EXTRA CREDIT: MULTIPLICATION AS TRANSFORMATIONS AND ROTATIONSName: The operation of multipliation an !e thou"ht of a# a pro$etion %either len"thenin" or #hortenin"& of a num!er on a num!er line' For in#tane( if )e multipl* the num!er + !* ,( )e han"e it# ma"nitu-e %len"th& !* a fator of ,( )ithout han"in" it# -iretion' If )e multipl* the num!er + !* .( a"ain )e -o not han"e it# -iretion( !ut onl* it# len"th' DRAW TWO DIAGRAMS ILLUSTRATING THESE OPERATIONS ON THE TWO NUMBER LINES BELOW.
If )e om!ine the#e t)o operation#( then the net re#ult i# that there i# NO C/AN0E' For in#tane( if )e INCREASE the ma"nitu-e !* a fator f ator of ,( an- then a fator of .( or !* a fator of .( an- then a fator of ,( the re#ult i# + in !oth a#e#' The or-er -oe# not matter' Thi# illu#trate# the ommutati1e propert* of multipliation( a# )ell a# the propert* that the in1er#e operation of multipliation i# multipliation !* the reiproal of a num!er %+ -i1i-e- !* the num!er&' A--itionall*( if )e tr* an- 2thin3 !a3)ar-#4 to ome up )ith an operation that i# i# the oppo#ite of multipliation( )e an thin3 of DECREASIN0 the ma"nitu-e !* a fator of ,( to "o from , to +' Thi# "i1e# u# the operation of -i1i#ion( )hih i# the oppo#ite of multipliation' It al#o #ho)# ho) *ou an "o from , to + EIT/ER !* multipl*in" !* .( or !* -i1i-in" !* ,' In the #ame )a*( *ou an "o from + to , either !* multipl*in" !* ,( or -i1i-in" !* .' Thi# in-iate# that multipl*in" !* a num!er i# the #ame a# -i1i-in" !* it# reiproal' So far( the#e re#ult# are 5uite #tan-ar- an- o!1iou#( !ut the* an l ea- to #ome intere#tin" re#ult#' One of the#e i# that rather than multipl*in" a num!er !* pro$etin" it alon" the num!er line( it i# po##i!le to atuall* C/AN0E T/E SCALE of the num!er line it#elf' it#elf' For in#tane( to multipl* the num!er , !* +6( in#tea- of atuall* -ra)in" a pro$etion that enlar"e# the num!er from , to ,6( )e oul- #impl* han"e the #ale of the num!er line it#elf !* multipl*in" all of the num!er# !* +67ro##in" out +( ,( an- 8( et' an- )ritin" +6( ,6( an- 86 in#tea- %in e9et( )hat u#e- to !e + i# no) +6( )hat u#e- to !e , i# no) ,6( et'&' Of our#e( the #ame i# al#o po##i!le to perform )ith -i1i#ion'
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DIAGRAM THE OPERATION OF MULTIPLYING THE NUMBER ‘5’ BY ‘2’ ON THE NUMBER LINE BELOW WITHOUT MAKING A PROJECTION (CHANGE THE SCALE OF THE NUMBER LINE INSTEAD).
DIAGRAM THE OPERATION OF DIVIDING THE NUMBER ‘’ BY ‘2’ ON THE NUMBER LINE BELOW WITHOUT MAKING A PROJECTION (CHANGE THE SCALE OF THE NUMBER LINE INSTEAD).
