Paso a paso: Cómo solicitar una decisión de cobertura

ApodeiknÔetai apì ta axi¸mata twn pragmatik¸n arijm¸n ìti to sÔnolo R èqei thn epiplèon idiìthta

16. Eˆn x < y tìte upˆrqei pragmatikìc arijmìc z tètoioc ¸ste x < z kai z < y.

Eˆn x < z kai z < y grˆfoume x < z < y. H idiìthta 16 ekfrˆzei thn puknìthta twn pragmatik¸n arijm¸n. To apotèlesma autì odhgeÐ sto na jewroÔme touc pragmatikoÔc arijmoÔc san ta shmeÐa mÐac eujeÐac. Eˆn jewr soume ta sÔmbola −∞ kai +∞, antÐstoiqa, meÐon ˆpeiro (minus infinity) kai sÔn ˆpeiro (plus infinity), tètoia ¸ste −∞ < x < +∞, gia kˆje pragmatikì arijmì x, mporoÔme na grˆfoume R = (−∞, +∞).

Eˆn x kai y eÐnai pragmatikoÐ arijmoÐ me x < y, tìte me ta diast mata (x, y), [x, y), (x, y], [x, y]

sumbolÐzoume anÐstoiqa to sÔnolo twn pragmatik¸n arijm¸n z pou ikanopoioÔn tic sqèseic

x < z < y, x ≤ z < y, x < z ≤ y, x ≤ z ≤ y.

'Omoia eˆn x eÐnai ènac pragmatikìc arijmìc me ta hmiˆpeira diast mata (−∞, x), (−∞, x], (x, +∞), [x, +∞)

sumbolÐzoume anÐstoiqa to sÔnolo twn pragmatik¸n arijm¸n z pou ikanopoioÔn tic sqèseic

z < x, ≤ x, z > x, z ≥ x.

Parˆdeigma 5.1. Eˆn S = (1, 2), tìte to 2 kaj¸c kai kˆje pragmatikìc arijmìc x ≥ 2 eÐnai èna ˆnw frˆgma tou S. 'Omoia eˆn S = (1, 2], tìte kˆje x ≥ 2 eÐnai èna ˆnw frˆgma tou

S. ParathroÔme ìti eˆn S = (1, 2), tìte sup S = 2 kai sup S /∈ S, en¸ an S = (1, 2], tìte sup S = 2 _{kai sup S ∈ S, dhlad  to elˆqisto ˆnw frˆgma sunìlou mporeÐ na an kei   na mhn} an kei sto sÔnolo. ParathroÔme ìti to sup(1, +∞) den upˆrqei (giati?).

A, τότε

• 'Ena sÔnolo S ⊂ R lègetai kˆtw fragmèno (bounded below) eˆn upˆrqei pragmatikìc arijmìc x tètoioc ¸ste gia kˆje y ∈ S na isqÔei y ≥ x. To x lègetai èna kˆtw frˆgma (lower bound) tou S. Eˆn s eÐnai ènac pragmatikìc arijmìc tètoioc ¸ste s ≥ x gia kˆje kˆtw frˆgma x tou S, tìte o s lègetai mègisto kˆtw frˆgma (greatest lower bound)   infimum tou S kai sumbolÐzetai me inf S.

SÔmfwna me ton orismì tou megÐstou kˆtw frˆgmatoc blèpoume ìti inf(1, 2) = 1, inf[1, 2) = 1, en¸ to inf(−∞, 2) den upˆrqei. MporeÐ na apodeiqjeÐ h akìloujh prìtash. Gia thn apìdeixh blèpe tic ask seic.

Prìtash 5.1. Kˆje kˆtw fragmèno sÔnolo pragmatik¸n arijm¸n èqei elˆqisto kˆtw frˆg- ma.

