Realna števila so pravzaprav vsa števila, ki si jih lahko... Show more
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Updated Apr 18, 2026
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Osnove Realnih števil
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Uvod v realna števila
Realna števila so kot velika družina, ki vključuje vsa števila, s katerimi se srečuješ v matematiki. Predstavljaj si jih kot vse možne točke na neskončni številski premici.
Ta tema povezuje vse, kar si se naučil o številih doslej. Od preprostega štetja do kompleksnejših problemov v geometriji in algebri - vse temelji na razumevanju različnih vrst števil.
Najbolj zanimivo je, da vsaka skupina števil rešuje določene probleme. Naravna števila so odlična za štetje, cela števila pa potrebuješ, ko se srečaš z dolgovi ali temperaturami pod ničlo.
💡 Pomni: Vsako število, ki ga lahko postaviš na številsko premico, je realno število!
Številske množice - od naravnih do realnih
Naravna števila (N) so osnova vsega - to so števila 1, 2, 3, s katerimi šteješ. Cela števila (Z) dodajo še negativne vrednosti in ničlo, kar ti omogoča opisati dolgove ali temperature.
Racionalna števila (Q) so vsa števila, ki jih lahko zapišeš kot ulomek. Sem spadajo tudi decimalna števila, kot je 0,25 (ker je ¼) ali neskončna, ki se ponavljajo, kot 0,333... (ker je ⅓).
Iracionalna števila (I) so prava skrivnost matematike! Teh ne moreš zapisati kot ulomek. Njihove decimalne števke se nadaljujejo v neskončnost brez vzorca - kot √2 ali π.
💡 Trik za teste: Če je število pod korenom popoln kvadrat (4, 9, 16...), je koren racionalen. Sicer je iracionalen!
Kako se številske množice povezujejo
Predstavljaj si številske množice kot ruske matruške - ena je znotraj druge. Naravna števila so znotraj celih, cela so znotraj racionalnih, racionalna pa skupaj z iracionalnimi tvorijo realna števila.
Decimalni zapis ti hitro pove, s katero vrsto števila imaš opravka. Če se decimalne števke končajo ali ponavljajo v vzorcu, imaš racionalno število. Če se nadaljujejo brez vzorca, je iracionalno.
Prav vsako realno število ima svoje mesto na številski premici. Ne obstaja "luknja" ali manjkajoča točka - premico popolnoma zapolnijo.
💡 Za teste: Vsako število pripada najmanjši možni množici, vendar hkrati tudi vsem večjim. Število 5 je naravno, celo, racionalno IN realno!
Reševanje primerov - razvrščanje števil
Poglejmo si, kako razvrščaš števila v practice. Vzemimo -7, 12/4, √9 in √10. Najprej vedno poenostavi, če je mogoče!
12/4 postane 3, torej je naravno število. √9 je enako 3, tudi naravno. -7 pa ni naravno (je negativno), vendar je celo število.
√10 je drugačen - 10 ni popoln kvadrat, torej je √10 iracionalno število. Njegove decimalne števke se nadaljujejo brez ponavljajočega vzorca.
Ko razvrščaš, vedno začni z najmanjšo možno množico. Če je število naravno, ne reci, da je "samo" realno - to ni dovolj natančno za teste!
💡 Strategija: Najprej poenostavi ulomke in korenine, šele nato določi najmanjšo množico, ki ji število pripada.
Prikaz na številski premici
Za uspešno risanje na številsko premico si zapomni ključne približke: π ≈ 3,14, √2 ≈ 1,41, √3 ≈ 1,73. Ti bodo olajšali umeščanje iracionalnih števil.
Decimalna števila, kot je 2,5, lahko enostavno umestiš natančno na polovico med 2 in 3. Za negativna števila se premakneš v levo od ničle.
Iracionalna števila zahtevajo malce več pozornosti. √2 umestiš med 1 in 2, bližje 1,5. Za -√2 greš enako razdaljo v levo od ničle.
Če potrebuješ natančen prikaz √2, lahko uporabiš Pitagorov izrek - narišeš pravokotni trikotnik s stranicama 1 in s šestilom preneseš hipotenuzo na premico.
