post-48307571715561744562018-09-02T09:57:00.001-07:002018-09-02T09:57:51.969-07:00CALCOLO DELLA FUNZIONE DI EULERO ON LINE<div class="separator" style="clear: both; text-align: center;">
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<a href="http://www.javascripter.net/math/calculators/eulertotientfunction.htm" target="_blank">CLICCA QUI</a></div>
profsergiolamalfahttp://www.blogger.com/profile/17034912698828182534noreply@blogger.com0tag:blogger.com,1999:blog-1637330636993085995.post-58189031018496261862017-11-07T09:48:00.001-08:002017-11-07T09:48:23.139-08:00Crivello di Eratostene<span style="font-size: large;"><span style="font-family: Arial,Helvetica,sans-serif;"><br /><br />e' un procedimento per trovare tutti i numeri primi minori di n ed è il seguente: si scrivono tutti i numeri naturali a partire da 2 fino n . Poi si cancellano (setacciano) tutti i multipli del primo numero del setaccio (escluso lui stesso). Si prende poi il primo numero non cancellato maggiore di 2 e si ripete l'operazione con i numeri che seguono, proseguendo fino a che non si applica l'operazione all'ultimo numero non cancellato. I numeri che restano sono i numeri primi minori o uguali a n .</span></span><br />
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profsergiolamalfahttp://www.blogger.com/profile/17034912698828182534noreply@blogger.com0tag:blogger.com,1999:blog-1637330636993085995.post-45267964823126694442017-09-24T01:50:00.003-07:002017-09-24T01:50:32.082-07:00DISUGUAGLIANZE CLASSICHE<div class="separator" style="clear: both; text-align: center;">
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profsergiolamalfahttp://www.blogger.com/profile/17034912698828182534noreply@blogger.com0tag:blogger.com,1999:blog-1637330636993085995.post-4237631034014030742016-09-03T07:55:00.002-07:002016-09-03T07:55:12.415-07:00PICCOLO TEOREMA DI FERMAT<!--[if gte mso 9]><xml>
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<span style="font-family: Arial,Helvetica,sans-serif;">Il piccolo teorema di Fermat afferma che se a è coprimo a p :
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<span style="font-family: Arial,Helvetica,sans-serif;"><span style="font-size: large;"><span style="line-height: 107%;">a<sup>p-1</sup>=1 mod p</span></span></span></div>
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<span style="font-family: Arial,Helvetica,sans-serif;"><span style="font-size: small;"><span style="line-height: 107%;">può servire
per dimostrare che un numero NON è primo. In pratica dice che il periodo delle
potenze con base a <span> </span>mod p è un divisore
di <span> </span>p-1 e poi si ripetono. Ad esempio le
potenze di 5,6,7… mod 11 si ripetono ogni 10 potenze. Così 5<sup>0</sup>=1, 5<sup>1</sup>=5,
5<sup>2</sup>=3, 5<sup>3</sup>=4, 5<sup>4</sup>=9, 5<sup>5</sup>=1</span></span></span></div>
<span style="font-family: Arial,Helvetica,sans-serif;"><span style="font-size: small;">
</span></span><div class="MsoNormal">
<span style="font-family: Arial,Helvetica,sans-serif;"><span style="font-size: small;"><span style="line-height: 107%;">E’
sicuramente molto utile quando si devono calcolare potenze modulo p primo.</span></span></span></div>
<span style="font-family: Arial,Helvetica,sans-serif;"><span style="font-size: small;">
</span></span><div class="MsoNormal">
<span style="font-family: Arial,Helvetica,sans-serif;"><span style="font-size: small;">Ad esempio: calcolare il resto di 4<sup>40</sup> diviso per 19.
</span></span></div>
<span style="font-family: Arial,Helvetica,sans-serif;"><span style="font-size: small;">
</span></span><div class="MsoNormal">
<span style="font-family: Arial,Helvetica,sans-serif;"><span style="font-size: small;">4 è coprimo a 19 quindi 4<sup>18</sup>=1 . Scrivo 4<sup>40</sup>=
(4<sup>18</sup>)<sup>2</sup>4<sup>4</sup>=4<sup>4</sup>=(16)²=(-3)²=9</span></span><span style="font-size: 12.0pt; line-height: 107%;"></span></div>
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<![endif]-->profsergiolamalfahttp://www.blogger.com/profile/17034912698828182534noreply@blogger.com0tag:blogger.com,1999:blog-1637330636993085995.post-29757870882802583562016-09-02T05:59:00.000-07:002016-09-02T11:45:42.334-07:00POTENZE CICLICHE MODULO n: cifra delle unità o ultime cifreLe potenze modulo n sono clicliche con periodo d<br />
ESEMPIO: <b>in modulo 10</b> ( cifra delle unità)<br />
potenze di 2<br />
2^0=1<br />
2^1=2<br />
2^2=4<br />
2^3=8<br />
2^4=6<br />
2^5=2<br />
2^6=4<br />
SI RIPETONO OGNI 4 termini<br />
quindi possiamo calcolare la cifra delle unità di qualunque potenza di 2<br />
2^134= ? 134=2 mod 4 allora 2^134=2^2=4<br />
L' unità della potenza è 4<br />
<br />
ALTRE POTENZE MOD 10<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEij668fczcL9uLJElgvT1lRGJbLR8Q_yWMJk3b0CIsndJpNxzvFh10zyYjkxvzp0b2-qmabOT4hMYlg0QaYM6eZWAm3I7n8R71z9xzm9t0t0PVfEG_exwdF0Tya4MBUls1lmINNHFA1uQ/s1600/potenze+mod10.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEij668fczcL9uLJElgvT1lRGJbLR8Q_yWMJk3b0CIsndJpNxzvFh10zyYjkxvzp0b2-qmabOT4hMYlg0QaYM6eZWAm3I7n8R71z9xzm9t0t0PVfEG_exwdF0Tya4MBUls1lmINNHFA1uQ/s1600/potenze+mod10.JPG" /></a></div>
le cifre che si ripetono sono :<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiKULbSyC0tyacDHHnTb3-TDzN1s5pIK594458KnTIW_RhZc_AllZ1_La2kCQ1ejDJQaf_SXuFrCmK4HePdHBuMUHxn-aiKfPD8eIO2OsbCynHSWx7Xs-w6cJR9U_2gzbhjZ2sZg_a9Ag/s1600/cifre+unit%25C3%25A0.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="154" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiKULbSyC0tyacDHHnTb3-TDzN1s5pIK594458KnTIW_RhZc_AllZ1_La2kCQ1ejDJQaf_SXuFrCmK4HePdHBuMUHxn-aiKfPD8eIO2OsbCynHSWx7Xs-w6cJR9U_2gzbhjZ2sZg_a9Ag/s320/cifre+unit%25C3%25A0.JPG" width="320" /></a></div>