Anafèroume, gia plhrìthta, ta parakˆtw apotelèsmata. H Prìtash 5.2 eÐnai apìrrroia tou axi¸matoc 15, en¸ h Prìtash 5.3 eÐnai sunèpeia twn Protˆsewn 5.1 kai 5.2. Gia thn apìdeixh parapèmpoume sto sÔggramma [2].

Prìtash 5.2 (AxÐwma tou Arqim dh). Eˆn x kai y eÐnai jetikoÐ pragmatikoÐ arijmoÐ, tìte upˆrqei fusikìc arijmìc n tètoioc ¸ste y < nx.

Prìtash 5.3. Eˆn x kai y eÐnai pragmatikoÐ arijmoÐ me x < y, tìte upˆrqei rhtìc arijmìc

r tètoioc ¸ste x < r < y.

KleÐnontac thn parˆgrafo anafèroume ìti kˆpoioc xekin¸ntac me thn paradoq  ìti upˆrqei èna sÔnolo pou ikanopoieÐ ta axi¸mata 1 15 mporeÐ na apodeÐxei th monadikìthta tou, me thn ènnoia tou isomorfismoÔ, kai sth sunèqeia na kataskeuˆsei pr¸ta touc fusikoÔc arijmoÔc kai metˆ touc akeraÐouc kai touc rhtoÔc. Gia th prosèggish aut  parapèmpoume sta suggrˆmmata [1] kai [8].

Ask seic

5.2.1 [8] Na apodeiqjoÔn oi parakˆtw idiìthtec thc ˆlgebrac twn pragmatik¸n arijm¸n: (aþ) Eˆn x + y = x, tìte y = 0.

(bþ) 0x = 0. Upìdeixh: 0 = 0 + 0. (gþ) −0 = 0.

(dþ) −(−x) = x. Upìdeixh: O −(−x) eÐnai o antÐjetoc tou −x. (eþ) x(−y) = −(xy) = (−x)y.

(þ) (−1)x = −x.

(zþ) x(y − z) = xy − xz.

(hþ) −(x + y) = −x − y, −(x − y) = −x + y. (jþ) Eˆn x 6= 0 kai xy = 1, tìte y = 1.

(iþ) Eˆn x 6= 0, tìte x/x = 1. (iaþ) x/1 = x.

(ibþ) Eˆn x 6= 0 kai y 6= 0, tìte xy 6= 0.

(igþ) (1/x)(1/y) = 1/(xy), eˆn x 6= 0 kai y 6= 0. (idþ) (w/x)(y/z) = (wy)/(xz), eˆn x 6= 0 kai z 6= 0.

(ieþ) (w/x) + (y/z) = (wz + yx)/(xz), eˆn x 6= 0 kai z 6= 0. (iþ) Eˆn x 6= 0, tìte 1/x 6= 0.

(izþ) Eˆn x 6= 0, tìte (x−1₎−1 _{= x.}

(ihþ) 1/(x/y) = y/x, eˆn x 6= 0 kai y 6= 0.

(ijþ) (w/x)/(y/z) = (wz)/(xy), eˆn x 6= 0, y 6= 0 kai z 6= 0. (kþ) (xy)/z = x(y/z), eˆn z 6= 0.

(kaþ) (−x)/y = x/(−y) = −(x/y), eˆn y 6= 0.

5.2.2 [8] Na apodeiqjoÔn oi parakˆtw idiìthtec twn anisot twn pragmatik¸n arijm¸n: (aþ) Eˆn w ≤ x kai y ≤ z, tìte w + y ≤ x + z.

(bþ) Eˆn x ≤ y kai z ≥ 0, tìte zx ≤ zy. (gþ) x ≤ 0 eˆn kai mìnon eˆn −x ≥ 0. (dþ) x ≤ y eˆn kai mìnon eˆn −x ≥ y. (eþ) Eˆn x ≤ y kai z ≤ 0, tìte zx ≥ zy. (þ) Eˆn x 6= 0, tìte x2 _{> 0, ìpou x}2 _{= xx.}

(zþ) −1 < 0 < 1.

