0.7 Phy1040: scientific notation and significant figures  (Page 4/8)

• Leading zeros where they serve merely as placeholders to indicate the scale of the number (.00356 for example).
• Spurious digits introduced, for example, by calculations carried out to greater accuracy than that of the original data, or measurements reported to a greater precision than the equipment supports.

A popular physics textbook provides a more complete set of rules for identifying the significant figures in a number:

1. Nonzero digits are always significant.
2. Final or ending zeros written to the right of the decimal point are significant.
3. Zeros written to the right of the decimal point for the purpose of spacing the decimal point are not significant.
4. Zeros written to the left of the decimal point may be significant, or they may only be there to space the decimal point. For example, 200 cm could have one,two, or three significant figures; it's not clear whether the distance was measured to the nearest 1 cm, to the nearest 10 cm, or to the nearest 100 cm. Onthe other hand, 200.0 cm has four significant figures (see rule 5). Rewriting the number in scientific notation is one way to remove the ambiguity.
5. Zeros written between significant figures are significant.

Ambiguity of the last digit in scientific notation

Again, according to Wikipedia , it is customary in scientific measurements to record all the significant digits from the measurements, and to guess one additional digit if there is any information at all available to the observer to make a guess. The resulting number is considered more valuable than it would be without that extra digit, and it is considered a significant digit because it contains some information leading to greater precision in measurements and in aggregations of measurements (adding them or multiplying them together).

Examples of significant digits

Referring back to the physics textbook mentioned earlier, Figure 1 shows:

• Four different numbers
• The number of significant figures in each number.
• The default JavaScript exponential representation of each number.

Note that the default JavaScript exponential representation fails to display the significant trailing zeros for the numbers on row 2 and row 5. Iwill show you some ways that you may be able to deal with this issue later but you may not find them to be very straightforward.

Discussion and sample code

Beyond knowing about scientific notation and significant figures from a formatting viewpoint, you need to know how to perform arithmetic whilesatisfying the rules for scientific notation and significant figures.

Performing arithmetic involves three main rules :

1. For addition and subtraction, the final result should be rounded so as to have the same number of decimal places as the number that was included inthe sum that has the smallest number of decimal places. In accordance with the discussion early in this module, this is the least precise number.
2. For multiplication and division, the final result should be rounded to have the same number of significant figures as the number that was includedin the product with the smallest number of significant figures.
3. The two rules listed above should not be applied to intermediate calculations. For intermediate calculations, keep as many digits aspractical. Round to the correct number of significant figures or the correct number of decimal places in the final result.