e i periodi sono :<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi5VJODWigUeTKS-ZUja5WalTRzYDOprpH-b8HI2PGBZcLc-1K1OGig7KALaNoG2sxVPieEwa117ak-WQ7o1RP26sjGi24QkrUyxh3nQgTEL_ODfukCZmSUqBskxlIfMtyE4hxbWBmYhg/s1600/periodi+unit%25C3%25A0.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi5VJODWigUeTKS-ZUja5WalTRzYDOprpH-b8HI2PGBZcLc-1K1OGig7KALaNoG2sxVPieEwa117ak-WQ7o1RP26sjGi24QkrUyxh3nQgTEL_ODfukCZmSUqBskxlIfMtyE4hxbWBmYhg/s1600/periodi+unit%25C3%25A0.JPG" /></a></div>
<br />
Per ottenere le ultime due cifre di una potenza bisogna scriverlo mod 100<br />
<br />profsergiolamalfahttp://www.blogger.com/profile/17034912698828182534noreply@blogger.com0tag:blogger.com,1999:blog-1637330636993085995.post-57490946552754865242016-08-31T11:51:00.000-07:002016-08-31T11:51:34.561-07:00Risolutore di equazioni ON LINE<a href="http://it.numberempire.com/equationsolver.php" target="_blank"><span style="font-family: Arial,Helvetica,sans-serif;">RISOLVI EQUAZIONI ALGEBRICHE E TRASCENDENTI ON LINE</span></a>profsergiolamalfahttp://www.blogger.com/profile/17034912698828182534noreply@blogger.com0tag:blogger.com,1999:blog-1637330636993085995.post-48176529828935156872016-08-30T07:01:00.000-07:002016-08-30T07:01:07.497-07:00ALGORITMO DI EUCLIDE: CALCOLO MCD<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhdVxoxiJDG6rhW5uglU-vphysrhyphenhyphen5znLdzASkVaC1t0OaH_agsBMvkX8a4Srs8cgx2t14G7iNXQUQmvWCMgzQ-msfTc0c_-lxZTwXxCSkH6LNOPLZUXqynqEkL2tOjRTRzGG46FE6LHw/s1600/algoritmo+di+Euclide.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="474" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhdVxoxiJDG6rhW5uglU-vphysrhyphenhyphen5znLdzASkVaC1t0OaH_agsBMvkX8a4Srs8cgx2t14G7iNXQUQmvWCMgzQ-msfTc0c_-lxZTwXxCSkH6LNOPLZUXqynqEkL2tOjRTRzGG46FE6LHw/s640/algoritmo+di+Euclide.JPG" width="640" /></a></div>
<div style="text-align: center;">
<br /></div>
profsergiolamalfahttp://www.blogger.com/profile/17034912698828182534noreply@blogger.com1tag:blogger.com,1999:blog-1637330636993085995.post-19665598832447660422016-08-26T10:01:00.000-07:002016-08-26T10:01:35.052-07:00CRITERI DI DIVISIBILITA'Per dimostrare i criteri di divisibilità si unano le <a href="http://allenamentidimatematica.blogspot.com/2016/08/aritmetica-modulare.html" target="_blank">congruenze mod n</a> .<br />
<br />
<b>DIVISIBILITA' PER 3</b><br />
la somma delle cifre è 0 oppure è multiplo di 3<br />
Infatti:<br />
10=1 mod 3<br />
100=10x10=1x1=1 mod3<br />
1000= 1 mod 3 per lo stesso motivo<br />
:<br />
<div style="direction: ltr; line-height: 80%; margin-bottom: 0pt; margin-left: 0.38in; margin-top: 7.68pt; text-align: left; text-indent: -0.38in; unicode-bidi: embed; vertical-align: baseline;">
<span style="font-size: small;"><span style="color: red; font-family: "Arial Unicode MS";">[10]</span><span style="color: red; font-family: "Arial Unicode MS"; vertical-align: super;">n</span><span style="color: red; font-family: "Arial Unicode MS";">=[1] </span><span style="color: red; font-family: "Arial Unicode MS";">mod</span><span style="color: red; font-family: "Arial Unicode MS";"> 3</span><span style="color: black; font-family: "Arial Unicode MS";"><span> </span>per ogni n</span></span></div>
<div style="direction: ltr; line-height: 80%; margin-bottom: 0pt; margin-left: 0.38in; margin-top: 7.68pt; text-align: left; text-indent: -0.38in; unicode-bidi: embed; vertical-align: baseline;">
<br /></div>
un numero ABC = 100A+10B+C=A+B+C mod 3<br />
<br />
Esempio : 1455=1000+4x100+5x10+5= 1+4+5+5=15 ok è multiplo di 3<br />
<br />
<br />
<b>DIVISIBILITA' PER 4</b><br />
<br />
<i>un numero è divisibile per 4 se lo è il numero formato dalle sue due ultime cifre</i><b> </b><br />
ad esempio 5734 è div per 4 se lo è 34 Allora non è div. <b><br /></b><br />
<b>consideriamo le potenze di 10 mod 2 </b><br />
<b>10=2</b><br />
<b>
</b><br />
<div style="direction: ltr; line-height: 80%; margin-bottom: 0pt; margin-left: 0.38in; margin-top: 5.76pt; text-align: left; text-indent: -0.38in; unicode-bidi: embed; vertical-align: baseline;">
<span style="font-size: small;"><span style="color: black; font-family: "Arial Unicode MS";">[10</span><span style="color: black; font-family: "Arial Unicode MS"; vertical-align: super;">0</span><span style="color: black; font-family: "Arial Unicode MS";">]=[1]</span></span></div>
<span style="font-size: small;">
</span><div style="direction: ltr; line-height: 80%; margin-bottom: 0pt; margin-left: 0.38in; margin-top: 5.76pt; text-align: left; text-indent: -0.38in; unicode-bidi: embed; vertical-align: baseline;">
<span style="font-size: small;"><span style="color: black; font-family: "Arial Unicode MS";">[10]=[2]</span></span></div>
<span style="font-size: small;">
</span><div style="direction: ltr; line-height: 80%; margin-bottom: 0pt; margin-left: 0.38in; margin-top: 5.76pt; text-align: left; text-indent: -0.38in; unicode-bidi: embed; vertical-align: baseline;">
<span style="font-size: small;"><span style="color: black; font-family: "Arial Unicode MS";">[10²]=[0]</span></span></div>
<span style="font-size: small;">
</span><div style="direction: ltr; line-height: 80%; margin-bottom: 0pt; margin-left: 0.38in; margin-top: 5.76pt; text-align: left; text-indent: -0.38in; unicode-bidi: embed; vertical-align: baseline;">
<span style="font-size: small;"><span style="color: black; font-family: "Arial Unicode MS";">[10³]=[8]=[0] </span><span style="color: black; font-family: "Arial Unicode MS"; font-style: italic;">e poi è sempre 0</span><span style="color: black; font-family: "Arial Unicode MS";"><span> </span></span></span></div>
<span style="font-size: small;">
</span><div style="direction: ltr; line-height: 80%; margin-bottom: 0pt; margin-left: 0.38in; margin-top: 5.76pt; text-align: left; text-indent: -0.38in; unicode-bidi: embed; vertical-align: baseline;">
<span style="font-size: small;"><span style="color: black; font-family: "Arial Unicode MS";">Quindi 1234 : 4 ha resto 2 ossia
[1234]=[34]=[2] </span><span style="color: black; font-family: "Arial Unicode MS";">mod</span><span style="color: black; font-family: "Arial Unicode MS";">
4</span></span></div>
<br />
