Notes for March 31 Fields: A field is a set of umbers with two (biary) operatios (usually called additio [+] ad multiplicatio [ ]) such that the followig properties hold: Additio: Name Descriptio Commutativity Associativity Idetity Iverses For all umbers a ad b a+b = b+a For all umbers a, b ad c a+(b+c) = (a+b)+c here exists a umber (which will be deoted 0) such that for every umber a a+0 = a For every umber a there exists a additive iverse (which will be deoted -a) such that a + -a = 0 +3 = 3+ 3+(6+17) = (3+6)+17 7+0 = 7 8 + -8 = 0 Multiplicatio: Name Descriptio Commutativity Associativity Idetity Iverses for No- Zero Numbers For all umbers a ad b a b = b a For all umbers a, b ad c a (b c) = (a b) c here exists a umber (which will be deoted 1) such that for every umber a a 1 = a For every o-zero umber a there exists a multiplicative iverse (which will be deoted a -1 ) such that a a -1 = 1 3 = 3 3 (6 17) = (3 6) 17 7 1 = 7 8 8-1 = 1 Additio ad Multiplicatio: Name Descriptio Distributivity For all umbers a, b ad c a (b+c) = a b + a c (3+4) = 3 + 4
ad No- A. Whole Numbers (1,, 3, 4,...) with usual +, is NO a field Idetity V Idetity Iverse V Iverse V Additio/Multiplicatio Distributive 1. here is ot a additive idetity elemet (0) i the set of whole umbers.. Because there are o egatives i the set of whole umbers there are ot ay additive iverses i the set of whole umbers for example 5 does ot have a additive iverse i the set of whole umbers 3. Because there are o fractios i the set of whole umbers there are ot, i geeral, ay multiplicative iverses i the set of whole umbers for example 5 does ot have a multiplicative iverse i the set of whole umbers B. Itegers (... -4, -3, -, 0, 1,, 3, 4,...) with usual +, is NO a field Idetity Idetity Iverse Iverse V Additio/Multiplicatio Distributive 1. Because there are o fractios i the set of itegers there are ot, i geeral, ay multiplicative iverses i the set of itegers for example 5 does ot have a multiplicative iverse i the set of itegers
C. Real Numbers with usual +, IS a field Idetity Idetity Iverse Iverse Additio/Multiplicatio Distributive Additioally, the set of ratioal umbers Q = { p/q p, q are itegers with q 0 } with the usual +, IS a field D. Plae ú = { (x,y) x,y are real umbers } with additio ad multiplicatio as follows: (a,b) + (c,d) = (a+c, b+d) (a,b) (c,d) = (a c, b d) is NO a field Idetity (0,0) Idetity (1,1) Iverse Iverse V Additio/Multiplicatio Distributive For this multiplicatio (5,0) is ot the zero elemet of the set ad it does ot have a multiplicative iverse.
E. Complex Plae (Complex Numbers) = { (x,y) x,y are real umbers } with additio ad multiplicatio as follows: Write (a,b) = (a,0) + (0,b) / a + b i. he, we ca thik of the horizotal axis which is the set of all { (a,0) a is real } as beig the same as the set of real umbers ad we will call it the real axis. Also, we will call the vertical axis which is the set of all { (0,b) = b i b is real } the imagiary axis. he symbol i = (0,1). We will defie multiplicatio of i with itself by i = -1. he: (a,b) + (c,d) = (a+b i) + (c+d i) = (a+c) + (b+d) i = (a+c, b+d) (a,b) (c,d) = (a+b i) (c+d i) = (a c + b d i ) + (a d + b c) i = (a c - b d) + (a d + b c) i = (a c - b d, a d + b c) IS a field. Idetity Idetity Iverse Iverse Additio/Multiplicatio Distributive Fields are importat sets because i a field (real umbers, ratioal umbers or the complex umbers) all of the usual properties ad rules of algebra for maipulatig expressios ad solvig equatios are true. For example, i a field there is a property that a b = 0 implies that either a = 0 or else b = 0 (or both) Aother example, a b = a c ad a 0 implies b = c (cacellatio rule)
History of Number Systems ad Solvig Equatios: Whole umbers: History: Whole umbers were the first umbers used because they represeted the process of coutig (sheep, wives, childre, rocks, etc.) hey were associated with the otio of the actual physical amout or presece of a give quatity. Equatios: Liear equatios of the form x = a could be solved usig oly whole umbers, but ot equatios of the form x + a = 0 i this umber system. (Here a is a whole umber.) Itegers: History: As society became more complex ad iteractios betwee idividuals developed, so did the eed for expadig the types of umbers eeded to represet iformatio. o represet, say debt, egative umbers were eeded. he morally weighted ames positive ad egative were imposed o umbers to say somethig about the perceived reality of them. Equatios: Liear equatios of the form x + a = 0 could be solved usig oly itegers, but ot equatios of the form a x + b = 0 i this umber system. (Here a, b are itegers). Ratioal Numbers: History: As society became more complex, its eeds to represet fractios or proportios as umbers developed, e.g., a father s estate beig divided amog his heirs; surveyig to establish property lies i the Nile valley after the aual