So they come on the 2nd day, and I ask them -

"Could you find the value of Square root of 3?"

People who have not yet read the previous part of this post (Part-1), then it is suggested to read that first, so as to get the context of how we landed up here and what are the mistakes, conversations & learnings that happened in that process. Here is the link

http://rupeshgesota.blogspot.com/2023/02/whats-value-of-square-root-of-3-part-1.html

"Could you find the value of Square root of 3?"

"We tried.. But we didn't get"

Means?

"1.73 squared gives less than 3 and 1.74 squared gives more than 3."

I was glad that they worked with numbers having 2 digits after the decimal point, rather than just stopping at 1.7 and 1.8. But then I also wondered how come they didn't go beyond that. So I asked them -

"So what does that mean?"

"It means, value of root 3 is between 1.73 and 1.74"

"And how do you find that now?"

Puzzled look.

------

"We don't know what to do now..."

After a while, I drew their attention to the shorter first column & told them to complete this list too - write the numbers above 1.6.... They wrote till 1.1

"So these numbers are between ?"

"1 and 2"

"How did you get these numbers between 1 and 2?"

"By dividing the range into 10 parts"

"Ok.. And how did you get the numbers 1.71, 1.72, 1.73 etc.? I am asking this because I don't see them in the first column?

"We knew its between 1.7 & 1.8 So we divided the range 1 to 2 into 100 parts now"

** I circled the two numbers 1.7 and 1.8 when they said this **and told them -

*"So can we say this second column is kind of Zoomed-In-picture between 1.7 and 1.8 ? ... numbers which were present but not visible earlier have become visible now because you have divided the range into smaller i.e more (100) parts.. Its like you have kept a magnifying glass now on the two numbers 1.7 and 1.8", while pointing my finger from the circled part of 1.7 and 1.8 towards the second column.*

I paused to help them understand this new analogy being presented now.

"So now you say that the answer, the number, is between 1.73 and 1.74. What can we do now?"

They said - "We will divide the range further - into 1000 parts now - so that the numbers between 1.73 and 1.74 become visible" while saying this he also circled the pair 1.73 and 1.74

They started enlisting from 1.731, 1.732 and so on till 1.740. So I asked them what does 1.740 represent. They said its same as 1.74. So then I told them to include another form of 1.73 too because they have circled / zoomed this number too?

So in that column, he wrote 1.730 above 1.731

How about completing the first column too this way?

He checked and wrote 1.70 above 1.71

Now I drew their attention to the two circled pairs and the list of numbers next to each pair. So that they can also actually / easily see (& not just visualize) that

(1.7, 1.8) expands to range of numbers from (1.70 to 1.80) next to it, and

(1.73, 1.74) expands to range of numbers from (1.730 to 1.740) next to it.

The picture started looking like this in some time....

The squares of 1.731, 1.732 and 1.733 were calculated by them manually using std. algorithm, but when it came to testing the squares of numbers in other columns (one with more digits after the DP), then I became their assistant and helped them getting & giving the squares of numbers which they wanted, with the help of my phone calculator.

Things had gone into auto-pilot mode now and they were sort of thrilled / enjoying this process, totally surprised as this hunting never seemed to stop, against their expectation. They said that they had never thought that square root of a number (that too such a smaller one like 3) will have so many digits :-))

I also shared with them that they don't need to enlist all the numbers in a column but can use dotted lines to indicate that. After some time, I stopped them and asked them what do they think about this process ?

"Sir, it seems this is never going to stop.... We are just reaching closer and closer to the answer...."

How do you know this?

"Square comes out to be 2.9999.... or 3.0000 and few other digits after 9 and 0 .... And then number of 9's and 0's keep increasing...."

I asked them if they can be very sure of at least some digits in the square root of 3?

They looked for a while in all the columns and noticed the growing & unchanging section of digits. As you can see above , they have written the value of square root of 3 as 1.73205_......

I asked them if some one tells square root of 3 equals 1.732 , then is it correct?

They said - **"No.. Answer is closer to 1.732, but not equal to 1.732"**

Looking at their facial expressions and body language, It was clear that this exercise was no less than an adventure ride for them :-))

So now it was time to plug-in this correct value of root-3 into the expression they had arrived at (remember the previous post? :)

'x' and 'y' in the above equations represent the respective lengths of hypotenuse and side opposite to angle measuring 60 degrees in Right Angled Triangle.

They had observed that side length opposite to angle measuring 30 is half the hypotenuse and next attempt was to find the relation between hypotenuse and side opposite to angle measuring 60.

They were told to construct two Right Triangles (with angles and hypotenuse lengths given) and they had then measured the lengths of other two sides in both triangles.

They were delighted to see that their measured lengths matched the lengths given by the formula. Further, the value of Square root 3 was also figured out by them.