<b>DIVISIBILITA' PER 7</b><br />
<b>
</b><br />
<div style="direction: ltr; margin-bottom: 0pt; margin-top: 0pt; text-align: left; unicode-bidi: embed; vertical-align: baseline;">
<i><span style="font-size: small;"><span style="color: black; font-family: "Arial Unicode MS";">Un
intero n è divisibile per 7, cioè n </span><span style="color: black; font-family: "Arial Unicode MS";">mod</span><span style="color: black; font-family: "Arial Unicode MS";"> 7 = 0, se e solo se lo è il numero
ottenuto</span></span></i></div>
<i><span style="font-size: small;">
</span></i><div style="direction: ltr; margin-bottom: 0pt; margin-top: 0pt; text-align: left; unicode-bidi: embed; vertical-align: baseline;">
<i><span style="font-size: small;"><span style="color: black; font-family: "Arial Unicode MS";">dall’intero
iniziale senza la cifra delle unità sottraendo due volte la cifra delle unità.</span></span></i></div>
<div style="direction: ltr; margin-bottom: 0pt; margin-top: 0pt; text-align: left; unicode-bidi: embed; vertical-align: baseline;">
<i><span style="font-size: small;"><span style="color: black; font-family: "Arial Unicode MS";">
</span></span></i></div>
<div style="direction: ltr; margin-bottom: 0pt; margin-top: 0pt; text-align: left; unicode-bidi: embed; vertical-align: baseline;">
<br /></div>
<div style="direction: ltr; margin-bottom: 0pt; margin-top: 0pt; text-align: left; unicode-bidi: embed; vertical-align: baseline;">
<span style="font-size: x-small;"><span style="font-family: Arial,Helvetica,sans-serif;"><span style="color: black;">Per esempio, 3367882 </span><span style="color: black;">mod</span><span style="color: black;"> 7 = 0 in </span></span></span></div>
<span style="font-size: x-small;"><span style="font-family: Arial,Helvetica,sans-serif;">
</span></span><div style="direction: ltr; margin-bottom: 0pt; margin-top: 0pt; text-align: left; unicode-bidi: embed; vertical-align: baseline;">
<span style="font-size: x-small;"><span style="font-family: Arial,Helvetica,sans-serif;"><span style="color: black;">quanto 336788 − 2 · 2 = 336784, e
ripetendo</span></span></span></div>
<span style="font-size: x-small;"><span style="font-family: Arial,Helvetica,sans-serif;">
</span></span><div style="direction: ltr; margin-bottom: 0pt; margin-top: 0pt; text-align: left; unicode-bidi: embed; vertical-align: baseline;">
<span style="font-size: x-small;"><span style="font-family: Arial,Helvetica,sans-serif;"><span style="color: black;">33678−2 · 4 = 33670, 3367−2 · 0 =
3367, 336−2 · 7 = 322, 32−2 · 2 = 28; ma dato che l’ultimo è divisibile per 7,
lo sono anche tutti gli altri e in particolare il numero di partenza</span></span></span></div>
<div style="direction: ltr; margin-bottom: 0pt; margin-top: 0pt; text-align: left; unicode-bidi: embed; vertical-align: baseline;">
<br /></div>
<br />
<b>DIVISIBILITA' PER 8 e in generale per 2^n POTENZE DI 2</b><br />
<br />
<i>un numero è divisibile per 8=2^3 se lo è il numero formato dalle sue ultime TRE cifre</i><br />
<i>un numero è divisibile per 8=2^n se lo è il numero formato dalle sue ultime n cifre</i><i> </i><b> </b><br />
<div style="direction: ltr; margin-bottom: 0pt; margin-top: 0pt; text-align: left; unicode-bidi: embed; vertical-align: baseline;">
<span style="font-size: small;"><span style="color: black; font-family: "Arial Unicode MS";">Esempio:
</span></span></div>
<span style="font-size: small;">
</span><div style="direction: ltr; margin-bottom: 0pt; margin-top: 0pt; text-align: left; unicode-bidi: embed; vertical-align: baseline;">
<span style="font-size: small;"><span style="color: black; font-family: "Arial Unicode MS";">[123156]=[156]<span> </span></span><span style="color: black; font-family: "Arial Unicode MS";">mod</span><span style="color: black; font-family: "Arial Unicode MS";"> 8= </span><span style="color: black; font-family: "Arial Unicode MS";">mod</span><span style="color: black; font-family: "Arial Unicode MS";"> 2³</span></span></div>
<span style="font-size: small;">
</span><div style="direction: ltr; margin-bottom: 0pt; margin-top: 0pt; text-align: left; unicode-bidi: embed; vertical-align: baseline;">
<span style="font-size: small;"><span style="color: black; font-family: "Arial Unicode MS";">[123
564]=[64] </span><span style="color: black; font-family: "Arial Unicode MS";">mod</span><span style="color: black; font-family: "Arial Unicode MS";">
4</span></span></div>
<span style="font-size: small;">
</span><div style="direction: ltr; margin-bottom: 0pt; margin-top: 0pt; text-align: left; unicode-bidi: embed; vertical-align: baseline;">
<span style="font-size: small;"><span style="color: black; font-family: "Arial Unicode MS";">[123
675]=[3675] </span><span style="color: black; font-family: "Arial Unicode MS";">mod</span><span style="color: black; font-family: "Arial Unicode MS";">
16</span></span></div>
<br />
<b>DIVISIBILITA' PER 9</b><br />
<b>
</b><br />
<div style="direction: ltr; line-height: 80%; margin-bottom: 0pt; margin-left: 0.38in; margin-top: 7.68pt; text-align: left; text-indent: -0.38in; unicode-bidi: embed; vertical-align: baseline;">
<span style="font-size: small;"><span style="color: black; font-family: "Arial Unicode MS";"><i>un numero è divisibile per 9 se e solo se la somma delle cifre è divisibile per 9</i> </span></span></div>
<div style="direction: ltr; line-height: 80%; margin-bottom: 0pt; margin-left: 0.38in; margin-top: 7.68pt; text-align: left; text-indent: -0.38in; unicode-bidi: embed; vertical-align: baseline;">
<span style="font-size: small;"><span style="color: black; font-family: "Arial Unicode MS";">Infatti:</span></span></div>
<span style="font-size: small;">
</span><div style="direction: ltr; line-height: 80%; margin-bottom: 0pt; margin-left: 0.38in; margin-top: 7.68pt; text-align: left; text-indent: -0.38in; unicode-bidi: embed; vertical-align: baseline;">