(hþ) Eˆn xy > 0, tìte oi x kai y eÐnai kai oi dÔo jetikoÐ   kai oi dÔo arnhtikoÐ. (jþ) Eˆn x > 0, tìte 1/x > 0.

(iþ) Eˆn 0 < x ≤ y, tìte 1/y ≤ 1/x. (iaþ) Eˆn x < y, tìte x < (x + y)/2 < y.

5.2.3 Na brejoÔn ìloi oi pragmatikoÐ arijmoÐ gia touc opoÐouc isqÔei (aþ) (x − 1)(x − 2) > 0. (bþ) x2_{+ x + 1 > 2.} (gþ) 1 x + x−11 > 0. (dþ) x−1 x+1 > 0.

5.2.4 Na apodeiqjoÔn oi parakˆtw idiìthtec thc apìluthc tim c: (aþ) |x| ≥ 0 kai |x| = 0, eˆn kai mìnon eˆn x = 0.

(bþ) |xy| = |x||y|. (gþ) |x| = | − x|

(dþ) |x + y| ≤ |x| + |y|. (eþ) ||x| − |y|| ≤ |x − y|.

5.2.5 Na brejoÔn ìloi oi pragmatikoÐ arijmoÐ gia touc opoÐouc isqÔei (aþ) |x − 2| > 8. (bþ) |x − 2| < 8. (gþ) |x − 1| + |x − 2| > 1. (dþ) |x − 1| + |x + 1| < 2. (eþ) |x − 1||x + 2| = 0. (þ) |x − 1||x + 2| = 3.

5.2.6 Eˆn S ⊂ R, tìte orÐzoume −S = {−x : x ∈ S}, dhlad  −[1, 2) = (−2, −1]. Na apodeiqjoÔn oi isqurismoÐ:

(aþ) 'Ena sÔnolo S ⊂ R eÐnai ˆnw fragmèno eˆn kai mìnon eˆn to −S eÐnai kˆtw frag- mèno.

(bþ) 'Ena sÔnolo S ⊂ R eÐnai kˆtw fragmèno eˆn kai mìnon eˆn to −S eÐnai ˆnw frag- mèno. Upìdeixh: −(−S) = S.

(gþ) sup S = − inf(−S) kai inf S = − sup(−S).

5.2.7 Na brejoÔn, eˆn autˆ upˆrqoun ta inf S kai sup S, ìpou (aþ) S = ½ 1 n : n ∈ N ¾ . (bþ) S = ½ (−1)n₊ 1 n : n ∈ N ¾ .

5.2.8 'Estw ìti A eÐnai èna tuqaÐo sÔnolo efodiasmèno me dÔo eswterikèc prˆxeic ⊕ kai ¯ tètoiec ¸ste na ikanopoioÔntai oi idiìthtec/axi¸mata

1. x ⊕ y = y ⊕ x, gia kˆje x, y ∈ A.

2. (x ⊕ y) ⊕ z = x ⊕ (y ⊕ z), gia kˆje x, y, z ∈ A.

3. Upˆrqei stoiqeÐo 0 ∈ A, ètsi ¸ste x ⊕ 0 = x, gia kˆje x ∈ A. 4. Gia kˆje x ∈ A upˆrqei stoiqeÐo x0 _{∈ A, ètsi ¸ste x ⊕ x}0 _{= 0.}

5. (x ¯ y) ¯ z = x ¯ (y ¯ z), gia kˆje x, y, z ∈ A. 6. x ¯ (y ⊕ z) = x ¯ y ⊕ x ¯ z kai

H triˆda (A, ⊕, ¯) lègetai daktÔlioc (ring). Na deiqjeÐ ìti to sÔnolo twn akeraÐwn efodiasmèno me tic prˆxeic thc prìsjeshc kai tou pollaplasiasmoÔ, (Z, +, ·), apoteleÐ daktÔlio.

