# 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.

I need the answers to my worksheet -_-

thx for this I am ready to my test tomorrow

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

Thanks

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.

Thanks alot

Thanks alot

Who is just looking down in the comments while there watching the video 😝 maybe it is just me okay 👌🏻

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

I guess you should my math teacher

your very helpful, my math teach……not so much! XD

Let's talk a little bit..rational..numbers 😀

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

4:04 Iiiiiration Iraa iratinol nubberrs nubsrs

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

والله مافهمت شي

😂😂😑

I understand EVERYTHING

Wait, so did you just say that 0.6 is an integer, because it isn’t. Don’t hate on me guys because I must have not heard correctly on that line.

Btw I’m on 7th grade honors.

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 .

I have a test on Monday…Thanks! 🙂

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

I did not understand anything

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

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

Maths hater are mad

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

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

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

Irrational numbers like twenty times😂😂

Thank u ….

But isn't pi 22/7??

This is an introduction?! He doesn’t even explain how he came up with the answers.

How many times does he say irrational

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

Khan: " Today were gonna learn about rational numbers… rational numbersss.."

Student: "Okay I get it rational num- "

Khan: " Rational numberss… rationallll numbers…"

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

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!"

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! 🌻

My teachers at school made me learn for years but still i failed in this topic

Today morning I watched this

Wow i passed!

Love you Khan Academy

Youuuuu so great!

I don't have words, I am so happy

Thanks!

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

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

is "i" from irrational number ?

I understand nothing 🙁

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

Irrational numbers is an example of a fallacie in Western mathematics

IM IN FIFTH GRADE Y'ALL

..

Would 12.85714286 be rational or irrational

this is good but there is loud and suddenly low volume

Pie is rational though pie*google/1google

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

⅝ = ¼

Your voice is very irritating

Well explained

Who are seeing comment without watching video are great mathematician😂😂just for lol😄

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

I wish i had a pause button for my teacher

Thank goodness I needed that

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

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

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

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

Thanks to teach better than my maths teacher thanks to all

Want with aim

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 🌎 one day . I’ll play an important role in making the quality of life better for all life , I won’t stop 🛑 😤

Wrong method hai

Dusare method se sikho

Nahi to complent karunga police ko

Hindi bake du kare

I wish you were my teacher ;-;

the first khan academy video that isn’t helping me:(

Super great video it helped me a lot! thanks bro.

🙂

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

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

Thank you Khan academy!!!!!

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

Anyone in college looking this video

I hate math.

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

This was mad helpful!

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

👌👌👌👌

Next question:

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

Such an ASMR voice

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

Great video,knocked me out fast.

👌👍❤️

I still don't understand🤦🏽♀️

1:54 ur math is wrong 3×4= 12

I enjoy math

I enjoy math

I love math I love math

rocks back and forth😳i love you

What software is he using to do this?

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

good teaching

What does that mean? The ratio of two intergers?

Good

Why is Phi a golden ratio?

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