Thi# i-ea an !e e;ten-e- into t)o -imen#ion#' For e;ample( here i# the "raph of y < x ,:
If )e ta3e the entire ri"ht #i-e of thi# e5uation an- multipl* it !* .( #o )e ha1e y < . x ,( the e9et i# that e1er* y = 1alue i# onl* half a# muh a# it )oul- ha1e !een !efore( #o the "raph i# 2>atter4' /o)e1er( it i# po##i!le to mo-if* thi# "raph )ithout han"in" the line of the e5uation at all' In#tea-( )e an u#e the pre1iou# i-ea to C/AN0E T/E NUM?ER LINE ITSELF' If )e multipl* the num!er# on the y = a;i# !* .( then )e ha1e e9eti1el* 2tran#forme-4 thi# e5uation )ith a 2#alin" fator4'
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CHANGE THE GRAPH OF Y ! " 2 ABOVE TO Y ! 2"2 BY CHANGING ONLY THE NUMBERS ON THE "# OR Y# A"IS (YOU CAN CROSS OUT THE NUMBERS AND PUT NEW NUMBERS IN THEIR PLACE TO CHANGE THE SCALING FACTOR). No)( let4# "o !a3 to thin3in" a!out one -imen#ion !rie>*( !ut thin3 a!out )hat happen# )hen )e appl* thi# #ame han"e T@ICE' For in#tane( if )e appl* the han"e of multipl*in" a num!er %+( for in#tane& !* 8( an- then 8 a"ain( )e an ima"ine the operation of SUARIN0 a num!er' @e an either han"e the #ale of the num!er line t)ie( or )e an pro$et or mo-if* the #alin" of the num!er on the line t)ie %from + to 8( an- then from 8 to B( in eah a#e&' If )e )ant to thin3 ?AC@ARDS( )e are in e9et ta3in" a SUARE ROOT' For in#tane( if )e ta3e the num!er ,( an- tr* to -e-ue )hat han"e )e mu#t appl* T@ICE to reah it7either a pro$etion or han"e in #alin" of ' Therefore( the #5uare root of , i# ' No)( let4# 3eep thi# i-ea in min-( !ut tr* an- "ure out ho) to u#e a pro$etion to multipl* !* a NE0ATIE NUM?ER' %Note that it i# po##i!le to han"e the #alin" to !e ne"ati1e on the ri"ht an- po#iti1e on the left( !ut that i# unon1entional an- impratial&'
DIAGRAM THE OPERATION OF MULTIPLYING THE NUMBER ‘$’ BY ‘#2’ ON THE NUMBER LINE BELOW BY MAKING A PROJECTION.
NOW DIAGRAM THE OPERATION OF MULTIPLYING THE NUMBER ‘$’ BY ‘#2’ TWICE ($ % #2 % #2) ON THE NUMBER LINE BELOW BY MAKING A PROJECTION.
IS THE RESULT OF MULTIPLYING BY #2 TWICE THE SAME AS THAT FROM MULTIPLYING BY 2 TWICE&
WHAT DOES THIS TELL US ABOUT THE S'UARE ROOT(S) OF & HOW MANY S'UARE ROOTS DOES HAVE AND WHAT ARE THEY&
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Notie ho) e1er* time *ou multipl* !* a ne"ati1e num!er( *ou S@ITC/ TO T/E OT/ER SIDE of the num!er line' Doin" thi# t)ie in a ro) %or an* EEN num!er of time#& )ill *iel- a POSITIE num!er( an- -oin" it an ODD num!er of time# )ill *iel- a NE0ATIE num!er' Thi# i# important( an- )e an u#e thi# i-ea to thin3 of ho) ne"ati1e num!er# interat in , -imen#ion#'
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So( to re1ie)( multipl*in" !* a num!er )ill lea- to a pro$etion )ith the follo)in" propertie#: If the num!er i# po#iti1e( the pro$etion i# in the #ame -iretion( !ut if the num!er i# ne"ati1e( the pro$etion i# >ippe- +H6 o %in the oppo#ite -iretion&' If the num!er it#elf ha# a ma"nitu-e %-i#tane from 6& "reater than +( the pro$etion inrea#e# the ma"nitu-e( !ut if the num!er ha# a ma"nitu-e le## than +( then the ma"nitu-e -erea#e#'
FROM THE ABOVE WHAT HAPPENS TO THE PROJECTION IF A NUMBER IS MULTIPLIED BY #$&
In mathemati#( there i# no inte"er %in fat( no 2real4 num!er at all& on the , -imen#ional num!er line )hih( )hen multiplie- !* it#elf( *iel-# +' /o)e1er( no) that )e ha1e ome up )ith a )a* to mo-el multipliation !* +( all )e ha1e to -o i# to "o 2half)a*4 throu"h thi# proe## in or-er to n- )hat the #5uare root of + atuall* i#
ON THE GRAPH (NUMBER LINE WITH 2 DIMENSIONS) BELOW DIAGRAM AN OPERATION WHICH WHEN PERFORMED TWICE CONSECUTIVELY ON THE NUMBER ‘$’ WILL YIELD THE NUMBER ‘#$’. HINT* IF MULTIPLICATION BY #$ YIELDS A PROJECTION IN THE OPPOSITE DIRECTION ($+,) PERFORM HALF OF THIS PROCESS.