Kefˆlaio 6

Oi migadikoÐ arijmoÐ

6.1 To s¸ma twn migadik¸n arijm¸n

Eˆn x ∈ R tìte x2 _{≥ 0, opìte h exÐswsh x}2 _{+ 1 = 0 den èqei lÔsh stouc pragmatikoÔc}

arijmoÔc. Diatup¸netai loipìn to er¸thma katˆ pìson upˆrqei èna sÔsthma arijm¸n pou katˆ kˆpoia ènnoia epekteÐnei touc pragmatikoÔc arijmoÔc kai eÐnai tètoio ¸ste h exÐswsh

x2_{+ 1 = 0 na èqei lÔsh. ApodeiknÔetai ìti èna tètoio sÔsthma upˆrqei.}

Kataskeu . Sto sÔnolo R × R orÐzoume touc nìmouc thc prìsjeshc kai pollaplasiasmoÔ me tic sqèseic (x1, y1) + (x2, y2) = (x1+ x2, y1+ y2) (6.1) (x1, y1)(x2, y2) = (x1x2− y1y2, x1y2+ x2y1). (6.2) ParathroÔme ìti (x1, y1) + (0, 0) = (x1, y1) (6.3) (x1, y1) + (−x1, −y1) = (0, 0) (6.4) (x1, y1)(1, 0) = (x1, y1), (6.5)

dhlad  to (0, 0) eÐnai to oudètero stoiqeÐo thc prìsjeshc, to (−x1, −y1) eÐnai to antÐjeto

tou (x1, y1), en¸ to (1, 0) eÐnai oudètero stoiqeÐo tou pollaplasiasmoÔ. Eˆn (x, y) 6= (0, 0)

kai (a, b) eÐnai to antÐstrofo stoiqeÐo tou (x, y), eˆn autì upˆrqei, tìte ja prèpei (x, y)(a, b) = (xa − yb, xb + ya) = (1, 0).

UpenjumÐzoume ìti (x1, y1) = (x2, y2) eˆn kai mìnon eˆn x1 = x2 kai y1 = y2, tìte apì thn

parapˆnw isìthta prokÔptoun oi sqèseic xa − yb = 1 kai xb + ya = 0. LÔnontac to sÔsthma brÐskoume a = x x2_{+ y}2, b = −y x2_{+ y}2. 53

Oi arijmoÐ a kai b upˆrqoun, kajìson x2_{+ y}2 _{> 0 opoted pote (x, y) 6= (0, 0). Epomènwc to}

antÐstrofo tou (x, y) to opoÐo sumbolÐzoume me (x, y)−1 _{eÐnai to}

(6.6) (x, y)−1 ₌ µ x x2 _{+ y}2, −y x2_{+ y}2 ¶ .

EÐnai plèon eÔkolo na apodeiqjeÐ ìti to R×R efodiasmèno me touc nìmouc (6.1) kai (6.2) eÐnai s¸ma, blèpe Parat rhsh 5.2. OrÐzoume to s¸ma twn migadik¸n (complex) arijm¸n C na eÐnai to sÔnolo twn shmeÐwn z = (x, y) ∈ R × R efodiasmèno me touc nìmouc (6.1) kai (6.2).

Apìrroia twn prˆxewn (6.1) kai (6.2) eÐnai ìti (x, y) = (x, 0)+(0, y) kai (0, 1)(y, 0) = (0, y), ètsi kˆje migadikìc arijmìc mporeÐ na grafeÐ sth morf 

(6.7) (x, y) = (x, 0) + (0, 1)(y, 0).

