# Introduction to rational and irrational numbers | Algebra I | Khan Academy

So let’s talk a little bit
about rational numbers. And the simple way to think
about it is any number that can be represented as
the ratio of two integers is a rational number. So for example, any integer
is a rational number. 1 can be represented as 1/1 or
as negative 2 over negative 2 or as 10,000/10,000. In all of these cases, these are
all different representations of the number 1,
ratio of two integers. And I obviously can
have an infinite number of representations
of 1 in this way, the same number over
the same number. The number negative 7 could be
represented as negative 7/1, or 7 over negative 1, or
negative 14 over positive 2. And I could go on, and
on, and on, and on. So negative 7 is definitely
a rational number. It can be represented as
the ratio of two integers. But what about things
that are not integers? For example, let us imagine–
oh, I don’t know– 3.75. How can we represent that as
the ratio of two integers? Well, 3.75, you
could rewrite that as 375/100, which is the
same thing as 750/200. Or you could say, hey,
3.75 is the same thing as 3 and 3/4– so let
me write it here– which is the same
thing as– that’s 15/4. 4 times 3 is 12, plus 3 is
15, so you could write this. This is the same thing as 15/4. Or we could write this as
negative 30 over negative 8. I just multiplied the
numerator and the denominator here by negative 2. But just to be clear,
this is clearly rational. I’m giving you multiple
examples of how this can be represented as
the ratio of two integers. Now, what about
repeating decimals? Well, let’s take
maybe the most famous of the repeating decimals. Let’s say you have 0.333, just
keeps going on and on forever, which we can denote by
putting that little bar on top of the 3. This is 0.3 repeating. And we’ve seen–
and later we’ll show how you can convert
any repeating decimal as the ratio of two integers–
this is clearly 1/3. Or maybe you’ve seen things like
0.6 repeating, which is 2/3. And there’s many, many,
many other examples of this. And we’ll see any
repeating decimal, not just one digit repeating. Even if it has a million
digits repeating, as long as the pattern
starts to repeat itself over and over and
over again, you can always represent that as
the ratio of two integers. So I know what you’re
probably thinking. Hey, Sal, you’ve
just included a lot. You’ve included all
of the integers. You’ve included all of finite
non-repeating decimals, and you’ve also included
repeating decimals. What is left? Are there any numbers
that are not rational? And you’re probably
guessing that there are, otherwise people
wouldn’t have taken the trouble of trying to
label these as rational. And it turns out– as you
can imagine– that actually some of the most famous
numbers in all of mathematics are not rational. And we call these numbers
irrational numbers. And I’ve listed there
just a few of the most noteworthy examples. Pi– the ratio of
the circumference to the diameter of a circle–
is an irrational number. It never terminates. It goes on and on and on
forever, and it never repeats. e, same thing– never
terminates, never repeats. It comes out of continuously
compounding interest. It comes out of
complex analysis. e shows up all over the place. Square root of 2,
irrational number. Phi, the golden ratio,
irrational number. So these things that
really just pop out of nature, many of these
numbers are irrational. Now, you might say, OK,
are these irrational? These are just these
special kind of numbers. But maybe most
numbers are rational, and Sal’s just picked out
some special cases here. But the important thing to
realize is they do seem exotic, and they are exotic
in certain ways. But they aren’t uncommon. It actually turns out
that there is always an irrational number between
any two rational numbers. Well, we could go on and on. There’s actually
an infinite number. But there’s at least one,
so that gives you an idea that you can’t
really say that there are fewer irrational numbers
than rational numbers. And in a future
video, we’ll prove that you give me two rational
numbers– rational 1, rational 2– there’s going to be
at least one irrational number between those, which
is a neat result, because irrational
numbers seem to be exotic. Another way to think about it–
I took the square root of 2, but you take the square root
of any non-perfect square, you’re going to end up
with an irrational number. You take the sum
of an irrational and a rational number– and
we’ll see this later on. We’ll prove it to ourselves. The sum of an irrational
and a rational is going to be irrational. The product of an
irrational and a rational is going to be irrational. So there’s a lot, a lot, a
lot of irrational numbers out there.

