2 Presenting Chemical Data
Measurement |
Exact Number |
Significant Figures |
| A numeric quantity in which every digit, except the last, is certain. The last digit is estimated, leading to an uncertainty in the measurement. | A number with an unlimited amount of significant figures. | The number of non-place holding digits in a reported measurement. A larger amount of significant figures means a larger precision in the measurement. |
Measurement Reliability
The last digit in a reported scientific measurement is estimated. The instrument used for measuring dictates the number of digits used in the reported measurement.
Exact Numbers
Numbers that have no uncertainty, and an unlimited number of significant figures. Exact numbers do not affect the outcome of significant figures in a calculation. Exact numbers are derived in the following way:
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2.
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3.
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| There are 4 apples = 4.000000… apples | There are 60 seconds in 1 minute = 60.000000… s in 1.000000… min | The formula for kinetic energy is [latex]E_{k} = \frac{1}{2} mv^{2}[/latex]. In this case, [latex]\frac{1}{2}[/latex] is an exact number, whereas mass and velocity are measured numbers. |
Significant Figures
An accepted method for preserving the precision of a measurement when recording data or doing calculations.
| 1. | 2. | 3. | 4. |
| Non-zero digits are significant. | Exact numbers are significant. | Contained zeros are significant. | Leading zeros are not significant. |
| 5. | ||
| Trailing zeros have significance as follows: | ||
| a. | b. | c. |
| After a decimal point, significant | After a non-zero number and before a decimal point, significant | After a non-zero number that in not a decimal number, generally a place holder |
For an even more detailed breakdown on significant figures click the link below to view.
Significant Figures
Calculations with Significant Figures
Rounding |
Multiplication and Division |
Addition and Subtraction |
| When completing a series of calculations carry extra digits through to the final answer then round to the correct amount of significant figures. | The reported result of a calculation when multiplying or dividing significant figures has the same number of significant figures as the least precise number used in the calculation. | The reported result of a calculation when adding or subtracting significant figures has the same decimal place as the least precise number used in the calculation. |
In this context the least precise number has the smallest number of decimal places
Scientific Notation
A notation for expressing large and small numbers as a small decimal between one and ten multiplied by a power of ten.
How to write using scientific notation:
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1. |
2. |
2a. |
2a. |
| Move the decimal point to the left or right to reach a decimal number between one and ten. | Write the number obtained in step 1 multiplied by 10 raised to the number of places the decimal point was moved. |
If the decimal is moved to the left, the power is positive. Example: [latex]140000 = 1.4 \times 10^{5}[/latex] |
If the decimal is moved to the right, the power is negative. Example: [latex]0.000014 = 1.4 \times 10^{-5}[/latex] |
Calculations with Scientific Notation
Addition |
Multiplication |
Division |
Using a Scientific Calculator |
| Rewrite all of the numbers in the calculation such that the exponent is the same. Add or subtract the decimal number. | Multiply the decimal numbers and add the exponents. | Write out as a fraction, divide the decimal number and subtract the exponent in the denominator from the exponent in the numerator. |
Generally scientific calculators have an EE or EXP button they both mean: x . For n < 1, a negative sign is used before entering the value of n |
Examples
Problem Set
Below are two documents. One is practice problems, the second is the same problems with solutions.
They can be downloaded and changed to suit your needs.
| Problem Set | Problem Set Solutions |