Eˆn x eÐnai ènac pragmatikìc arijmìc, shmeÐo thc eujeÐac, mporeÐ na tautopoihjeÐ me to (x, 0), shmeÐo tou epipèdou. Epiplèon parathroÔme ìti

(x1, 0) + (x2, 0) = (x1 + x2, 0), (x1, 0)(x2, 0) = (x1x2, 0),

dhlad  to s¸ma twn migadik¸n arijm¸n epekteÐnei katˆ fusiologikì trìpo to s¸ma twn prag- matik¸n arijm¸n, kai upì to prÐsma thc tautopoÐhshc x ≡ (x, 0) mporoÔme na jewroÔme ìti R ⊂ C_{. Jètontac i = (0, 1)}i = (0, 1)i = (0, 1) _{sÔmfwna me thn parapˆnw tautopoÐhsh h (6.7) grˆfetai}

(6.8) (x, y) = x + iy.

O migadikìc arijmìc i lègetai fantastik  monˆda (imaginary unit). Eˆn z = (x, y) eÐnai ènac migadikìc arijmìc apì ed¸ kai sto ex c ja grˆfoume z = x + iy. Eˆn z1 = x1+ iy1 kai

z2 = x2 + iy2 eÐnai migadikoÐ arijmoÐ tìte to ˆjroisma z1 + z2 kai to ginìmeno z1z2 dÐnontai,

mèsw twn (6.1) kai (6.2), apì tic sqèseic

z1+ z2 = (x1+ iy1) + (x2+ iy2) = (x1 + x2) + i(y1+ y2)

(6.9)

z1z2 = (x1+ iy1)(x2+ iy2) = (x1x2− y1y2) + i(x1y2+ x2y1).

(6.10)

• 'Estw o migadikìc arijmìc z = x + iy, tìte apì ton orismì tou C èqoume ìti x ∈ R kai y ∈ R. O x lègetai pragmatikì mèroc (real part) tou z kai grˆfoume x = Re z, kai o y lègetai fantastikì mèroc (imaginary part) tou z kai grˆfoume y = Im z. 'Et- si eˆn z ∈ R tìte Re z = z kai Im z = 0, en¸ eˆn z = iy, me y ∈ R, tìte Re z = 0 kai Im z = y.

•Oi migadikoÐ arijmoÐ z1 = x1+ iy1 kai z2 = x2+ iy2 eÐnai Ðsoi kai grˆfoume z1 = z2, eˆn kai

mìnon eˆn x1 = x2 kai y1 = y2, isodÔnama Re z1 = Re z2 kai Im z1 = Im z2.

'Opwc kai stouc pragmatikoÔc arijmoÔc, epagwgikˆ orÐzoume z1 _{= z, z}2 _{= zz, z}3 _{= z}2_z,

zn+1 _{= z}n_{z gia kˆje fusikì arijmì n. ParathroÔme ìti i}2 _{= (0, 1)(0, 1) = (−1, 0)   i}2 _{= −1.}

kai (−i)2_{+ 1 = 0.}

• DeÐxame loipìn ìti to s¸ma twn migadik¸n arijm¸n C apoteleÐ mÐa fusiologik  epèktash twn pragmatik¸n arijm¸n, ìpou sto sÔsthma autì h exÐswsh z2_{+ 1 = 0 èqei lÔsh.}

• _{KleÐnoume aut  th parˆgrafo me mÐa parat rhsh. Den upˆrqei sto C mÐa diˆtaxh pou na}

eÐnai sumbibast  me tic prˆxeic thc prìsjeshc kai tou pollaplasiasmoÔ kai na epekteÐnei th gnwst  diˆtaxh tou R. An upojèsoume ìti mÐa tètoia upˆrqei kai an th sumbolÐsoume me ≤, tìte ja prèpei na isqÔei 0 ≤ 1, epeid  autì isqÔei kai stouc pragmatikoÔc arijmoÔc. EpÐshc èna apì ta dÔo eÐnai alhjèc: eÐte 0 ≤ i, eÐte 0 ≥ i. Eˆn 0 ≤ i, tìte ja eÐnai 0i ≤ i2_,  

isodÔnama 0 ≤ −1,   isodÔnama 0 ≥ 1 pou eÐnai ˆtopo. 'Omoia eˆn 0 ≥ i tìte ja eÐqame 0i ≤ i2_{,   isodÔnama 0 ≤ −1,   isodÔnama 0 ≥ 1 pou eÐnai ˆtopo.}

