This document discusses significant digits and how to properly express calculated numbers using them. It defines significant figures as how well a measurement was taken, and significant digits as how to express a calculated answer accounting for significant figures. The key rules are: - When multiplying or dividing, the answer should have the same number of significant digits as the factor with the least significant digits. - When adding or subtracting, the answer should have the same number of decimal places as the number with the least decimal places. - Trailing zeros and zeros between numbers are significant, leading zeros are placeholders. The document provides examples of applying these rules to calculated problems and expressions.

1. Everything you wanted to know
about Significant Digits…
… but were afraid to ask

2. What is a good way to express
calculated numbers?
 Are these?
 2 x 3.00 = 6.00
 14.00000 ÷ 2.1 = 6.66666
 How about these?
 2 x 3.00 = 6
 14.00000 ÷ 2.1 = 6.7

3. Calculated answers need to have a balance
between accuracy and precision to account for
the QUALITY of measure
 Accuracy: In science, engineering, industry and statistics,
accuracy is the degree of conformity of a measured or
calculated quantity to its actual, nominal, absolute, or some
other reference, value…It’s “correctness”
 Precision: The precision of a measurement or value
describes the number of digits that are used to express that
value. This might be the total number of digits (sometimes
called the significant digits)…How well was it measured

4. SOME clarification
 Significant Figure (sig-fig): How well a
measurement was taken
 Significant Digit (sig-dig): How well we
can express a calculated answer to account
for sig-figs
 Scientific notation: A base 10 method of
expressing large answers

5. Significant Figure
 A way to express the quality of a measured
value
Rule of thumb: Estimate one and only one
number beyond what you can read from the
instrument

6. Significant digit
 A way to express the quality of measured
value in a calculated answer using a balance
of precision and accuracy

7. Scientific Notation
1) One and only one integer left of the decimal
2) When converting from a number larger than one-move
decimal left and express exponent positively
3) When converting from a number less than one-move
decimal right and express exponent negatively

8. Rules for determining how many
significant digits exist in a
number
1) All Integers count
2) Zeros contained between integers count
3) All trailing zeros count as along as a decimal point is
expressed in the number. If there is no decimal, then
the trailing zeros are only significant if it is a
measurement.
4) Preceding zeros do not count as they are simply
placeholders

9. Rules for determining which
value dictates the expressed
number
 Multiplication and division: The number with the least sig-figs
determine how many numbers you will use to express your answer
Example: 4.44 ÷ 4 = 1.11 but expressed as 1
 Addition and subtraction: The number with the least number of
decimal places determines which number of sig figs, you will use to
express your answer in. If there is a tie, refer to multiplication/division
rule.
Example: 3.11-1.3-.055-1= .755 but expressed as 8 X 10-1

10. Rules for rounding values
 All rounding rules apply as usual UNLESS the number ends in 5
 When you have an answer which ends with a 5 and you need to
express it with one less integer, use these rules:
 1) If the number preceding the 5 is odd then round up
 2) If the number preceding the 5 is even then leave it alone.
 1.0975 is expressed as 1.098.
 1.0985 expressed 1.098
 THIS ONLY APPLIES IF THE NUMBER ENDS IN 5 AND YOU
HAVE TO EXPRESS IN ONE NUMBER LESS

11. Practice Time
 How many sig-figs are in the following
measured values?
 2.555
 .0000056
 1.000055
 45.9000000000

12. Answers
 2.555-4
 .0000056-2
 1.000055-7
 45.9000000000-12

13. Expression practice problem
 Based on the rules, what is the proper
expression of the answer in the problem
below?
 3.00 X 6 =

14. Answer
 The sig-digs are determined by the least
number of sig-figs when multiplying or
dividing
 3.00 X 6 = 18.000 but expressed in sig-digs
as 2 X 101
because you can’t express 18 as 1
sig-dig

15. One More
 Use the sig-figs in the problem below to
determine how many sig-digs to express the
answer.
 1.00 + 2.111 + 2,254 =

16. Answer
The sig-fig with the least decimal places
determines the number of sig-digs to
express the answer
 1.00 + 2.111 + 2,254 = 2257.111 but
expressed as 2,257 or 2.257 x 103
because
2254 has the least decimal places and has
sig-fig equal to 4

17. Disclaimer
 Sig-digs are NOT the most important
concept in chemistry…they do help you
express yourself more intelligently and thus
can add to your popularity