<span style="font-size: small;"><span style="color: red; font-family: "Arial Unicode MS";">[10]</span><span style="color: red; font-family: "Arial Unicode MS"; vertical-align: super;">n</span><span style="color: red; font-family: "Arial Unicode MS";">=[1] </span><span style="color: red; font-family: "Arial Unicode MS";">mod</span><span style="color: red; font-family: "Arial Unicode MS";"> 9</span><span style="color: black; font-family: "Arial Unicode MS";"><span> </span>per ogni n</span></span></div>
<span style="font-size: small;">
</span><div style="direction: ltr; line-height: 80%; margin-bottom: 0pt; margin-left: 0.38in; margin-top: 6.72pt; text-align: left; text-indent: -0.38in; unicode-bidi: embed; vertical-align: baseline;">
<span style="font-size: small;"><span style="color: black; font-family: "Arial Unicode MS";">1234 è divisibile per 9?</span></span></div>
<span style="font-size: small;">
</span><div style="direction: ltr; line-height: 80%; margin-bottom: 0pt; margin-left: 0.38in; margin-top: 6.72pt; text-align: left; text-indent: -0.38in; unicode-bidi: embed; vertical-align: baseline;">
<span style="font-size: small;"><span style="color: black; font-family: "Arial Unicode MS";">solo se lo è LA SOMMA DELLE SUE
CIFRE 1+2+3+4 perché:</span></span></div>
<span style="font-size: small;">
</span><div style="direction: ltr; line-height: 80%; margin-bottom: 0pt; margin-left: 0.38in; margin-top: 6.72pt; text-align: left; text-indent: -0.38in; unicode-bidi: embed; vertical-align: baseline;">
<span style="font-size: small;"><span style="color: black; font-family: "Arial Unicode MS";">[1234]=[1x10³+2x10²+3x10+4]=</span></span></div>
<span style="font-size: small;">
</span><div style="direction: ltr; line-height: 80%; margin-bottom: 0pt; margin-left: 0.38in; margin-top: 6.72pt; text-align: left; text-indent: -0.38in; unicode-bidi: embed; vertical-align: baseline;">
<span style="font-size: small;"><span style="color: black; font-family: "Arial Unicode MS";">=[1][10]³+[2][10]²+[3][10]+[4]=[1+2+3+4]=[1]</span></span></div>
<span style="font-size: small;">
<span style="color: black; font-family: "Arial Unicode MS";">Quindi
in particolare 1234:9 ha resto 1</span></span><br />
<b>DIVISIBILITA' PER 11</b><br />
<b>
</b><br />
<div style="direction: ltr; line-height: 80%; margin-bottom: 0pt; margin-left: 0.38in; margin-top: 7.68pt; text-align: left; text-indent: -0.38in; unicode-bidi: embed; vertical-align: baseline;">
<span style="font-size: small;"><span style="color: black; font-family: "Arial Unicode MS";"><i><span style="font-family: Arial,Helvetica,sans-serif;"><span style="color: black;">un
numero è divisibile per 11 se e solamente se la somma delle sue cifre a segni
alterni è divisibile per 11. </span></span></i><span style="color: black; font-family: "Arial Unicode MS"; font-size: 20.0pt; font-style: italic; font-weight: bold; language: it; mso-ascii-font-family: "Arial Unicode MS"; mso-bidi-font-family: "Arial Unicode MS"; mso-color-index: 1; mso-fareast-font-family: "Arial Unicode MS"; mso-font-kerning: 12.0pt; mso-style-textfill-fill-alpha: 100.0%; mso-style-textfill-fill-color: black; mso-style-textfill-fill-themecolor: text1; mso-style-textfill-type: solid;"> </span></span></span></div>
<div style="direction: ltr; line-height: 80%; margin-bottom: 0pt; margin-left: 0.38in; margin-top: 7.68pt; text-align: left; text-indent: -0.38in; unicode-bidi: embed; vertical-align: baseline;">
<span style="font-size: small;"><span style="color: black; font-family: "Arial Unicode MS";">Infatti:</span></span></div>
<span style="font-size: small;"></span>
<div style="direction: ltr; margin-bottom: 0pt; margin-top: 0pt; text-align: left; unicode-bidi: embed; vertical-align: baseline;">
<span style="font-size: small;"><span style="color: black; font-family: "Arial Unicode MS";">10</span><span style="color: black; font-family: "Arial Unicode MS"; vertical-align: super;">0</span><span style="color: black; font-family: "Arial Unicode MS";"> = 1 è congruo a 1 modulo 11. La
potenza</span></span></div>
<span style="font-size: small;">
</span><div style="direction: ltr; margin-bottom: 0pt; margin-top: 0pt; text-align: left; unicode-bidi: embed; vertical-align: baseline;">
<span style="font-size: small;"><span style="color: black; font-family: "Arial Unicode MS";">10</span><span style="color: black; font-family: "Arial Unicode MS"; vertical-align: super;">1</span><span style="color: black; font-family: "Arial Unicode MS";"> = 10 è invece congrua a −1 modulo
11. Ne segue che</span></span></div>
<span style="font-size: small;">
</span><div style="direction: ltr; margin-bottom: 0pt; margin-top: 0pt; text-align: left; unicode-bidi: embed; vertical-align: baseline;">
<span style="font-size: small;"><span style="color: black; font-family: "Arial Unicode MS";">10</span><span style="color: black; font-family: "Arial Unicode MS"; vertical-align: super;">2</span><span style="color: black; font-family: "Arial Unicode MS";"> = 10 · 10<span> </span>(−1) · (−1) = 1 (</span><span style="color: black; font-family: "Arial Unicode MS";">mod</span><span style="color: black; font-family: "Arial Unicode MS";">
11),</span></span></div>
<span style="font-size: small;">
</span><div style="direction: ltr; margin-bottom: 0pt; margin-top: 0pt; text-align: left; unicode-bidi: embed; vertical-align: baseline;">
<span style="font-size: small;"><span style="color: black; font-family: "Arial Unicode MS";">e
quindi</span></span></div>
<span style="font-size: small;">
</span><div style="direction: ltr; margin-bottom: 0pt; margin-top: 0pt; text-align: left; unicode-bidi: embed; vertical-align: baseline;">
<span style="font-size: small;"><span style="color: black; font-family: "Arial Unicode MS";">10</span><span style="color: black; font-family: "Arial Unicode MS"; vertical-align: super;">3</span><span style="color: black; font-family: "Arial Unicode MS";"> = 10 · 10</span><span style="color: black; font-family: "Arial Unicode MS"; vertical-align: super;">2</span><span style="color: black; font-family: "Arial Unicode MS";"><span>