sprig floods. Equatios: Liear equatios of the form a x + b = 0 could be solved usig oly ratioal umbers, but ot quadratic equatios like x = i this umber system. (Here a, b are itegers.) Real Numbers: History: As the Greeks developed geometry, they courted a philosophy that ratioal umbers (to which they assiged moral values) could be used to describe the uiverse. However, with the developmet of the Pythagorea heorem, they came to a crisis. hey could show that certai physical, measurable legths could ot be ratioal. (I a 45-45 -90 triagle with two sides of legth 1, the hypoteuse must be legth.) he morally weighted ames ratioal ad irratioal were imposed o umbers to say somethig about the perceived reality of them. At this poit, the preset day otio of the real umber lie existed ad was valid: ay legth measuremet alog a scaled, ordered lie correspods to a real umber ad ay real umber correspods to a legth measuremet alog a scaled, ordered lie. Equatios: Quadratic equatios like x = could be solved, but ot equatios like x + 1 = 0 i this umber system. Complex Numbers: History: o represet solutios for complex problems arisig i physics, mechaics ad astroomy, the eed arose to exted the type of available umbers or else certai equatios which represeted real world problems would ot be solvable. he morally weighted ames real ad imagiary were imposed o umbers to say somethig about the perceived reality of them. Equatios: Quadratic equatios like x + 1 = 0 could be solved. Give the above
progressio of eedig to add ew (abstract) sets of umbers to our existig umber system every time a ew type of equatio was itroduced could suggest that if we ext wated to solve other types of quadratics or cubics or quartics (ad so forth) that we would agai each time eed to expad our umber system. However, the Fudametal heorem of Algebra was proved (discovered) which i effect says that we ca stop, that at this poit we have a complete eough set of umbers i which solutios for every type of equatio ca be foud. Fudametal heorem of Algebra: 1 Every o-costat polyomial px ( ) = a x + a 1x +... + a x + ax 1 + a0 has a root (i ). (Here the coefficiets a, a -1,..., a, a 1, a 0 are itegers or real umbers or complex umbers.) Alterate versio. Every o-costat polyomial px ( ) = a x + a x +... + a x + ax+ a 1 1 1 0 has a liear factor of the form (a x + b), with a 0, i its factorizatio. (Here agai the coefficiets a, a -1,..., a, a 1, a 0 are itegers or real umbers or complex umbers.) A root or a zero or a solutio of a polyomial p(x) all three terms mea the same thig is a value x that solves the equatios p(x) = 0, that is, is a umber x which whe substituted i makes p(x) = 0. p(1 + i) = 0. Let px ( ) = x 4x 5. he, the umber 5 is a root of p(x), because p(5) = 0. 3 Let px ( ) = x + 1. he, the umber -1 is a root of p(x), because p(-1) = 0. Let px ( ) = x x+ 5. he, the umber 1 + i is a root of p(x), because A liear factor is a factor for the form (a x + b). Note that whe attemptig to solve a liear equatio of the form a x + b = 0, with a 0, it is always possible to fid a solutio, amely, x = -b/a. What the Fudametal heorem of Algebra does ot say, is that the root of p(x) will be i the same umber system as the coefficiets of p(x). Usually, that is ot true. Let irratioal umbers (, - ). 6 6 px ( ) = x 6. he, while the coefficiets of p(x) are itegers, the roots are Let px ( ) = x + 1. he, agai while the coefficiets of p(x) are itegers, the roots are complex umbers ( i, -i ).
Oe of the first cosequeces of the Fudametal heorem of Algebra is that every ocostat polyomial px ( ) = ax + a 1x +... + ax + ax 1 + a0 of degree (the degree of a 1 polyomial is the degree of its highest expoet), has exactly roots. Alterate versio. Every 1 o-costat polyomial px ( ) = ax + a 1x +... + ax + ax 1 + a0 of degree (the degree of a polyomial is the degree of its highest expoet), ca be factored ito a product of exactly liear factors. A quadratic polyomial has always exactly roots. A cubic polyomial has always exactly 3 roots. A fifth degree polyomial has always exactly 5 roots. Etc. Let px ( ) = x 4x 5. he, p(x) has roots (5 ad -1). 3 Let px ( ) = x + 1. he, p(x) has 3 roots (-1, ½ (1+ 3 i ), ½ (1-3 i ) ). Let px ( ) = x x+ 5. he, p(x) has roots (1 + i, 1 - i ). Fially, oe last ote about the roots of polyomials. Note that if a + b i is a complex umber we call a - b i the complex cojugate. From the heory of Equatios we have the followig: 1 heorem: Let px ( ) = ax a x ax ax a be a polyomial with real + 1 +... + + 1 + 0 coefficiets. he, ay complex roots of p(x) must occur i cojugate pairs. Let px ( ) = x 4x 5. he theorem does ot apply because i this case p(x) does ot happe to have ay complex roots. 3 Let px ( ) = x + 1. he, ½ (1+ 3 i ) is a complex root of p(x) ad so is its cojugate ½ (1- i ) 3 Let cojugate 1 - i px ( ) = x x+ 5. he, 1 + i is a complex root of p(x) ad so is its