💡 Praktičen nasvet: Za teste si vnaprej zapomni, da je √4 = 2, √9 = 3, √16 = 4... Tako hitreje prepoznaš, kdaj je koren racionalen!
Povzetek za hitro ponavljanje
Zapomniti si moraš hierarhijo: N ⊂ Z ⊂ Q ⊂ R. To pomeni, da je vsako naravno število tudi celo, vsako celo tudi racionalno, in vsa pa realna števila.
Racionalna in iracionalna števila skupaj tvorita realna števila , vendar se med seboj ne prekrivajo. Število je bodisi racionalno bodisi iracionalno, ne more biti oboje.
Ključ za razpoznavanje je decimalni zapis: če se konča ali ponavlja, je racionalno. Če se nadaljuje brez vzorca, je iracionalno.
Vsa realna števila lahko prikažeš na številski premici - to je njihova najpomembnejša lastnost. S tem konceptom boš delal še v gimnaziji in na fakulteti!
💡 Za izpite: Če si negotov, ali je število racionalno, poskusi ga zapisati kot ulomek. Če ne gre, je verjetno iracionalno!
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4.7/5
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The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan S
iOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha Klich
Android user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
Anna
iOS user
Best app on earth! no words because it’s too good
Thomas R
iOS user
Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.
Basil
Android user
This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.
Rohan U
Android user
I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.
Xander S
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE Knowunity AI. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
Paul T
iOS user
The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan S
iOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha Klich
Android user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
Anna
iOS user
Best app on earth! no words because it’s too good
Thomas R
iOS user
Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.
Basil
Android user
This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.
Rohan U
Android user
I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.
Xander S
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE Knowunity AI. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
Paul T
iOS user
Osnove Realnih števil
Realna števila so pravzaprav vsa števila, ki si jih lahko predstavljaš na številski premici - od običajnih števil za štetje do bolj skrivnostnih, kot je π. Ta tema ti bo pomagala razumeti, kako se vsa ta števila povezujejo med seboj... Show more
Uvod v realna števila
Realna števila so kot velika družina, ki vključuje vsa števila, s katerimi se srečuješ v matematiki. Predstavljaj si jih kot vse možne točke na neskončni številski premici.
Ta tema povezuje vse, kar si se naučil o številih doslej. Od preprostega štetja do kompleksnejših problemov v geometriji in algebri - vse temelji na razumevanju različnih vrst števil.
Najbolj zanimivo je, da vsaka skupina števil rešuje določene probleme. Naravna števila so odlična za štetje, cela števila pa potrebuješ, ko se srečaš z dolgovi ali temperaturami pod ničlo.
💡 Pomni: Vsako število, ki ga lahko postaviš na številsko premico, je realno število!
Sign up to see the contentIt's free!
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Številske množice - od naravnih do realnih
Naravna števila (N) so osnova vsega - to so števila 1, 2, 3, s katerimi šteješ. Cela števila (Z) dodajo še negativne vrednosti in ničlo, kar ti omogoča opisati dolgove ali temperature.
Racionalna števila (Q) so vsa števila, ki jih lahko zapišeš kot ulomek. Sem spadajo tudi decimalna števila, kot je 0,25 (ker je ¼) ali neskončna, ki se ponavljajo, kot 0,333... (ker je ⅓).
Iracionalna števila (I) so prava skrivnost matematike! Teh ne moreš zapisati kot ulomek. Njihove decimalne števke se nadaljujejo v neskončnost brez vzorca - kot √2 ali π.
💡 Trik za teste: Če je število pod korenom popoln kvadrat (4, 9, 16...), je koren racionalen. Sicer je iracionalen!
Sign up to see the contentIt's free!
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Kako se številske množice povezujejo
Predstavljaj si številske množice kot ruske matruške - ena je znotraj druge. Naravna števila so znotraj celih, cela so znotraj racionalnih, racionalna pa skupaj z iracionalnimi tvorijo realna števila.
Decimalni zapis ti hitro pove, s katero vrsto števila imaš opravka. Če se decimalne števke končajo ali ponavljajo v vzorcu, imaš racionalno število. Če se nadaljujejo brez vzorca, je iracionalno.