</span>(−1) · 1 = −1 (</span><span style="color: black; font-family: "Arial Unicode MS";">mod</span><span style="color: black; font-family: "Arial Unicode MS";">
11).</span></span></div>
<span style="font-size: small;">
</span><div style="direction: ltr; margin-bottom: 0pt; margin-top: 0pt; text-align: left; unicode-bidi: embed; vertical-align: baseline;">
<span style="font-size: small;"><span style="color: black; font-family: "Arial Unicode MS";">Allora
:</span></span></div>
<div style="direction: ltr; margin-bottom: 0pt; margin-top: 0pt; text-align: left; unicode-bidi: embed; vertical-align: baseline;">
<span style="font-size: small;"><span style="color: black; font-family: "Arial Unicode MS";">ESEMPIO </span></span></div>
<span style="font-size: small;">
</span><div style="direction: ltr; margin-bottom: 0pt; margin-top: 0pt; text-align: left; unicode-bidi: embed; vertical-align: baseline;">
<span style="font-size: small;"><span style="color: black; font-family: "Arial Unicode MS";">6131829
= 9·10</span><span style="color: black; font-family: "Arial Unicode MS"; vertical-align: super;">0</span><span style="color: black; font-family: "Arial Unicode MS";">+2·10</span><span style="color: black; font-family: "Arial Unicode MS"; vertical-align: super;">1</span><span style="color: black; font-family: "Arial Unicode MS";">+8·10</span><span style="color: black; font-family: "Arial Unicode MS"; vertical-align: super;">2</span><span style="color: black; font-family: "Arial Unicode MS";">+1·10</span><span style="color: black; font-family: "Arial Unicode MS"; vertical-align: super;">3</span><span style="color: black; font-family: "Arial Unicode MS";">+3·10</span><span style="color: black; font-family: "Arial Unicode MS"; vertical-align: super;">4</span><span style="color: black; font-family: "Arial Unicode MS";">+1·10</span><span style="color: black; font-family: "Arial Unicode MS"; vertical-align: super;">5</span><span style="color: black; font-family: "Arial Unicode MS";">+6·10</span><span style="color: black; font-family: "Arial Unicode MS"; vertical-align: super;">6</span><span style="color: black; font-family: "Arial Unicode MS";"><span> </span>=9−2+8−1+3−1+6 = 22<span> </span>= 0 (</span><span style="color: black; font-family: "Arial Unicode MS";">mod</span><span style="color: black; font-family: "Arial Unicode MS";"> 11).</span></span></div>
profsergiolamalfahttp://www.blogger.com/profile/17034912698828182534noreply@blogger.com0tag:blogger.com,1999:blog-1637330636993085995.post-43409836612153171772016-08-19T08:48:00.001-07:002016-08-26T06:46:19.428-07:00ARITMETICA MODULARE<span style="font-family: "arial" , "helvetica" , sans-serif;"><span style="font-size: small;"><br /></span></span>
<span style="font-family: "arial" , "helvetica" , sans-serif;"><span style="font-size: small;">Consideriamo i numeri interi Z. Si dice che due numeri interi a,b sono congrui modulo n e si scrive</span></span><br />
<b><span style="font-family: "arial" , "helvetica" , sans-serif;"><span style="font-size: small;">a=b mod n</span></span></b><br />
<span style="font-family: "arial" , "helvetica" , sans-serif;"><span style="font-size: small;">se <span style="color: red;">a-b è un multiplo di n</span></span></span><br />
<br />
<span style="font-family: "arial" , "helvetica" , sans-serif;"><span style="font-size: small;">Ad esempio : 4=1 mod 3 e ande 7=1 mod 3</span></span><br />
<span style="font-family: "arial" , "helvetica" , sans-serif;"><span style="font-size: small;">se a=b mod n allora a e b hanno lo stesso resto se vengono divisi per n</span></span><br />
<span style="font-family: "arial" , "helvetica" , sans-serif;"><span style="font-size: small;">Ad esempio i numeri che sono uguali modulo 3 sono 1 - 4 - 7 - 10</span></span><br />
<span style="font-family: "arial" , "helvetica" , sans-serif;"><span style="font-size: small;">Si pone [1] mod3 per indicare l'insieme di tutti i numeri congrui a 1 mod3</span></span><br />
<br />
<span style="font-family: "arial" , "helvetica" , sans-serif;"><span style="font-size: small;">In generale <span style="color: red;">[a] mod n</span> indica l'insieme di tutti i numeri interi che divisi per n hanno resto a e si dice <b>CLASSE di a modulo n</b>.</span></span><br />
<span style="font-family: "arial" , "helvetica" , sans-serif;"><span style="font-size: small;"><br /></span></span>
<span style="font-family: "arial" , "helvetica" , sans-serif;"><span style="font-size: small;">Fissato n le sue classi sono n : [0], [1], [2]....[n-1]</span></span><br />
<span style="font-family: "arial" , "helvetica" , sans-serif;"><span style="font-size: small;">La congruenza è una relazione di equivalenza e l'unione delle classi (che sono disgiunte) è tutto l'insieme Z degli interi. </span></span><br />
<br />
<span style="font-family: "arial" , "helvetica" , sans-serif;"><span style="font-size: small;">Nell'insieme delle classi mod n si può definire l'operazione di somma e prodotto. Risulta:</span></span><br />
<span style="font-family: "arial" , "helvetica" , sans-serif; font-size: large;"><span style="font-size: x-small;">[a]+[b]=[a+b]</span></span><br />
<span style="font-family: "arial" , "helvetica" , sans-serif; font-size: large;"><span style="font-size: x-small;">[a][b]=[ab] </span></span><br />
<br />
<span style="font-size: small;"><span style="font-family: "arial" , "helvetica" , sans-serif;">APPLICAZIONI:</span></span><br />
<span style="font-family: "arial" , "helvetica" , sans-serif;"><span style="font-size: small;"><span style="font-family: "arial" , "helvetica" , sans-serif;">Se siamo nell'aritmetica <b>modulo 7</b></span></span></span><br />
<span style="font-family: "arial" , "helvetica" , sans-serif;"><span style="font-size: small;"><span style="font-family: "arial" , "helvetica" , sans-serif;">[10]=[3]</span></span></span><br />