6.2 'Algebra twn migadik¸n arijm¸n

To sÔnolo twn migadik¸n arijm¸n efodiasmèno me tic prˆxeic thc prìsjeshc kai tou pol- laplasiasmoÔ ìpwc autèc orÐzontai stic sqèseic (6.1) kai (6.2)   (6.9) kai (6.10) eÐnai s¸ma dhlad  isqÔoun oi nìmoi

1. z1+ z2 = z2+ z1, gia kˆje z1, z2 sto C.

2. (z1+ z2) + z3 = z1+ (z2+ z3), gia kˆje z1, z2, z3 sto C.

3. Upˆrqei o monadikìc migadikìc arijmìc 0 = (0, 0) = 0 + i0, ètsi ¸ste z + 0 = z, gia kˆje z ∈ C.

4. Gia kˆje migadikì arijmì z upˆrqei monadikìc migadikìc arijmìc −z, ètsi ¸ste z + (−z) = 0.

5. z1z2 = z2z1, gia kˆje z1, z2 sto C.

6. (z1z2)z3 = z1(z2z3), gia kˆje z1, z2, z3 sto C.

7. Upˆrqei o monadikìc migadikìc arijmìc 1 = (1, 0) = 1 + i0, ètsi ¸ste z · 1 = z, gia kˆje

z ∈ C_.

8. Gia kˆje migadikì arijmì z 6= 0 upˆrqei monadikìc migadikìc arijmìc z−1 _{ètsi ¸ste} z · z−1 _{= 1.}

9. z1(z2+ z3) = z1z2+ z1z3, gia kˆje z1, z2, z3 sto C.

Parat rhsh 6.1. Ac jewr soume ton migadikì arijmì z = x+iy. Apì ton antimetajetikì nìmo (nìmoc 5) èqoume iy = yi opìte o mporoÔme na grˆfoume

Epeid  i(−y) = (−y)i = (−1)yi = (−1)iy kai i(−y) + iy = i(−y + y) = i0 = 0, sunduˆzontac ta dÔo apotelèsmata sumperaÐnoume ìti

i(−y) = (−1)iy = −iy.

'Etsi apì tic (6.8), (6.4) kai (6.6) èpetai ìti oi −z kai z−1_{, efìson z 6= 0, dÐnontai antÐstoiqa}

apì tic sqèseic

−z = −x + i(−y) = −x − iy (6.11) z−1 ₌ x x2_{+ y}2 + i −y x2_{+ y}2 = x x2_{+ y}2 − i y x2_{+ y}2 (6.12)

Parat rhsh 6.2. 'Estw z1 = x1+ iy1 kai z2 = x2+ iy2, tìte kˆnontac qr sh tou nìmou

9 (epimeristik  idiìthta tou pollaplasiasmoÔ wc proc thn prìsjesh) upologÐzoume

z1z2 = (x1+ iy1)(x2+ iy2) = x1(x2+ iy2) + iy1(x2 + iy2) (nìmoc 9) = x1x2+ x1iy2+ iy1x2+ iy1iy2 (nìmoc 9) = x1x2+ ix1y2+ iy1x2+ i2y1y2 (nìmoc 5) = x1x2+ ix1y2+ iy1x2− y1y2 (i2 = −1) = (x1x2− y1y2) + i(x1y2+ x2y1) (nìmoc 9)

pou eÐnai h (6.10). O pollaplasiasmìc dhlad , migadik¸n arijm¸n mporeÐ na ektelesjeÐ me qr sh thc oikeÐac, apì touc pragmatikoÔc arijmoÔc, epimeristik c idiìthtac.