Prav vsako realno število ima svoje mesto na številski premici. Ne obstaja "luknja" ali manjkajoča točka - premico popolnoma zapolnijo.
💡 Za teste: Vsako število pripada najmanjši možni množici, vendar hkrati tudi vsem večjim. Število 5 je naravno, celo, racionalno IN realno!
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
Join milions of students
Reševanje primerov - razvrščanje števil
Poglejmo si, kako razvrščaš števila v practice. Vzemimo -7, 12/4, √9 in √10. Najprej vedno poenostavi, če je mogoče!
12/4 postane 3, torej je naravno število. √9 je enako 3, tudi naravno. -7 pa ni naravno (je negativno), vendar je celo število.
√10 je drugačen - 10 ni popoln kvadrat, torej je √10 iracionalno število. Njegove decimalne števke se nadaljujejo brez ponavljajočega vzorca.
Ko razvrščaš, vedno začni z najmanjšo možno množico. Če je število naravno, ne reci, da je "samo" realno - to ni dovolj natančno za teste!
💡 Strategija: Najprej poenostavi ulomke in korenine, šele nato določi najmanjšo množico, ki ji število pripada.
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
Join milions of students
Prikaz na številski premici
Za uspešno risanje na številsko premico si zapomni ključne približke: π ≈ 3,14, √2 ≈ 1,41, √3 ≈ 1,73. Ti bodo olajšali umeščanje iracionalnih števil.
Decimalna števila, kot je 2,5, lahko enostavno umestiš natančno na polovico med 2 in 3. Za negativna števila se premakneš v levo od ničle.
Iracionalna števila zahtevajo malce več pozornosti. √2 umestiš med 1 in 2, bližje 1,5. Za -√2 greš enako razdaljo v levo od ničle.
Če potrebuješ natančen prikaz √2, lahko uporabiš Pitagorov izrek - narišeš pravokotni trikotnik s stranicama 1 in s šestilom preneseš hipotenuzo na premico.
💡 Praktičen nasvet: Za teste si vnaprej zapomni, da je √4 = 2, √9 = 3, √16 = 4... Tako hitreje prepoznaš, kdaj je koren racionalen!
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
Join milions of students
Povzetek za hitro ponavljanje
Zapomniti si moraš hierarhijo: N ⊂ Z ⊂ Q ⊂ R. To pomeni, da je vsako naravno število tudi celo, vsako celo tudi racionalno, in vsa pa realna števila.
Racionalna in iracionalna števila skupaj tvorita realna števila , vendar se med seboj ne prekrivajo. Število je bodisi racionalno bodisi iracionalno, ne more biti oboje.
Ključ za razpoznavanje je decimalni zapis: če se konča ali ponavlja, je racionalno. Če se nadaljuje brez vzorca, je iracionalno.
Vsa realna števila lahko prikažeš na številski premici - to je njihova najpomembnejša lastnost. S tem konceptom boš delal še v gimnaziji in na fakulteti!
💡 Za izpite: Če si negotov, ali je število racionalno, poskusi ga zapisati kot ulomek. Če ne gre, je verjetno iracionalno!
We thought you’d never ask...
What is the Knowunity AI companion?
Our AI Companion is a student-focused AI tool that offers more than just answers. Built on millions of Knowunity resources, it provides relevant information, personalised study plans, quizzes, and content directly in the chat, adapting to your individual learning journey.
Where can I download the Knowunity app?
You can download the app from Google Play Store and Apple App Store.
Is Knowunity really free of charge?
That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.
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4.6/5
App Store
4.7/5
Google Play
The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan S
iOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha Klich
Android user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
Anna
iOS user
Best app on earth! no words because it’s too good
Thomas R
iOS user
Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.
Basil
Android user
This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.
Rohan U
Android user
I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.
Xander S
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE Knowunity AI. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
Paul T
iOS user
The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan S
iOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha Klich
Android user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
Anna
iOS user
Best app on earth! no words because it’s too good
Thomas R
iOS user
Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.
Basil
Android user
This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.
Rohan U
Android user
I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.
Xander S
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE Knowunity AI. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
Paul T
iOS user