<span style="font-family: "arial" , "helvetica" , sans-serif;"><span style="font-size: small;"><span style="font-family: "arial" , "helvetica" , sans-serif;">[100]=[10][10] =[9]=<span style="font-family: "arial" , "helvetica" , sans-serif;">[<span style="font-family: "arial" , "helvetica" , sans-serif;">2]</span></span></span></span></span><br />
<span style="font-family: "arial" , "helvetica" , sans-serif;"><span style="font-size: small;"><span style="font-family: "arial" , "helvetica" , sans-serif;"><span style="font-family: "arial" , "helvetica" , sans-serif;"><span style="font-family: "arial" , "helvetica" , sans-serif;">[1000]=[10][100]=[3][2]=[6]</span></span></span></span></span><br />
<span style="font-family: "arial" , "helvetica" , sans-serif;"><span style="font-size: small;"><span style="font-family: "arial" , "helvetica" , sans-serif;"><span style="font-family: "arial" , "helvetica" , sans-serif;"><span style="font-family: "arial" , "helvetica" , sans-serif;"><span style="font-family: "arial" , "helvetica" , sans-serif;"><span style="font-family: "arial" , "helvetica" , sans-serif;">Quanto vale il resto di 1546 per 7? E' divisibile per 7?</span> </span></span></span></span></span></span><br />
<span style="font-family: "arial" , "helvetica" , sans-serif;"><span style="font-size: small;"><span style="font-family: "arial" , "helvetica" , sans-serif;"><span style="font-family: "arial" , "helvetica" , sans-serif;"><span style="font-family: "arial" , "helvetica" , sans-serif;"><span style="font-family: "arial" , "helvetica" , sans-serif;">allora [1546]=[1000]+5[100<span style="font-family: "arial" , "helvetica" , sans-serif;">]+4[10]+6=[6]+<span style="font-family: "arial" , "helvetica" , sans-serif;">5[2]+4[3]+6=[6]+[3]+[5]+[6]=[20]=[6]</span></span></span></span></span></span></span></span><br />
<span style="font-family: "arial" , "helvetica" , sans-serif;"><span style="font-family: "arial" , "helvetica" , sans-serif;"><span style="font-size: small;"><span style="font-family: "arial" , "helvetica" , sans-serif;"><span style="font-family: "arial" , "helvetica" , sans-serif;"><span style="font-family: "arial" , "helvetica" , sans-serif;"><span style="font-family: "arial" , "helvetica" , sans-serif;"><span style="font-family: "arial" , "helvetica" , sans-serif;"><span style="font-family: "arial" , "helvetica" , sans-serif;">è 6 </span></span></span> </span></span></span></span></span></span><br />
<span style="font-family: "arial" , "helvetica" , sans-serif;">quindi 1540 è divisibile per 7!</span><br />
<br />
<span style="font-family: "arial" , "helvetica" , sans-serif;"><span style="font-size: small;">L'aritme<span style="font-family: "arial" , "helvetica" , sans-serif;">tica dell'orologio è modulo 12<span style="font-family: "arial" , "helvetica" , sans-serif;"> <span style="font-family: "arial" , "helvetica" , sans-serif;">se consideriamo la lancetta delle ore.</span></span></span></span></span><br />
<span style="font-family: "arial" , "helvetica" , sans-serif;"><span style="font-size: small;"><span style="font-family: "arial" , "helvetica" , sans-serif;">[1]=[13<span style="font-family: "arial" , "helvetica" , sans-serif;">] mod 12</span></span></span></span><br />
<span style="font-family: "arial" , "helvetica" , sans-serif;"><span style="font-size: small;"><span style="font-family: "arial" , "helvetica" , sans-serif;"><span style="font-family: "arial" , "helvetica" , sans-serif;"> <span style="font-family: "arial" , "helvetica" , sans-serif;">S</span>e sono le 14 e passano 13 ore allora la lancetta segnerà le <span style="font-family: "arial" , "helvetica" , sans-serif;">3: </span>[14]+[<span style="font-family: "arial" , "helvetica" , sans-serif;">13]=[27]=[3] </span> </span> </span></span></span><br />
<br />
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgJu4ZKxYEW2PsG03yJ9Oeb_MGL_soGyWA3hmyUb1unIY8ExEtieyPAGwUou6p-_AlrUkbNxJWI82-ke2nkQzp3kQ8Uv3zIHUcZXiyCLNc_lLhX7GB8HO4N9sF1px_p3irpaWgM62Spuw/s1600/modgiorni.JPG" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="220" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgJu4ZKxYEW2PsG03yJ9Oeb_MGL_soGyWA3hmyUb1unIY8ExEtieyPAGwUou6p-_AlrUkbNxJWI82-ke2nkQzp3kQ8Uv3zIHUcZXiyCLNc_lLhX7GB8HO4N9sF1px_p3irpaWgM62Spuw/s640/modgiorni.JPG" width="640" /></a><span style="font-family: "arial" , "helvetica" , sans-serif;"><span style="font-size: small;"><span style="font-family: "arial" , "helvetica" , sans-serif;"> </span></span></span><br />
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</span></span></span><span style="font-family: "arial" , "helvetica" , sans-serif;"><span style="font-size: small;"> </span></span>profsergiolamalfahttp://www.blogger.com/profile/17034912698828182534noreply@blogger.com0tag:blogger.com,1999:blog-1637330636993085995.post-18308411523992020452016-08-19T07:52:00.003-07:002016-08-19T08:36:13.891-07:00RETTA DI EULERO<span style="font-family: "arial" , "helvetica" , sans-serif;"><span style="font-size: large;">TEOREMA:</span></span><span style="font-family: "arial" , "helvetica" , sans-serif;"><span style="font-size: large;">L’ortocentro
(H), il baricentro (G) e il circocentro
(O) di un triangolo qualsiasi sono sempre allineati.</span></span><br />
<span style="font-family: "arial" , "helvetica" , sans-serif;"><span style="font-size: large;"><b> </b> </span></span><br />
<span style="font-family: "arial" , "helvetica" , sans-serif;"><span style="font-size: large;">Si dice retta di Eulero di un triangolo
la retta che unisce il circocentro <b>O</b><a href="https://www.blogger.com/null">,
il baricentro <b>G</b> e l’ortocentro <b>H</b></a>
del triangolo </span></span><br />
<br />
<span style="font-family: "arial" , "helvetica" , sans-serif;"><span style="font-size: large;">PROPRIETA': Il baricentro G divide il segmento OH in due parti una doppia dell'altra: HG=2OG </span></span><br />
