{"id":1007,"date":"2020-08-10T21:33:53","date_gmt":"2020-08-10T19:33:53","guid":{"rendered":"http:\/\/matma.com.pl\/?p=1007"},"modified":"2020-08-12T09:01:36","modified_gmt":"2020-08-12T07:01:36","slug":"liczby-calkowite","status":"publish","type":"post","link":"http:\/\/matma.com.pl\/?p=1007","title":{"rendered":"Liczby ca\u0142kowite"},"content":{"rendered":"<p><span style=\"font-family: 'book antiqua', palatino, serif;\">Znamy ju\u017c liczby naturalne <img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?\\mathbb{N}=\\left&amp;space;\\{&amp;space;0,1,2,...\\right&amp;space;\\}\" alt=\"\\mathbb{N}=\\left \\{ 0,1,2,...\\right \\}\" align=\"absmiddle\" \/>. Jest to zbi\u00f3r liczb dodatnich razem z zerem. <span style=\"text-decoration: underline;\">Zero nie jest ani dodatnie ani ujemne<\/span>. Je\u015bli do liczb naturalnych dodamy wszystkie ich liczby przeciwne to otrzymamy <span style=\"color: #4472c4;\"><strong>zbi\u00f3r liczb ca\u0142kowitych<\/strong><\/span>(oznaczamy <img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?\\mathbb{Z}\" alt=\"\\mathbb{Z}\" align=\"absmiddle\" \/> ). <span style=\"color: #4472c4;\"><strong>Liczby przeciwne<\/strong><\/span> s\u0105 to dwie liczby, kt\u00f3rych suma daje nam zero (liczb\u0105 przeciwn\u0105 do <img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?1\" alt=\"1\" align=\"absmiddle\" \/>\u00a0jest liczba\u00a0<img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?-1\" alt=\"-1\" align=\"absmiddle\" \/> (odczytujemy &#8222;minus jeden&#8221;), liczba przeciwna do <img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?2\" alt=\"2\" align=\"absmiddle\" \/> jest liczba <img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?-2\" alt=\"-2\" align=\"absmiddle\" \/> (odczytujemy &#8222;minus dwa&#8221;), i tak dalej).<\/span><\/p>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\">Liczby przeciwne znajduj\u0105 si\u0119 na osi liczbowej w takiej samej odleg\u0142o\u015bci od zera.<\/span><\/p>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\"><img decoding=\"async\" class=\"wp-image-1019 aligncenter\" src=\"http:\/\/matma.com.pl\/wp-content\/uploads\/2020\/08\/odleg\u0142o\u015bc-od-zera-300x56.png\" alt=\"\" width=\"472\" height=\"88\" srcset=\"http:\/\/matma.com.pl\/wp-content\/uploads\/2020\/08\/odleg\u0142o\u015bc-od-zera-300x56.png 300w, http:\/\/matma.com.pl\/wp-content\/uploads\/2020\/08\/odleg\u0142o\u015bc-od-zera-768x143.png 768w, http:\/\/matma.com.pl\/wp-content\/uploads\/2020\/08\/odleg\u0142o\u015bc-od-zera-800x151.png 800w, http:\/\/matma.com.pl\/wp-content\/uploads\/2020\/08\/odleg\u0142o\u015bc-od-zera.png 809w\" sizes=\"(max-width: 472px) 100vw, 472px\" \/><\/span><\/p>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\">Zaznaczmy zbi\u00f3r liczb ca\u0142kowitych na osi liczbowej.<\/span><\/p>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\"><img fetchpriority=\"high\" decoding=\"async\" class=\"wp-image-1018 aligncenter\" src=\"http:\/\/matma.com.pl\/wp-content\/uploads\/2020\/08\/liczby-dodatnie-liczby-ujemne-300x74.png\" alt=\"\" width=\"458\" height=\"113\" srcset=\"http:\/\/matma.com.pl\/wp-content\/uploads\/2020\/08\/liczby-dodatnie-liczby-ujemne-300x74.png 300w, http:\/\/matma.com.pl\/wp-content\/uploads\/2020\/08\/liczby-dodatnie-liczby-ujemne-768x190.png 768w, http:\/\/matma.com.pl\/wp-content\/uploads\/2020\/08\/liczby-dodatnie-liczby-ujemne.png 788w\" sizes=\"(max-width: 458px) 100vw, 458px\" \/><\/span><\/p>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\"><img decoding=\"async\" class=\"aligncenter\" src=\"http:\/\/latex.codecogs.com\/gif.latex?\\mathbb{Z}=\\left&amp;space;\\{&amp;space;...,-4,-3,-2,-1,0,1,2,3,4,...