The #i"niane of thi# re#ult i# that it repre#ent# the #5uare root of +' Thi# 5uantit* i# 3no)n a# i( an- i# alle- an 2ima"inar* num!er4 to -i#tin"ui#h it from the 2real num!er#4 on the num!er line' Sine it i# a rotation to a -i9erent
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-iretion an- not $u#t a pro$etion in the #ame -iretion( it mo1e# the 1alue onto the y = a;i#' It i# !etter repre#ente- on a "raph that ha# the y a;i# %1ertial a;i#& la!elle- )ith ima"inar* num!er# %)hih are #impl* i multiplie- !* real num!er#&' Num!er# )ith ?OT/ a 2real4 %horiKontal& AND 2ima"inar*4 %1ertial& omponent are alle- 2omple;4 num!er#'
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ON THE ABOVE COMPLE" PLANE SHOW HOW BOTH i 2 ! #$ AND (#i )2 ! #$. USE 2 CONSECUTIVE ROTATIONS ONE COUNTERCLOCKWISE AND ONE CLOCKWISE.
NOW SHOW THAT
i 5 ! i USING
5 OF THE SAME ROTATION IN A ROW.
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no)in" that i < i i# a 1er* important re#ult that allo)# u# to ome to the follo)in" onlu#ion:
ANY NUMBER (REAL IMAGINARY OR COMPLE") THAT RESULTS IN A NET TURN OF -+, AFTER 5 ROTATIONS (THIS INCLUDES GOING $ OR MORE TIMES AROUND PREVIOUSLY/0+, 1 -+, ! 5+, 2+, 1 -+, ! $+, ETC) AND HAS A MAGNITUDE (DISTANCE FROM + THE ORIGIN) OF $ IS A 5TH ROOT OF THE IMAGINARY NUMBER i . There are #uh num!er#( e1enl* #pae- aroun- the ori"in )ith a ma"nitu-e of +( an- the an"le#: B6o < $,( %8J6B6o& < -+,( %,6B6o& < $02,( %+6H6B6o& < 2/,( an%+GG6oB6o& < /+0,
USING THIS PATTERN WRITE THE ANGLES OF THE 0 DIFFERENT 034 ROOTS OF i *
Al#o( )e an 3no) that AN NUM?ER %real( ima"inar*( or omple;& an !e repre#ente- a# a 2proe##4 %either a 2pro$etion4 or 2rotation4& in the follo)in" form#: +' A ?i( )here A i# the real part an- ? i# the ima"inar* part( an- if !oth A an- ? are not 6( the num!er i# 2omple;4' ,' An an"le( an- a -i#tane( )here the an"le i# ta3en from the po#iti1e x a;i# an- i# ounterlo3)i#e in the po#iti1e -iretion( an- the -i#tane i# the ma"nitu-e %a!#olute 1alue or -i#tane from 6& of the num!er'
IN THIS SECOND FORM AT WHAT 2 ANGLES IN THE COMPLE" PLANE ARE REAL NUMBERS FOUND&
AT WHAT 2 ANGLES IN THE COMPLE" PLANE ARE IMAGINARY NUMBERS FOUND AT&
From thi#( it i# lear that multipliation an !e thou"ht of a# a tran#formation %either a pro$etion( rotation( or !oth& in a oor-inate pl ane( an- i# more than #impl* arithmeti' @hile a num!er line i# not an 2inorret4 repre#entation( a--in" ima"inar* num!er# in or-er to form a 2omple; plane4 a--# to our po##i!ilitie# an- allo)# u# to perform operation# that )ere other)i#e impo##i!le'
WHAT INSIGHTS HAVE YOU GAINED FROM THIS ASSIGNMENT& WHAT HAS INTERESTED YOU& (W637 83 978:3 /#5 :7;37;<7:=:7 347 >8