Parat rhsh 6.3. Eˆn z1 = x1+ iy1 kai z2 = x2+ iy2, eÐnai migadikoÐ arijmoÐ, ìpwc stouc

pragmatikoÔc arijmoÔc, h afaÐresh kai to phlÐko orÐzontai, antÐstoiqa, me tic sqèseic

z1− z2 = z1+ (−z2) = (x1 + iy1) + (−x2+ i(−y2)) = (x1− x2) + i(y1− y2) (6.13) z1 z2 = z1z2−1 = (x1+ iy1) µ x2 x2 2+ y22 + i −y2 x2 2+ y22 ¶ = x1x2 + y1y2 x2 2+ y22 + i−x1y2+ x2y1 x2 2+ y22 . (6.14)

ParathroÔme ìti gia z1 = 1 = 1 + i0 kai z2 = z = x + iy apì thn teleutaÐa sqèsh èpetai

(6.15) 1_z = x

x2_{+ y}2 + i

−y x2 _{+ y}2 = z

−1_.

Epakìloujo thc teleutaÐac aut c sqèshc eÐnai h

(6.16) z_z1 2 = z1 1 z2 .

Ask seic

(aþ) (−3 + i)(1 − i2) (bþ) _{9 + i2}1 .

(gþ) (√7 + i√3)(√7 − i√3)_. (dþ) _{3 + i5}7 − i.

(eþ) (3 + i2)2_.

6.2.2 Na deiqjeÐ ìti oi arijmoÐ 1 ± i ikanopoioÔn thn exÐswsh z2_{− 2z + 2 = 0.}

6.2.3 Eˆn z ∈ C kai w ∈ C na deiqjeÐ ìti: (i) z2 _{+ 1 = (z + i)(z − i) kai (ii) z}2 _{+ w}2 ₌

(z + iw)(z − iw).

6.2.4 Eˆn x kai y eÐnai pragmatikoÐ arijmoÐ na brejoÔn oi timèc touc se kˆje mÐa apì tic efrˆseic:

(aþ) 5x + i6 = −8 + i2y

(bþ) i(2x − 4y) = 4x + 2 + i3y. (gþ) (3x + i)2 _{= 8 + iy.}

6.2.5 Na brejoÔn oi lÔseic thc exÐswshc z2_{+z +1 = 0. Upìdeixh: Jètoume z = x+iy sthn}

exÐswsh kai afoÔ kˆnoume prˆxeic koitˆzoume xeqwristˆ to pragmatikì kai fantastikì mèroc.

6.2.6 Na deiqjeÐ ìti o arijmìc a eÐnai pragmatikìc eˆn kai mìnon eˆn Re a = a. 6.2.7 Eˆn z ∈ C na deiqjeÐ ìti: (i) Re(iz) = − Im z kai (ii) Im(iz) = Re z.

6.2.8 Na upologisjoÔn oi dunˆmeic in_{, gia kˆje fusikì arijmì n. Upìdeixh: i}1 _{= i, i}2 _{= −1,}

i3 _{= i}2_{i = −i.}

6.2.9 Na deiqjeÐ ìti (1 + z)2 _{= 1 + 2z + z}2_.

6.2.10 Me qr sh thc majhmatik c epagwg c na deiqjeÐ ìti (1 + z)n_{= 1 +} µ n 1 ¶ z + µ n 2 ¶ z2_{+ · · · +} µ n k ¶ zk_{+ · · · +} µ n n − 1 ¶ zn−1_{+ z}n_.

6.2.11 Eˆn z, w, v kai u eÐnai migadikoÐ arijmoÐ na apodeiqjoÔn oi isìthtec: (aþ) _zw1 = 1 z 1 w. (bþ) z + w_v = z v + w v, v 6= 0. (gþ) zw_vu = z v w u, v 6= 0 kai u 6= 0. (dþ) zw_zv = w v, z 6= 0 kai v 6= 0.

In document Evidencia de Cobertura: (página 197-200)