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profsergiolamalfahttp://www.blogger.com/profile/17034912698828182534noreply@blogger.com0tag:blogger.com,1999:blog-1637330636993085995.post-28018450608968747292016-08-19T05:46:00.003-07:002016-08-23T11:40:50.369-07:00TEOREMA DI CEVA<div class="separator" style="clear: both; text-align: center;">
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<i><span style="font-family: Times,"Times New Roman",serif;">SPIEGAZIONE: Quindi la dimostrazione si basa sul fatto che due triangoli che hanno la stessa altezza hanno le aree proporzionali alle basi. Quindi le lunghezze presenti nei rapporti sono le basi di triangoli con la stessa altezza e queste aree si semplificano in croce.</span></i></div>
profsergiolamalfahttp://www.blogger.com/profile/17034912698828182534noreply@blogger.com0tag:blogger.com,1999:blog-1637330636993085995.post-60793818116463520752016-08-18T09:04:00.003-07:002016-08-19T05:47:35.502-07:00TEOREMA DI TOLOMEO<div class="separator" style="clear: both; text-align: center;">
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profsergiolamalfahttp://www.blogger.com/profile/17034912698828182534noreply@blogger.com0tag:blogger.com,1999:blog-1637330636993085995.post-47355299716316311692016-08-18T08:04:00.001-07:002016-08-20T09:09:05.860-07:00TUTTO SULLE MEDIANE e BARICENTRO<div class="separator" style="clear: both; text-align: center;">
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<span style="font-family: "arial" , "helvetica" , sans-serif;"><span style="font-size: large;">Da questa si possono ricavare le lunghezze delle mediane date le lunghezze dei lati si usa:</span></span><br />
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<span style="font-family: "arial" , "helvetica" , sans-serif;"><span style="font-size: large;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhXigNbD_w7VwxMM2hcks73-7XIFjDU7Jdo-DrMHLO_oTz-_od8McWPPAvKC5csnHF68JPZlnYoyiU-BH5h3KKOu-SFqZFjalnvKbcGPtilja1UH7yYz1KJqLfYkQc9IuQfcME2CU_Weg/s1600/LUNGHEZZE+MEDIANE+NOTI+I+LATI.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhXigNbD_w7VwxMM2hcks73-7XIFjDU7Jdo-DrMHLO_oTz-_od8McWPPAvKC5csnHF68JPZlnYoyiU-BH5h3KKOu-SFqZFjalnvKbcGPtilja1UH7yYz1KJqLfYkQc9IuQfcME2CU_Weg/s1600/LUNGHEZZE+MEDIANE+NOTI+I+LATI.JPG" /></a></span></span></div>
<span style="font-family: "arial" , "helvetica" , sans-serif;"><span style="font-size: large;">dove a,b,c sono i tre lati del triangolo</span></span></div>
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<span style="font-family: "arial" , "helvetica" , sans-serif;"><span style="font-size: large;">L'incontro delle mediane è il BARICENTRO</span></span><br />
<span style="font-family: "arial" , "helvetica" , sans-serif;"><span style="font-size: large;">Il baricentro divide ogni mediana in due parti di cui una doppia dell'altra.</span></span><br />
<span style="font-family: "arial" , "helvetica" , sans-serif;"><span style="font-size: large;">Per il baricentro vale:</span></span><br />
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<span style="font-family: "arial" , "helvetica" , sans-serif;"><span style="font-size: large;">la somma dei quadrati dei tre lati è uguale al triplo della somma dei quadrati delle distanze del baricentro dai tre vertici.</span></span><br />
<span style="font-family: "arial" , "helvetica" , sans-serif;"><span style="font-size: large;">Conseguenza della più generale</span></span><br />
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<span style="font-family: "arial" , "helvetica" , sans-serif;"><span style="font-size: large;"><span style="font-size: small;"><b>PROPRIETA'</b> : dato un punto interno P al triangolo ABC vale</span></span></span><span style="font-family: "arial" , "helvetica" , sans-serif;"><span style="font-size: large;"></span></span><br />
<span style="font-family: "arial" , "helvetica" , sans-serif;"><span style="font-size: large;"><img alt="" src="data:image/png;base64,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" /></span></span><br />
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profsergiolamalfahttp://www.blogger.com/profile/17034912698828182534noreply@blogger.com0tag:blogger.com,1999:blog-1637330636993085995.post-55537965060222017782016-08-13T10:48:00.002-07:002016-08-18T06:06:39.759-07:00QUANTI PERCORSI?<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjNwcFcxhZMPjpNYzu9qFkmNwzhWd3dCy70Aj2vkl4dD7O3ZKzLvoRjKCxmcKIEl2qvoIFLZx1FqBhDUx80lD6A9JMyBmQTPlESkozggUXqj_9fXz-JTfQ_xcYTWwnvFfTXp3m8S8xAMw/s1600/percorsi.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="296" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjNwcFcxhZMPjpNYzu9qFkmNwzhWd3dCy70Aj2vkl4dD7O3ZKzLvoRjKCxmcKIEl2qvoIFLZx1FqBhDUx80lD6A9JMyBmQTPlESkozggUXqj_9fXz-JTfQ_xcYTWwnvFfTXp3m8S8xAMw/s400/percorsi.JPG" width="400" /></a></div>
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profsergiolamalfahttp://www.blogger.com/profile/17034912698828182534noreply@blogger.com0tag:blogger.com,1999:blog-1637330636993085995.post-79625371701380220662016-08-01T07:24:00.001-07:002016-08-18T06:09:51.134-07:00PERMUTAZIONI SENZA PUNTI FISSI<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiqOOtPURB2M5lrY3u8KB64Hsc5kpjjbMQu_eetGb-hbGS-aGlkg5WqzMLQDTYvPO1kzA1rXwNrcujGpUxhN6hWyHhr9kra_lG1BFXFPcytuMk0iVnh72RAWDPoaY2EI1aK6YFHuh7jcg/s1600/permutazione+senza+punti+fissi.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiqOOtPURB2M5lrY3u8KB64Hsc5kpjjbMQu_eetGb-hbGS-aGlkg5WqzMLQDTYvPO1kzA1rXwNrcujGpUxhN6hWyHhr9kra_lG1BFXFPcytuMk0iVnh72RAWDPoaY2EI1aK6YFHuh7jcg/s1600/permutazione+senza+punti+fissi.png" /></a></div>
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profsergiolamalfahttp://www.blogger.com/profile/17034912698828182534noreply@blogger.com0tag:blogger.com,1999:blog-1637330636993085995.post-20970528527131877862016-07-10T10:12:00.000-07:002016-08-01T07:10:22.363-07:00SPERANZA MATEMATICA<span style="font-family: "times new roman" , "serif"; font-size: 14pt; line-height: 115%;"></span><span style="font-size: large;"><span style="font-family: "arial" , "helvetica" , sans-serif;">Il valore atteso (o speranza matematica) di una
