Realna števila so pravzaprav vsa števila, ki si jih lahko... Show more
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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!
We thought you’d never ask...
What is the Knowunity AI companion?
Our AI companion is specifically built for the needs of students. Based on the millions of content pieces we have on the platform we can provide truly meaningful and relevant answers to students. But its not only about answers, the companion is even more about guiding students through their daily learning challenges, with personalised study plans, quizzes or content pieces in the chat and 100% personalisation based on the students skills and developments.
Where can I download the Knowunity app?
You can download the app in the Google Play Store and in the 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.
Most popular content in Matematika
Most popular content
Can't find what you're looking for? Explore other subjects.
Students love us — and so will you.
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
I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️
Thomas R
iOS user
Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades
Brad T
Android user
Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍
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
I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend
Aubrey
iOS user
Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀
Marco B
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 app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!
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
I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️
Thomas R
iOS user
Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades
Brad T
Android user
Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍
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
I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend
Aubrey
iOS user
Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀
Marco B
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 app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!
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!
Access to all documents
Improve your grades
Join milions of students
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!
Access to all documents
Improve your grades
Join milions of students
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 specifically built for the needs of students. Based on the millions of content pieces we have on the platform we can provide truly meaningful and relevant answers to students. But its not only about answers, the companion is even more about guiding students through their daily learning challenges, with personalised study plans, quizzes or content pieces in the chat and 100% personalisation based on the students skills and developments.
Where can I download the Knowunity app?
You can download the app in the Google Play Store and in the 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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Students love us — and so will you.
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
I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️
Thomas R
iOS user
Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades
Brad T
Android user
Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍
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
I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend
Aubrey
iOS user
Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀
Marco B
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 app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!
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
I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️
Thomas R
iOS user
Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades
Brad T
Android user
Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍
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
I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend
Aubrey
iOS user
Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀
Marco B
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 app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!
Paul T
iOS user