Okay, last time we talked about

how Newton’s method works in general. Now we’re gonna use it to compute √2. So suppose you’re left

in the Dark Ages, like 1970 or so,

before calculators were common. And somebody said, “Okay,

I need you to tell me what √2 is “to 10 decimal places,

and you can only do it by hand.” If you had no calculus, this is actually

not nearly that bad a problem. It may sound tough, but you can

do it in about 5 minutes. So let’s see. What we’re looking for is places— a value of x for which x^2=2. In other words,

x^2 – 2=0. So that’s our function f(x). And the derivative of that is just 2x. And we set up the table

like we talked about last time. Let’s take as our first guess

something really rough. We’re gonna guess x=1. That’s not all that close to √2,

but it’s a good enough starting point. Then f(x)=1^2 – 2,

that’s -1. And f'(x)=2. And our new and improved guess

is gonna be 1 – (-1/2), so that’s 3/2. So that’s 1.5, okay? That does a little bit better. We try 3/2. And we say okay, f(x),

(3/2)^2 is 9/4, -2 leaves 1/4. f'(x) is 2(3/2), that’s 3. And so we want (3/2) – (1/4)/(3). That winds up being 17/12.

Okay? Let’s keep going.

17/12… You square that,

and you get 289/144, which is 2 and 1/144. So we’re getting pretty good. f'(x) is 17/6. And now we wind up getting

17/12 – (1/144)/(17/6). And the arithmetic gets

a little bit ugly, but you can do it. The answer turns out to be 577/408. And yes, I did this all by hand

without a calculator. You plug in 577/408,

you square that, you subtract 2, and you wind up

getting 1/166,464. The derivative—

577/204. And you take the difference, and it winds up being

665,857 / 470,832. Okay, it takes it a little while

to crank that out. But now we’ve got something

that is actually good… good—really accurate. If we do this in decimals,

this is 1.5. Our next guess was

1.41666. That’s good to two decimal places. √2 is actually closer to 1.414. So these first two

decimal places are good, the third one’s not so good. This one turns out to be

1.4142156, and so on. That’s good to 1, 2, 3, 4, 5 decimals;

this 5 is a little bit off. This winds up being

1.414213562376. And so on. All these decimal places are right,

it’s wrong on the 11th decimal place. So Newton’s method gives you

really accurate answers. And in fact, if you’re wondering

how does your calculator actually compute √2,

this is how it does it. It uses an algorithm that’s

equivalent to Newton’s method, and it zooms in on this number. Okay, next time we’ll come

to the modern age, assume that you do have a calculator,

and we’ll solve a problem that you can’t directly

solve on a calculator, but that you can solve

with Newton’s method.

## 14 Comments

## Justin Koenig

great video

#newton #math #useful

## SoKoS

i am sorry i did it with a calc and showed me this "1.4142135623730950488016887242097"

the calc is wrong or you?b t w nice thx man +1

## Jiayang Wang

the most clear one I've found! Many thanks

## Capbium TEOI

I thought it was complicated. Thanks!

## Hunar Omar

Ok .. but how does a computer find the Derivative .. i think that there are more easier ways for a computer to find square root

## Nitish garg

cool trick ; easy to program

## Riteeka Rathod

many thanks sir 🙂

## Glen Millard

Excellent video! Thanks much for this sir 🙂

## Huize M

Did you do the calculation all by yourself?

## Huize M

But you didn't explain why you can do that

## abdul wafir

can we start with x != 1?

## viraltaco

Ok. Thank you so much. At first I was thinking "wow this is way over my abilities" then you explained it and I couldn't help but laugh because of the euphoria. This is absolutely amazing. I can't believe I didn't learn that in school.

## samuel gebru

1970 dark ages. that is pretty harsh lol

## Farhan CustomMail

Can you explain that 665857/47083. Also, can I use the same technic to find root 3?