The question is why when we round the number 404.4 to two significant figures we get 4.0*10^2 why not 40*10?

 

The question is why when we round the number 404.4 to two significant figures we get 4.0*10^2 why not 40*10?

What does it mean to specify a given number of significant digits. Some examples might help.

123.44 to four significant figures is 123.4.
123.45 to four significant figures is 123.5.
What does this tell us.
First, that we want to limit the number digits and second that rounding might be involved.

For both numbers above we could do this:
M=floor(N*10+0.5)/10

and
L=floor(log10(N)+1)
L=3

See if you can figure it out from there.

The min idea though is that the precision is limited to the number of digits specified, and because we end up limiting the number of digits sometimes rounding is needed as well. Rounding is sometimes needed because we dont want to simply truncate the number.
So perhaps you need to rethink your process in finding the right result.

 

From: https://www.nku.edu/~intsci/sci110/worksheets/rules_for_significant_figures.html

To determine the number of significant figures in a number use the following 3 rules:

• Non-zero digits are always significant.
• Any zeros between two significant digits are significant.
• A final zero or trailing zeros in the decimal portion ONLY are significant.

It's apparently because a "rule" says that is so. Rules do change.

Specifically in your example, 4.0*10^2 makes the zero significant as it trails the decimal. However, ask yourself whether the product of a 1-significant figure value and 3- significant figure value (for example) can have more significant figures than the value with the least number of significant figures?

From: loc. cit.

For multiplication and division use the following rule:
The LEAST number of significant figures in any number of the problem determines the number of significant figures in the answer. (You are now looking at the entire number, not just the decimal portion)

Thus, it has to be assumed that writing it as an exponent, while clumsy, avoids that problem since the exponent denotes an exact value.

 

Reminds me of my slide rule days... With the advent of calculators, doing calculations, I typically use engineering notation, E3, E6, E-9, etc. and when I do multiple calculations I may notate the step to 2 decimal places (w/ Engr Notat.). However, when doing multistep calculations, I let the calculator handle all the digits it can and then present the result on paper as a 2 decimal place value with Engineering Notation if needed. This is not what is taught as "significant digits" in school but what works for me. Especially working with values from digital instruments that can only yield 2 to 3 decimal place values. Believe me, it's a whole lots better than the slide rule was. YMMV

 

The question is why when we round the number 404.4 to two significant figures we get 4.0*10^2 why not 40*10?

Because the first form explicitly shows the number of significant digits, while the second form is ambiguous.

Consider a reported value of 4000. How many of its digits are meaningful? We don't know unless there is accompanying verbiage. Using scientific notation, we can make it explicit in the number representation itself:

4000 = 4000 * 10^0 (four sig-digs)
4000 = 400 * 10^1 (three sig-digs)
4000 = 40 * 10^2 (two sig-digs)
4000 = 4 * 10^3 (one sig-dig)