Significant Figures

Significant Figures A significant figure is a measured or counted quantity that is written to show its accuracy.

Rules for determining Sig Figs • All counted quantities are exact. • All measured quantities have some degree of error. (Error from the accuracy of the measurement tool and from the reader) • All digits (numbers from 1 to 9) are significant. • Whether a zero is significant depends on the job it is doing in the number.

When are Zeros Significant? • Embedded zeros or “zeros with digit neighbors” are always counted as significant figures. If you can measure the place before it and the place after it with accuracy then it must also represent an accurate number. ie. 4067 101 20,506

When are Zeros Significant? • Trailing zeros without a decimal point are NEVER counted as significant figures. These zeros trailing behind at the end of a number are merely place holders to give you information about the size of a very rough number. Those places are not measured accurately and so are not significant. ie. 850 15,000 1,000,000 2 sig figs 2 sig figs 1 sig fig

When are Zeros Significant? • Trailing zeros followed by or after a decimal point when there is a digit in front are always counted as significant figures. These zeros are written because they have been accurately measured and provide an exact answer. ie. 4.00 100. 28.400

When are Zeros Significant? • The leading zeros in decimal numbers (<1) do not count as significant until the first actual digit. ie. 0.0005 0.045 0.00006 • Trailing zeros behind that digit are always counted as significant figures. You wouldn’t add these zeros, if you weren’t able to measure them. ie. 0.000500 0.040500 0.0040

Scientific Notation & Significant Figures When a number is written in scientific notation, all of its significant figures must be recorded. This makes sense for the previous “leading zeros” rule as those zeros disappear in scientific notation. Consider those numbers again. ie. 0.0005 0.045 0.00006 ie. 0.000500 0.040500 0.0040

Scientific Notation & Significant Figures When a number is written in scientific notation, all of its significant figures must be recorded. This makes sense for the previous “leading zeros” rule as those zeros disappear in scientific notation. Now, rewrite each in scientific notation. ie. 0.0005 0.045 0.00006 decimal moves forward 4 times 5x10-4 ie. 0.000500 0.040500 0.0040

Scientific Notation & Significant Figures When a number is written in scientific notation, all of its significant figures must be recorded. This makes sense for the previous “leading zeros” rule as those zeros disappear in scientific notation. Consider those numbers again. Rewrite each in scientific notation. ie. 0.0005 0.045 0.00006 5x10-4 4.5x10-2 6x10-5 ie. 0.000500 0.040500 0.0040 5.00x10-4 4.0500x10-2 4.0x10-3

Addition & Subtraction using Significant Figures • When adding or subtracting numbers, the correct answer should be recorded to the same place value as the least accurate number used in the equation. • Secretly known as the “Magic Ruler”. ie. 650 75.3 Add as normal. 142.75 + 300

Addition & Subtraction using Significant Figures • When adding or subtracting numbers, the correct answer should be recorded to the same place value as the least accurate number used in the equation. ie. 650 Add as normal. 75.3 142.75 Everyone got this answer?? + 300 1168.05

Addition & Subtraction using Significant Figures • When adding or subtracting numbers, the correct answer should be recorded to the same place value as the least accurate number used in the equation. ie. 650 Now take your magic ruler, place 75.3 it at the last column, move forward 142.75 until every number has a sig fig + 300 in that column. 1168.05

Addition & Subtraction using Significant Figures • When adding or subtracting numbers, the correct answer should be recorded to the same place value as the least accurate number used in the equation. ie. 650 Add as normal. 75.3 142.75 Now take your magic ruler, place + 300 it at the last column, move forward 1168.05 until every number has a sig fig

Addition & Subtraction using Significant Figures • When adding or subtracting numbers, the correct answer should be recorded to the same place value as the least accurate number used in the equation. ie. 650 This is the last significant place 75.3 for the calculated answer. 142.75 Your answer cannot be more + 300 exact than the least accurate 1168.05 number used in its solution!

Addition & Subtraction using Significant Figures • When adding or subtracting numbers, the correct answer should be recorded to the same place value as the least accurate number used in the equation. ie. 650 Use your rules for rounding 75.3 to properly record your answer. 142.75 + 300 The rule for rounding says 1168.05 = 1200 “5 or bigger, round it up”

Addition & Subtraction using Significant Figures Sometimes your answer has more significant figures than any of the numbers used 85.6 3 sig figs 7.25 3 sig figs 95.7 3 sig figs + 0.5 1 sig fig 189.05 = 189.1 4 sig figs

Addition & Subtraction using Significant Figures Sometimes your answer has less 15.6 3 sig figs - 7.27 3 sig figs 8.33 = 8.3 2 sig figs The rule for rounding says “less than 5, just throw it away”

Multiplication & Division using Significant Figures • When multiplying or dividing numbers, the answer should be written to the same number of significant figures as the least precise value used in the calculation. • This rule includes squaring, roots, and log functions. ie. (3.25)(8.6) = 42 = 21.00 = 3 √125 = 3.00

Multiplication & Division using Significant Figures • When multiplying or dividing numbers, the answer should be written to the same number of significant figures as the least precise value used in the calculation. • This rule includes squaring, roots, and log functions. ie. (3.25)(8.6) = 27.95 ≈ 28 42 = 16 ≈ 20 21.00 = 7.003 √125 = 5.00 3.00

Significant Figures