<span style="text-decoration: underline;">variabile casuale discreta,</span> se la distribuzione è <span style="text-decoration: underline;">finita</span>, è un numero reale dato dalla somma dei prodotti di ogni
valore della variabile casuale per la rispettiva probabilità, cioè:</span></span><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhjMzX20Al2UX3QaCdNO7wMOxyrtSVrjnuwr2ELq8uvrYApo-JIgEgvFiY6X2iOZdgg3TVJdsDQeMIbuikBdWfL70WME9KQnTLmG3Ua2YxJzaJTgXQXMIv2YeJ4e5h_nzs6w3uynOcR5Q/s1600/valore_atteso.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="161" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhjMzX20Al2UX3QaCdNO7wMOxyrtSVrjnuwr2ELq8uvrYApo-JIgEgvFiY6X2iOZdgg3TVJdsDQeMIbuikBdWfL70WME9KQnTLmG3Ua2YxJzaJTgXQXMIv2YeJ4e5h_nzs6w3uynOcR5Q/s640/valore_atteso.jpg" width="640" /></a></div>
<span style="font-size: large;"><span style="font-family: "arial" , "helvetica" , sans-serif;"> ESEMPIO1: </span></span><br />
<span style="font-size: large;"><span style="font-family: "arial" , "helvetica" , sans-serif;">X lancio di un dado </span></span><br />
<span style="font-size: large;"><span style="font-family: "arial" , "helvetica" , sans-serif;">x valore della variabile= valore del dado</span></span><br />
<span style="font-size: large;"><span style="font-family: "arial" , "helvetica" , sans-serif;">Qual è il valore atteso nel lancio del dado?</span></span><br />
<span style="font-size: large;"><span style="font-family: "arial" , "helvetica" , sans-serif;">E(X)=1*1/6+2*1/6+3*1/6+4*1/6+5*1/6+6*1/6=1/6(1+2..+6)=</span></span><br />
<span style="font-size: large;"><span style="font-family: "arial" , "helvetica" , sans-serif;">1/6* 7*6/2=21/6=7/2=3,5</span></span><br />
<span style="font-size: large;"><span style="font-family: "arial" , "helvetica" , sans-serif;">I valori del dado si accumolano intorno a 3,5</span></span>profsergiolamalfahttp://www.blogger.com/profile/17034912698828182534noreply@blogger.com0tag:blogger.com,1999:blog-1637330636993085995.post-3890715348769272422013-09-04T07:38:00.001-07:002013-09-26T08:38:26.752-07:00SCOMPONI UN NUMERO INTERO IN NUMERI PRIMI <div class="separator" style="clear: both; text-align: center;">
<i><span style="font-size: x-large;">SCOMPOSIZIONE IN NUMERI PRIMI </span></i></div>
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<a href="http://www.softschools.com/math/factors/prime_factors/" target="_blank"><img border="0" height="230" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiTWtrorCQlrgaywIuWh_i-oSUy4w45iFIPqDzEwiMXFkO377KLjUIuWdL3q7MlXefqV3aDuCa9OEX4Rt2Mno5eG4GkjzAvKQOLmZequ8v6f_lnALeEf0Ap_FNcIJwlR0LBx-QVYOTcjA/s320/capture-20130904-163404.png" width="320" /> </a></div>
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<i>scrivi il numero e il programma ti fornisce i fattori primi</i></div>
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<i>altro programma <a href="http://web.liceomendrisio.ch/~marsan/matematica/materiale_vario/fattorizzazione.php" target="_blank">CLICCA QUI</a> </i></div>
<br />profsergiolamalfahttp://www.blogger.com/profile/17034912698828182534noreply@blogger.com2tag:blogger.com,1999:blog-1637330636993085995.post-6584673160573508582013-09-04T07:31:00.002-07:002013-09-04T07:31:34.624-07:00<div style="text-align: center;">
<i><span style="font-size: large;"><b>GIOCHI CON LE FRAZIONI</b></span></i></div>
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<a href="http://www.softschools.com/math/fractions/games/" target="_blank"><img border="0" height="250" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgOzJrI4PErkEMQHTmN4t9Jul4Jx801cfG5gMHyTkfMpy-AcxOGyLhV34pQ-U87EpxNZ64v-70u_IasT-U3AKuc7d8WrOA_RJxnzax7_c7M7Fz8nUpXAYR5T7SmvErr9-85IiBlzEqw7A/s320/capture-20130904-162523.png" width="320" /> GIOCO</a></div>
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Allena a riconoscere le frazioni</div>
profsergiolamalfahttp://www.blogger.com/profile/17034912698828182534noreply@blogger.com0tag:blogger.com,1999:blog-1637330636993085995.post-10538154690382596642013-02-19T09:07:00.000-08:002013-02-19T09:07:01.805-08:00SIMULATORE DI DADI<a href="http://www.zambros.it/just-for-fun/dadi-online-flash-ita" target="_blank">SIMULATORE CON DADI POLIEDRICI</a>profsergiolamalfahttp://www.blogger.com/profile/17034912698828182534noreply@blogger.com0tag:blogger.com,1999:blog-1637330636993085995.post-17905767681554674022012-12-17T09:32:00.001-08:002012-12-17T09:32:33.063-08:00TEOREMA DI EULERO<span style="font-size: small;">TEOREMA di EULERO : <i>Se a è coprimo con n allora a elevato a <b>Φ(n) è congruo a 1 modulo n</b></i></span><br />
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<span style="font-size: small;"><i><span style="font-size: small;">ESEMPIO: </span></i></span><br />
<span style="font-size: small;"><i><span style="font-size: small;"> </span> </i></span><br />
n=10 =2²x5² e <br /><span style="font-size: small;"><i>Φ(<span style="font-size: small;">10</span>)<span style="font-size: small;">=4</span></i></span><br />
<span style="font-size: small;"><i><span style="font-size: small;">infatti sono coprimi con 10 : 1 3 <span style="font-size: small;">7 9 </span></span></i></span><br />
<span style="font-size: small;"><i><span style="font-size: small;"><span style="font-size: small;"> allora 3 ^ 4=1 mod<span style="font-size: small;">10 e infatti è 81=1 mod<span style="font-size: small;">10</span></span> </span></span></i></span><br />
<b><span style="font-size: small;"><i>E anche 7^4=1 mod10</i></span></b><br />
<b><span style="font-size: small;"><i><span style="font-size: small;">oppure 9^4=1 mod 10</span> </i></span></b>profsergiolamalfahttp://www.blogger.com/profile/17034912698828182534noreply@blogger.com0