&amp;space;\\right&amp;space;\\}\" alt=\"\\mathbb{Z}=\\left \\{ ...,-4,-3,-2,-1,0,1,2,3,4,... \\right \\}\" align=\"absmiddle\" \/><\/span><\/p>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\">Liczby dodatnie znajduj\u0105 si\u0119 na prawo od zera, natomiast liczby ujemne na lewo od zera. <\/span><\/p>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\">Gdzie maj\u0105 zastosowanie liczby dodatnie i ujemne?<\/span><\/p>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\"> Pierwsze skojarzenie to oczywi\u015bcie termometr. Je\u015bli jest ciep\u0142o, temperatura jest powy\u017cej zera, natomiast je\u015bli jest mro\u017ano temperatura spada poni\u017cej zera. We\u017amy przyk\u0142ad z temperatur\u0105: na dworze mamy\u00a0 <img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?-10^{\\circ&amp;space;}C\" alt=\"-10^{\\circ }C\" align=\"absmiddle\" \/> . Je\u015bli temperatura spad o dwa stopnie to b\u0119dziemy mieli <img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?-12^{\\circ&amp;space;}C\" alt=\"-12^{\\circ }C\" align=\"absmiddle\" \/>. Je\u015bli natomiast temperatura wzro\u015bnie o <img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?4^{\\circ&amp;space;}C\" alt=\"4^{\\circ }C\" align=\"absmiddle\" \/>, w\u00f3wczas nasz termometr pokarze <img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?-6^{\\circ&amp;space;}C\" alt=\"-6^{\\circ }C\" align=\"absmiddle\" \/>. Prze\u0142\u00f3\u017cmy ten przyk\u0142ad na o\u015b liczbow\u0105. Tak wi\u0119c, je\u015bli temperatura spada to przesuwamy si\u0119 na osi w lew\u0105 stron\u0119, natomiast je\u015bli ro\u015bnie to w praw\u0105 stron\u0119.<\/span><\/p>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\"><img decoding=\"async\" class=\"wp-image-1049 aligncenter\" src=\"http:\/\/matma.com.pl\/wp-content\/uploads\/2020\/08\/termo-300x235.png\" alt=\"\" width=\"440\" height=\"345\" srcset=\"http:\/\/matma.com.pl\/wp-content\/uploads\/2020\/08\/termo-300x235.png 300w, http:\/\/matma.com.pl\/wp-content\/uploads\/2020\/08\/termo.png 765w\" sizes=\"(max-width: 440px) 100vw, 440px\" \/><\/span><\/p>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\">Liczby\u00a0 ujemne maj\u0105 zastosowanie w wielu dziedzinach naszego \u017cycia. Kolejnym przyk\u0142adem ich u\u017cycia jest zaci\u0105gni\u0119ty d\u0142ug. Nie da si\u0119 inaczej zapisa\u0107 d\u0142ugu, jak tylko za pomoc\u0105 liczby ujemnej, na przyk\u0142ad je\u015bli nasz d\u0142ug wynosi sto z\u0142otych to zapiszemy to <img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?-100\" alt=\"-100\" align=\"absmiddle\" \/> z\u0142.<\/span><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Znamy ju\u017c liczby naturalne . Jest to zbi\u00f3r liczb dodatnich razem z zerem. Zero nie jest ani dodatnie ani ujemne. Je\u015bli do liczb naturalnych dodamy wszystkie ich liczby przeciwne to otrzymamy zbi\u00f3r liczb ca\u0142kowitych(oznaczamy ). Liczby przeciwne s\u0105 to dwie liczby, kt\u00f3rych suma daje nam zero (liczb\u0105 przeciwn\u0105 do \u00a0jest liczba\u00a0 (odczytujemy &#8222;minus jeden&#8221;), liczba [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[6,12],"tags":[],"class_list":["post-1007","post","type-post","status-publish","format-standard","hentry","category-baza-wiedzy","category-klasa-v"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v20.1 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Liczby ca\u0142kowite - Matma.com.pl<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/matma.com.pl\/?p=1007\" \/>\n<meta property=\"og:locale\" content=\"pl_PL\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Liczby ca\u0142kowite - Matma.com.pl\" \/>\n<meta property=\"og:description\" content=\"Znamy ju\u017c liczby naturalne . 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