Irrational Numbers - Math Antics Extras

0:02 Have you ever thought that irrational numbers

0:04 are called irrational because they’re insane?

0:07 Well they’re not. They’re actually really cool

0:09 and really amazing.

0:10 To understand what an irrational number is,

0:13 it helps to know what a rational number is.

0:15 A rational number is any number that can

0:17 be written as a ratio of two integers.

0:20 Does that sound familiar? Yup, a ratio of two integers

0:24 is a fraction, like 1/2 or 2/3.

0:27 So any number that can be written

0:29 as a fraction is a rational number.

0:31 But can’t all numbers be written as a fraction?

0:34 Surprisingly the answer is no.

0:37 Sure, any whole number can be written as a

0:39 fraction by giving it 1 as the denominator.

0:41 And any decimal number that you can write,

0:43 can be written as a fraction by having the

0:45 proper power of 10 as the denominator.

0:48 So all of these are rational numbers.

0:50 And there are a lot more… infinitely more.

0:53 Any number that can be accurately written

0:55 as a fraction or a decimal number is a rational number.

0:59 But there are numbers that can’t be written

1:01 accurately as a fraction, or even as a decimal number.

1:04 Those numbers are called irrational numbers

1:06 because they are not rational numbers.

1:09 One of the most famous irrational numbers is pi.

1:12 And you might be confused because you may

1:13 have heard that you can represent pi with a

1:15 fraction like 22/7 or 355/113, but those are just

1:21 approximations for pi. Which means they get close

1:23 to the value of pi but they are not exactly equal.

1:26 You can’t get the exact value of pi with a fraction

1:29 no matter what numbers you use.

1:31 And you’ve probably seen pi written as a decimal number

1:34 like 3.14 or 3.14159 but those aren’t completely accurate

1:40 values of pi either. Again, they’re approximations.

1:44 If you tried to write a completely accurate decimal number

1:47 for pi or any other irrational number, the decimal digits

1:50 would never end and won’t repeat.

1:53 The statement that the decimal digits would never end

1:56 might be hard to believe, but it’s true

1:59 they just keep going and going—forever.

2:02 But the statement that the decimal digits won’t repeat

2:04 might be hard to understand.

2:06 It might even make you think something that’s not true.

2:10 You might think it means that you could never have 2

2:12 of the same digit next to each other.

2:14 Or you might think that a particular pattern of digits,

2:16 like 123, won’t appear more than once.

2:19 Or that it at least won’t be followed immediately

2:22 by the same pattern. But that’s not what it means.

2:25 To see that, let’s take a look at the first 500 digits of pi.

2:29 There are many places where there are 2 or more of

2:31 the same digit next to each other.

2:33 And here’s the pattern 360 and then a little further along

2:36 we have 360 again. And look at this, here we have

2:40 the digits 209 followed immediately by 209!

2:44 So if that’s not what’s meant by the

2:47 decimal digits not repeating then what is?

2:50 Well, remember that fraction that’s

2:51 often used to approximate pi? 22/7?

2:55 Let’s look at the first 500 digits of that number.

2:58 Notice how the sequence ‘142857’ keeps repeating?

3:03 Well, it keeps on going like this,

3:04 repeating that same sequence forever.

3:07 The digits of a rational number either end

3:09 or keep repeating like this.

3:11 And because of that, you can know exactly what all

3:13 of those decimal digits are, even when they go on forever.

3:17 Since this sequence of 6 digits repeats, you already know

3:21 what the 501st digit is going to be: ’4’.

3:24 But for pi, you don’t know what the 501st digit

3:27 is going to be until you calculate it.

3:29 So one way to understand the difference between

3:31 rational and irrational numbers is that the digits of a

3:34 rational number can be completely known,

3:36 but an irrational number always has

3:38 more digits that you don’t know…

3:40 they’re kind of mysterious that way!

3:42 There’s another cool thing about

3:44 irrational numbers that I want to show you.

3:46 In our video lesson about the Number Line we learned

3:48 that you can subdivide the space between

3:50 two consecutive marks on the number line.

3:52 And you can subdivide the smaller space between

3:54 two of those new marks.

3:56 And you can keep doing that, forever!

3:59 So let’s look at the value of pi on the number line.

4:01 It’s right about here, between the 3 and the 4.

4:04 And now let's zoom in. As we zoom in we keep

4:07 dividing the spaces into smaller and smaller spaces.

4:10 It might seem that if we kept zooming in,

4:13 eventually the value of pi would line up

4:15 exactly with one of our marks. But it won’t.

4:18 Since pi is an irrational number,

4:20 even if we could zoom in forever,

4:22 we would never get to a point where pi exactly lines up

4:24 with a mark on the number line.

4:26 It might get really close, but as we keep zooming in

4:29 we’ll see that it doesn’t quite line up and it never will.

4:33 Pretty cool, huh?

4:35 There’s one last thing I want to tell you

4:36 about irrational numbers.

4:38 It might seem like there are just a few special numbers

4:40 that happen to be irrational, but in reality,

4:43 there are infinitely many irrational numbers.

4:46 In fact there are more irrational numbers

4:48 than there are rational numbers!

4:50 I don’t know about you, but I find that amazing!!

4:54 Alright, hopefully you now understand

4:56 irrational numbers a bit better.

4:58 And you realize they aren’t called irrational

5:00 because they’re insane, they’re called

5:01 irrational because they are not rational.

5:04 They can’t be represented as a ratio of two integers.

5:07 So they can’t be accurately represented as a fraction.

5:10 And their decimal digits go on forever

5:12 and don’t end in a repeating pattern.

5:14 Alright, that’s it for this video.

5:16 Keep practicing and I’ll catch you in the next one.

5:19 And for better access to all of our material,

5:21 check out mathantics.com