How scientific measurements should be recorded and used. Significant Figures How scientific measurements should be recorded and used.

5.1 cm 2 significant figures

How to count significant figures The number of significant figures is determined by counting the first nonzero digit and all following digits in a number with a decimal. 1.34 has 3 sig figs In the United States use a period (.) not a comma (,) for a decimal point.

Estimate 5.11 cm for 3 sig figs

Internal zeroes Internal zeros (zeros between nonzero digits) are always significant. 4.0067 has 5 sig figs. 1203. has 4 sig figs.

A little shorter could be 5 A little shorter could be 5.09 cm The internal zero would be significant

How could we get another sig fig? We might get a better scale with scale markings between 5.1 and 5.2 so there are 10 divisions 5.11, 5.12, 5.13, etc. and pick the closest line again and estimate a 4th sig fig.

What if we changed the units? 5.11 cm = ? Km 5.11 cm = 5.11 cm 5.11 cm = 0.0000511 km How many sig figs is this?

Initial zeroes Zeros before the first nonzero digit are never significant. 0.0000511 km has 3 sig figs The measurement was 3 sig figs. Changing the units does not make the measurement better.

Final zeroes Final zeros in a decimal number are always significant. 4.500 has 4 sig figs. 0.0120 has 3 sig figs

Final zeroes when NO decimal Numbers without decimals may be: Exact: 1 mile = 5280 feet (by definition) Countable: 1 ream = 500 sheets Approximate: The distance to the sun is 93,000,000 miles Exact and countable numbers should be written without decimals.

Exponential Notation Scientific Notation 93,000,000 miles could be written as 93.x106 miles if it is meant to convey 2 sig figs Scientific Notation 93,000,000 miles could be written as 9.3x107 miles if it is meant to convey 2 sig figs or 9.30x107 miles if it is meant to convey 3 sig figs

Calculate the area of a rectangle 5. cm x 80.2 cm The area is 80.2 cm x 5. cm 401.0 cm² Is how many significant figures?

In multiplication and division In multiplication and division the calculated answer should have no more significant figures than the measurement with the least significant figures. 0.044134.56 = 5.92064  5.9 (2 sig figs) 134.56/0.044 = 3058.18  3.1x103 not 3100 5. cm x 80.2 cm = 401.0 cm² → 4.x10² cm²

Calculate the area of a rectangle 5. cm x 80.2 cm What we really want to know is how many boxes there are 1 cm x 1 cm

Calculate the area of a rectangle 5. cm x 80.2 cm To be 5. cm to 1 sig fig means it is between 4.5 and 5.5 To be 80.2 cm long to 3 sig figs means it is between 80.15 and 80.25 cm long.

Calculate the area of a rectangle 5. cm x 80.2 cm By error analysis the maximum possible area is: 5.5 cm x 80.25 cm = 441.375 cm² 4.5 cm x 80.15 cm = 360.675 cm² is the minimum Or 401. ± 40.375

Calculate the area of a rectangle 5. cm x 80.2 cm Area is 401. ± 40.375 cm² Because of the 1 sig fig width the number of cm² boxes could be as much as 40 more or 40 less than 401 cm². That would leave the answer good to only 1 sig fig.

Calculate the length of the perimeter of the rectangle 5. cm 80.2 cm This would be 5. cm 5. cm +80.2 cm 170.4 cm How many sig figs?

In addition and subtraction In addition and subtraction the calculated answer should carry as many decimal places (not significant figures) as the measurement with the fewest decimal places. 45.1145 + 2.36 47.4745 → 47.47 71.2 - 2.3618 68.8382 → 68.8

Calculate the length of the perimeter of the rectangle 5. cm 80.2 cm This would be 5. cm 5. cm +80.2 cm 170.4 cm → 170. cm

Again using error analysis, the perimeter is: 5.5 4.5 80.25 80.15 5.5 4.5 171.5 max 169.3 min or 170.4 ± 1.1 cm So it is surely not good to the nearest 0.1 cm. But it is nearly good to 3 sig figs. Significant figures approximate a more careful error analysis without nearly as much effort.

Round Numbers Correctly Round the answer to the nearest number having the correct number of significant figures. If the leftmost digit being removed is less than 5, drop this digit and all following digits. If the leftmost digit being removed is 5 or greater, drop this digit and all following digits AND increase that rightmost digit being retained by one unit.

Rounding Example 457.26451 is 457.2645 to 7 sig figs 4.6x102 to 2 sig figs

Round at the end of a calculation Calculate an answer to a problem using all given significant figures, then round the final answer to the correct significant figures. Consider each step along the way to determine correct significant figures, but do not round at each step along the way. (4.65 ─ 2.2)×3.51 = 8.5995 → 8.6