11

Significant FiguresSignificant Figures

Significant figures tell us Significant figures tell us the range of values to the range of values to expect for repeated expect for repeated measurementsmeasurements

The more significant The more significant figures there are in a figures there are in a measurement, the smaller measurement, the smaller the range of values is; the the range of values is; the more precise.more precise.

12.3 cm12.3 cmhas has 33 sig. figs. sig. figs. and its range isand its range is12.2 to 12.4 cm12.2 to 12.4 cm

12.30 cm12.30 cmhas has 44 sig. figs. sig. figs. and its range isand its range is

12.29 to 12.31 cm12.29 to 12.31 cm

22

Counting Significant FiguresCounting Significant Figures

All non-zero digits are significantAll non-zero digits are significant– 1.5 has 2 sig. figs.1.5 has 2 sig. figs.

Interior zeros are significantInterior zeros are significant– 1.05 has 3 sig. figs.1.05 has 3 sig. figs.

Trailing zerosTrailing zeros after a decimal point are after a decimal point are significantsignificant– 1.051.0500 has 4 sig. figs. has 4 sig. figs.

33

Counting Significant FiguresCounting Significant Figures

4.4. Leading zerosLeading zeros are are NOTNOT significant significant– 00..000010501050 has has 44 sig. figs. sig. figs.

1.050 x 101.050 x 10-3-3

5.5. Zeros at the end of a number are Zeros at the end of a number are significant figuressignificant figures

150 has 3 sig. figs. then 1.50 x 10150 has 3 sig. figs. then 1.50 x 1022

44

Example 2.4 – Determining the Number Example 2.4 – Determining the Number of Significant Figures in a Numberof Significant Figures in a Number

How many significant figures are in each How many significant figures are in each of the following numbers?of the following numbers?

0.00350.0035

1.0801.080

23712371

2.97 2.97 ×× 10 1055

55

Example 2.4 – Determining the Number Example 2.4 – Determining the Number of Significant Figures in a Numberof Significant Figures in a Number

How many significant figures are in each of the How many significant figures are in each of the following numbers?following numbers?

0.00350.0035 2 sig. figs. – leading zeros not sig.2 sig. figs. – leading zeros not sig.

1.0801.080 4 sig. figs. – trailing & interior zeros 4 sig. figs. – trailing & interior zeros sig.sig.

23712371 4 sig. figs. – all digits sig.4 sig. figs. – all digits sig.

2.97 2.97 ×× 10 1055 3 sig. figs. – only decimal parts 3 sig. figs. – only decimal parts count count sig.sig.

66

Multiplication and Division with Multiplication and Division with Significant FiguresSignificant Figures

When multiplying or dividing measurements When multiplying or dividing measurements with significant figures, the with significant figures, the result has the result has the same number of significant figures as the same number of significant figures as the measurement with the fewest number of measurement with the fewest number of significant figuressignificant figures; round final answer:; round final answer:

5.02 5.02 ×× 89,665 89,665 ×× 0.100.10 == 45.0118 45.0118 = = 4545 3 sig. figs.3 sig. figs. 5 sig. figs. 5 sig. figs. 2 sig. figs. 2 sig. figs.

2 sig. figs.2 sig. figs.

5.892 5.892 ÷÷ 6.106.10 = = 0.965900.96590 = = 0.9660.966 4 sig. figs. 3 sig. figs. 4 sig. figs. 3 sig. figs. 3 sig. figs. 3 sig. figs.

77

Rules for RoundingRules for RoundingWhen rounding to the correct number of When rounding to the correct number of significant figures, if the number after the significant figures, if the number after the place of the last significant figure is place of the last significant figure is

1.1. 0.0 to 0.4, round down0.0 to 0.4, round down– drop all digits after the last sig. fig. and leave drop all digits after the last sig. fig. and leave

the last sig. fig. alonethe last sig. fig. alone– add insignificant zeros to keep the value if add insignificant zeros to keep the value if

necessarynecessary

2.2. 0.5 to 0.9, round up0.5 to 0.9, round up– drop all digits after the last sig. fig. and drop all digits after the last sig. fig. and

increase the last sig. fig. by oneincrease the last sig. fig. by one– add insignificant zeros to keep the value if add insignificant zeros to keep the value if

necessarynecessary

88

RoundingRounding

Rounding to Rounding to 22 significant figures significant figures

2.34 rounds to 2.34 rounds to 2.32.3– because the 3 is where the last sig. fig. will be because the 3 is where the last sig. fig. will be

and the number after it is 4 or lessand the number after it is 4 or less

2.37 rounds to 2.37 rounds to 2.42.4– because the 3 is where the last sig. fig. will be because the 3 is where the last sig. fig. will be

and the number after it is 5 or greaterand the number after it is 5 or greater

2.349865 rounds to 2.349865 rounds to 2.32.3– because the 3 is where the last sig. fig. will be because the 3 is where the last sig. fig. will be

and the number after it is 4 or lessand the number after it is 4 or less

99

Determine the Correct Number of Determine the Correct Number of Significant Figures for each Calculation and Significant Figures for each Calculation and