## 100 thoughts on “Introduction to rational and irrational numbers | Algebra I | Khan Academy”

1. gay says:

I need the answers to my worksheet -_-

2. Miguel Hernandez says:

thx for this I am ready to my test tomorrow

3. Angela Redford says:

great

You said that 3 bar goes on and its rational but the number with a square root also goes on so what's the difference if both rational and irrational numbers goes on

5. Priya Cute says:

Thanks

6. Jasper Saures says:

I hate how alot of teachers teach math and it's so hard to understand them, but this is just easy to understand thanks so much.

7. Fareed Hassan says:

Thanks alot

8. Fareed Hassan says:

Thanks alot

9. Y 15 says:

Who is just looking down in the comments while there watching the video ๐ maybe it is just me okay ๐๐ป

10. Paige Chicklo says:

I am wondering….can you make up an irrational number, or do they exist on their own?

11. Vansh Chojar says:

I guess you should my math teacher

12. Pusheen UwU says:

13. Yaneth Acosta says:

Let's talk a little bit..rational..numbers ๐

14. Stirling Rutty says:

Are there any irrational numbers that contain more of one digit than another, i.e. the distribution of the ten digits is not uniform?

15. Stella Alexandropoulos says:

4:04 Iiiiiration Iraa iratinol nubberrs nubsrs

16. Anees Ur Rehman says:

0.3131131131111… Is a rational or irrational number ? Sir plzz give this answer

17. Abdullah Ballaith says:

ูุงููู ูุงูููุช ุดู

18. Abdullah Ballaith says:

๐๐๐

19. Nathalia Dereck Dawson says:

I understand EVERYTHING

20. Muhammed Kamruzzaman says:

Wait, so did you just say that 0.6 is an integer, because it isnโt. Donโt hate on me guys because I must have not heard correctly on that line.

Btw Iโm on 7th grade honors.

21. petit man says:

Just leave

Anyone think this dude is the smartest person in the world like whenever I go on yt to get math help his video always pops up first if this dude was my teacher I would the straight a's every test .

23. Allrhen Cerillo says:

I have a test on Monday…Thanks! ๐

24. gut viol says:

Why have you written pi as 3.14159265359 ??!!!! Pi to the same number of places has an 8 before the last nine!!!!

25. Hamda Alqallaf says:

I did not understand anything

26. Fatima Ajmal says:

3:50 he said irrational too times , btw we can understand it by one single time

27. DJY djy says:

Just saying that you got the golden ratio wrong. You wrote 1.61803399… but it is 1.6180339887…

28. MandG MUSIC AND GAME says:

29. salsa chips says:

w h a t t t t t t t, so confused ;-;

30. Milton Zhou says:

at 3:47, he wrote pi = 3.14159265359…. but it is really 3.1415926535 89 79……. Though he probably did it to round it.

31. WhatThe StopMotion says:

Your not explaining why you're just say oh this can do this

32. Nontlahla Dubeko says:

Irrational numbers like twenty times๐๐

33. Josjas Anitharaj says:

Thank u ….
But isn't pi 22/7??

34. Melissa Allred says:

This is an introduction?! He doesnโt even explain how he came up with the answers.

35. KingWolf says:

How many times does he say irrational

36. Olivia Hall says:

pre-calc got me messed up but this helped me a lot.

37. Zay Zay says:

Student: "Okay I get it rational num- "

Khan: " Rational numberss… rationallll numbers…"

38. AvengerXP says:

You say "integrity" or "integral" why would you soften the g in integer as intejer?

39. Makoto Tachibana says:

I felt like I was watching a mystery drama when he said, "are there even any irrational numbers!?", then he slid the page to reveal some of the most widely known numbers of all time. I was like, "oh snap. It was Pi! It was Pi this whole time!"

40. Lynx th3wuf says:

thank you so much! this helped a ton. your voice is so soothing to listen to and you're simple and straight to the point. I like it! ๐ป

41. Ghufran Ahmed says:

My teachers at school made me learn for years but still i failed in this topic
Today morning I watched this
Wow i passed!
Youuuuu so great!
I don't have words, I am so happy
Thanks!

42. Sugar Free says:

Somehow irrational people are not as easy to explain or easy to accept. But 3.14 is. I'm not so sure I like that

43. *nut* says:

When am I ever gonna do this as an adult? Being happy and financially stable is my goal in life

44. Elf Nanno says:

is "i" from irrational number ?

