How to calculate the root number 2 manually?

Let me introduce the continued fraction method to you, which is the most ingenious.

* * * * * * * is a placeholder, otherwise the format is chaotic.

√2

= 1+(√2- 1)

********* 1

****= 1+ -

********√2+ 1

********** 1

****= 1+ -

*********2+ √2- 1

********** 1

****= 1+ -

************* 1

*********2+ -

************√2+ 1

Please note that it appears again.

************* 1

*********** -

***********√2+ 1

So the following is repeated.

therefore

**************** 1

****= 1+ -

**************** 1

**********2+ -

******************** 1

***************2+ -

*********************2+ ....

That is, write a number as the sum of its integer part and decimal part.

Then write the decimal part as the reciprocal of its reciprocal.

You can do it indefinitely.

This number can be written as [1, 2, 2, 2, 2. . . . ]

How many digits do you want to be accurate? Just take as many 2 as you need and substitute them into the continued fraction to approximate it.

It can be proved that this is the fastest way.