I am no Euclid or Ramanujan but the mathematician in me was flabbergasted. What on earth did I just hear?!
Me: Noo, that can’t be. They are the same value, 1!
B: First of all, tell me what is the importance of scientific notation?
Me: (some shitty answer about increased readability, easier comparison blah blah)
B: Ok so there is always some amount of uncertainty in every measurement. Suppose you calculate the dimensions of an object using a vernier caliper and arrive at the value 1.41cm.
You’re completely sure of 1.4 but the 1 following 1.4 is uncertain.
How many significant digits does this value have?
Me: Three of course! (come on, that was easy peasy)
B: So in the next step you measure using screw gauge for extra precision.
The new measurement is say, 1.414cm.
Now you can be dead sure that 1.41 is correct and the 4 following the 1.41 is still uncertain.
How many significant digits are there now?
B: Right, in scientific notation it would be 1.41 x 100cm and 1.414 x 100cm.
If you were to calculate the dimensions of the same object in another world where the unit of measurement is kilometer,
1.41cm would be-?
Me : 0.00141km.
B: What would happen to the number of significant digits?
Me: It becomes six? (gaping and still not able to figure out what this has got to do with 1, 1.0 and 1.00)
B: The number of significant digits in any world should not change.
That is why we use scientific notation. So 1.41cm will be 1.41 x 10-5km, and thus the number of significant digits is the same in both cases. Three.
So now you know why 1, 1.0 and 1.00 are different.
Me: (bulb inside my head finally lit up) They have 1, 2 and 3 significant digits respectively!
Conversations with the Brother always leave me feeling intellectually impaired.