# What are rounding and significant figures?

Mar 26, 2014

WARNING: This is a long answer. It gives all the rules and many examples.

Significant figures are the digits used to represent a measured number. Only the digit farthest to the right is uncertain. The digit farthest to the right has some error in its value but is still significant.

Exact numbers have a value that is exactly known. There is no error or uncertainty in the value of an exact number. You can think of exact numbers as having an infinite number of significant figures.

Examples are numbers obtained by counting individual objects and defined numbers (e.g., there are 10 cm in 1 m) are exact.

Measured numbers have a value that is NOT exactly known due to the measuring process. The amount of uncertainty depends on the precision of the measuring device.

Examples are numbers obtained by measuring an object with some measuring device.

RULES FOR COUNTING SIGNIFICANT FIGURES:

1. Non-zero digits are always significant.
2. All zeroes between other significant digits are significant.
3. Leading zeroes are not significant.
4. Trailing zeroes are significant only if they come after a decimal point and have significant figures to the left.

Examples:

1. How many significant digits are in 0.077?
2. How many significant digits are in a measurement of 206 cm? Answer: Three. The zero is significant because it is between two significant figures. Trailing zeroes are significant only if they come after a decimal point and have significant figures to the left.
3. How many significant digits are in a measurement of 206.0 °C? Answer: Four. The first zero is significant because it is between two significant figures. The trailing zero is significant because it comes after a decimal point and has significant figures to its left.

Rounding means reducing the number of digits in a number according to certain rules.

RULES FOR ROUNDING:

1. When adding or subtracting numbers, find the number that is known to the fewest decimal places. Then round the result to that decimal place.
2. When multiplying or dividing numbers, find the number with the fewest significant figures. Then round the result to that many significant figures.
3. If either the unrounded result or the result rounded according to Rule 2 has 1 as its leading significant digit, and none of the operands has 1 as the leading significant digit, keep an extra significant figure in the result while making sure that the leading digit remains 1.
4. When squaring a number or taking its square root, count the number's significant figures. Then we round the result to that many significant figures.
5. If either the unrounded result or the result rounded according to Rule 4 has 1 as its leading significant digit, and the operand's leading significant digit isn't 1, keep an extra significant figure in the result.
6. Numbers obtained by counting and defined numbers have an infinite number of significant figures.
7. In order to avoid "round off error" during multistep calculations, keep an extra significant figure for intermediate results. Then round properly when you reach the final result.

EXAMPLES:

Round the answers to the correct number of significant figures:

1. 21.398 + 405 - 2.9; Answer = $423$. The 405 is known only to the ones place. Rule 1 says the result must be rounded to the ones place.
2. (0.0496 × 32.0)/478.8. Answer = $0.003 32$. Both 0.0496 and 32.0 are known to only three significant figures. Rule 2 says the result must be rounded to three significant figures.
3. 3.7 × 2.8; Answer = $10.4$. Following Rule 2 would give us 10. as our result. This is precise to only 1 part in 10. This is substantially less precise than either of the two operands. We err instead on the side of extra precision and write 10.4.
4. 3.7 × 2.8 × 1.6; Answer = $17$. This time, the 1.6 is known only to 1 part in 16, so the result should be rounded to 17 rather than 16.6.
5. 38 × 5.22; Answer = $198$. Rule 2 would give us 2.0 x 10² but, since the unrounded result is 198.36, Rule 3 says to keep an extra significant figure.
6. $\frac{7.81}{80}$. Answer = $0.10$. The 80 has one significant figure. Rule 2 says to round 0.097 625 to 0.1, at which point Rule 3 tells us to keep a second significant figure.
Writing 0.098 would imply an uncertainty of 1 part in 98. This is much too optimistic, since the 80 is uncertain by 1 part in 8. So we keep 1 as the leading digit and write 0.10.
7. (5.8)²; Answer = $34$. The 5.8 is known to two significant figures, so Rule 4 says the result must be rounded to two significant figures.
8. (3.9)²; Answer = $15.2$. Rule 4 predicts an answer of 15. The leading digit of 15 is 1, but the leading digit of 3.9 isn't 1. Rule 5 says we should keep an extra significant figure in the result.
9. √0.0144; Answer = $0.120$. The number 0.0144 has three significant figures. Rule 4 says the answer should have the same number of significant figures.
10. (40)²; Answer = 1.6 × 10³. The number 40 has one significant figure. Rule 4 would yield 2 x 10³, but the unrounded result has 1 as its leading digit, so Rule 5 says to keep an extra significant figure.
11. If ten marbles together have a mass of 265.7 g, what is the average mass per marble? Answer = $\frac{265.7 g}{10}$ = 26.57 g. The 10 has an infinite number of significant figures, so Rule 6 says the answer has four significant figures.
12. Calculate the circumference of a circle with measured radius 2.86 m. Answer: C = 2πr = 2 × π × 2.86 m = 17.97 m. The 2 is exact, and your calculator stores the value of π to many significant figures, so we invoke Rule 3 to obtain a result with four significant figures.