Round and Report the ResultRound and Report the Result

1.1. 1.01 1.01 ×× 0.12 0.12 ×× 53.51 53.51 ÷ 96 = 0.067556÷ 96 = 0.067556

2.2. 56.55 × 0.920 ÷ 34.2585 = 1.5186356.55 × 0.920 ÷ 34.2585 = 1.51863

1010

Determine the Correct Number of Determine the Correct Number of Significant Figures for each Calculation and Significant Figures for each Calculation and

Round and Report the ResultRound and Report the Result

1.1. 1.01 1.01 ×× 0.12 0.12 ×× 53.51 53.51 ÷ 96 = 0.06÷ 96 = 0.0677556 = 556 = 0.0680.068

2.2. 56.55 × 0.920 ÷ 34.2585 = 1.556.55 × 0.920 ÷ 34.2585 = 1.511863 = 863 = 1.521.52

3 sf3 sf 2 sf2 sf 4 sf4 sf 2 sf2 sf result should result should have 2 sfhave 2 sf

7 is in place 7 is in place of last sig. fig., of last sig. fig., number after number after

is 5 or greater, is 5 or greater, so round upso round up

4 sf4 sf 3 sf3 sf 6 sf6 sf result should result should have 3 sfhave 3 sf

1 is in place 1 is in place of last sig. fig., of last sig. fig., number after number after

is 5 or greater, is 5 or greater, so round upso round up

1111

Addition and Subtraction with Addition and Subtraction with Significant FiguresSignificant Figures

When adding or subtracting measurements When adding or subtracting measurements with significant figures, the result has the with significant figures, the result has the same number ofsame number of decimal placesdecimal places as the as the measurement with the measurement with the fewest number of fewest number of decimal placesdecimal places

5.74 + 0.823 +5.74 + 0.823 + 2.6512.651= = 9.2149.214 = 9.21= 9.21 2 dec. pl.2 dec. pl. 3 dec. pl. 3 dec. pl. 3 dec. pl. 3 dec. pl. 2 dec. pl.2 dec. pl.

4.8 4.8 -- 3.965 3.965 = = 0.8350.835 = = 0.80.8 1 dec. pl 3 dec. pl. 1 dec. pl 3 dec. pl. 1 dec. pl. 1 dec. pl.

1212

Determine the Correct Number of Determine the Correct Number of Significant Figures for each Calculation and Significant Figures for each Calculation and

Round and Report the ResultRound and Report the Result

1.1. 0.987 + 125.1 – 1.220.987 + 125.1 – 1.22 = 124.867 = 124.867

2.2. 0.764 – 3.449 – 5.98 = -8.6640.764 – 3.449 – 5.98 = -8.664

1313

Determine the Correct Number of Determine the Correct Number of Significant Figures for each Calculation and Significant Figures for each Calculation and

Round and Report the ResultRound and Report the Result

1.1. 0.987 + 125.1 – 1.220.987 + 125.1 – 1.22 = 124. = 124.8867 = 67 = 124.9124.9

2.2. 0.764 – 3.449 – 5.98 = -8.60.764 – 3.449 – 5.98 = -8.6664 = 4 = -8.66-8.66

3 dp3 dp 1 dp1 dp 2 dp2 dp result should result should have 1 dphave 1 dp

8 is in place of last 8 is in place of last sig. fig., number sig. fig., number after is 5 or greater, after is 5 or greater, so round upso round up

3 dp3 dp 3 dp3 dp 2 dp2 dp result should result should have 2 dphave 2 dp

6 is in place 6 is in place of last sig. fig., of last sig. fig., number after number after

is 4 or less, is 4 or less, so round downso round down

1414

Both Multiplication/Division and Both Multiplication/Division and Addition/Subtraction with Significant Addition/Subtraction with Significant

FiguresFigures

When doing different kinds of operations When doing different kinds of operations with measurements with significant figures, with measurements with significant figures, do whatever is in parentheses first, find the do whatever is in parentheses first, find the number of significant figures in the number of significant figures in the intermediate answer, then do the remaining intermediate answer, then do the remaining stepssteps

3.489 3.489 ×× (5.67 – 2.3) = (5.67 – 2.3) = 2 dp 1 dp2 dp 1 dp

3.489 3.489 × 3.4 =× 3.4 = 12 12 4 sf 1 dp & 2 sf4 sf 1 dp & 2 sf 2 sf 2 sf