Physics in English: Significant digits and rounding off

The significant digits represent the valid digits of a number. The following rules summarize the significant digits:

1. Nonzero digits are always significant.
2. All final zeros after the decimal points are significant.
3. Zeros between two other significant digits are always significant.
4. Zeros used solely for spacing the decimal point are not significant.

For example, the number of significant digits for the value 5,6 is two. The number of significant digits for the value 0,0017495 is five.

Exercises

1. What is the number of significant digits in 650046746830?
2. How many zeros are significant figures ina measured mass of 0,010010 g?
3. What is the number of significant digits in 0,00230300 m?

When the answer to a calculation contains too many significant figures, it must be rounded off.

There are 10 digits that can occur in the last decimal place in a calculation. One way of rounding off involves underestimating the answer for five of these digits (0, 1, 2, 3, and 4) and overestimating the answer for the other five (5, 6, 7, 8, and 9). This approach to rounding off is summarized as follows.

Rule 1. If the digit is smaller than 5, drop this digit and leave the remaining number unchanged.
Thus, 1.684 becomes 1.68.

Rule 2. If the digit is 5 or larger, drop this digit and add 1 to the preceding digit. Thus, 1.247 becomes 1.25.

In addition and subtraction, round up your answer to the least precise measurement. For example:

24.686+2.343+3.21=30.239≈30.24

$24.686+2.343+3.21=30.239\approx 30.2$

because 3.21 is the least precise measurement.

When measurements are multiplied or divided, the answer can contain no more significant figures than the least accurate measurement. For example:

1.435×7.23=10.37505≈10.4

$1.435\times 7.23=10.37505\approx 10.4$

because 7,23 is the least precise measurement.

In a problem with the mixture of addition, subtraction, multiplication or division, round up your answer at the end, not in the middle of your calculation. For example:

3.6×0.3+2.1=1.08+2.1≈3.2

$3.6\times 0.3+2.1=1.08+2.1\approx 3.2$

Exercise. Round off the answers of the following calculations.

1. 37.76 + 3.907 + 226.4 = ?
2. 319.15 – 32.614 = ?
3. 104.630 + 27.08362 + 0.61 = ?
4. 125 – 0.23 + 4.109 = ?
5. 2.02 × 2.5 = ?
6. 600.0 / 5.2302 = ?
7. 0.0032 × 273 = ?
8. 0.556 × (40 – 32.5) = ?

[Answers. 1) 268,1  2) 286,54  3) 132,32  4) 129  5) 5,0  6) 114,7  7) 0,87  8) 4]