45. benor says:

I understand nothing ๐

46. Neena Rathore says:

my son is in 8th grade and doing year 9 as well as year 10 thank you khan academy

47. chris someone says:

Irrational numbers is an example of a fallacie in Western mathematics

48. Ethan Griffeth says:

49. Ranbir Kapoor says:

..

50. Golden Customs says:

Would 12.85714286 be rational or irrational

51. amanthindhere says:

this is good but there is loud and suddenly low volume

53. minion crew says:

I feel like it was a bit over complicated in this video. Honest review

54. Cynthia Mauto says:

โ = ยผ

55. Ajay Shahi says:

56. Tariq Ali says:

Well explained

57. dhinakaran rmd says:

Who are seeing comment without watching video are great mathematician๐๐just for lol๐

58. smithavijay332003 says:

Awesome it became so easy for me and helped mea a lot thanks a lot

59. Courtney Wright says:

I wish i had a pause button for my teacher

60. Natalie Chamboko says:

Thank goodness I needed that

61. glelves says:

7th grade here we will have our exam on monday good luck to me

62. TheFhdude says:

How do we know that irrational numbers never repeat? May be they repeat after 100 million digits.

63. Christi Potgieter says:

Guys!! Sal said "irrational numbers" like four times in a row(4:00). Great video, BTW

64. pan pan says:

Violets are blue like buttons are to someone just hit my like button WAIT is it blue?

65. Lilly Therese says:

Thanks to teach better than my maths teacher thanks to all

66. Chand Shak says:

Want with aim

67. Praise Jah says:

Thank you Khan for the hashtag, I was put into remedial classes recently and i started to believe Natural Selection, I felt my existence was simply to be inferior by nature . Telling me that gives me just confidence and motivation to work harder and catch up , I would like to make a difference in the world ๐ one day . Iโll play an important role in making the quality of life better for all life , I wonโt stop ๐ ๐ค

68. Jatin Tigaya says:

Wrong method hai

69. Jatin Tigaya says:

Dusare method se sikho

70. Jatin Tigaya says:

Nahi to complent karunga police ko

Hindi bake du kare

72. L โข Lawliet says:

I wish you were my teacher ;-;

73. Lana_qaffaf lana says:

the first khan academy video that isnโt helping me:(

74. Kaliyah O says:

Super great video it helped me a lot! thanks bro.
๐

75. Freddy Jenkins says:

GOOD LUCK WITH YOUR HW/ TESTS. KNOW THAT YOU ARE LOVED AND YOU CAN DO/BE ANYTHING.

76. umar farooq says:

We can represent pi by 22/7 then why pi is irrational?

77. Riya Rohra says:

78. Richard Omelda says:

"Phi, the golden ratio, irRATIOnal number!."

79. Jamila says:

Anyone in college looking this video

80. sp0kd says:

I hate math.

81. Aseel 83 says:

Just 5 minutes, I have understand. My teacher is trying to explain this for 45 minutes and I didn't understand any word:)

82. Joselyn Rodezno says:

83. Aliyah Tate says:

Thanks I am really stuck on homework and this will be on my quiz this friday.you explained it way better than my teacher

84. Milan Baram says:

๐๐๐๐

85. acoow says:

Next question:

Why is it important to know what an irrational number is?

86. Kyle Weichert says:

Such an ASMR voice

87. Mr Fiest says:

what happens if you add an rational number to a irational number

88. Kirk Just Kirk says:

Great video,knocked me out fast.

89. Green JellyfishYT says:

๐๐โค๏ธ

90. Audio Stories says:

I still don't understand๐คฆ๐ฝโโ๏ธ

91. Salty Jv says:

1:54 ur math is wrong 3×4= 12

92. Natia N.N says:

I enjoy math
I enjoy math
I love math I love math rocks back and forth ๐ณ

93. Leen Tv says:

i love you

94. Math Help by Dan says:

What software is he using to do this?

95. Vasu Bhatt says:

but why pie is not rational ? i mean we can just devide the no. with 1????

96. Zahra Mahdi says:

good teaching

What does that mean? The ratio of two intergers?

98. Ahmed. ahmed says:

Good

99. Dorothy Plummer says:

Why is Phi a golden ratio?

100. Lillian Cuiๅดไพๅฉท says:

This man taught me more in five minutes than my math teacher does in an hour.