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Let’s Talk Numbers Uncertainties of Measurements, Scientific Notation, and Significant Figures

Uncertainties of Measurements Every measurement made, regardless of what instrument or technique is used or of who is performing the measurement, is an approximation and is subject to uncertainties Every measurement made, regardless of what instrument or technique is used or of who is performing the measurement, is an approximation and is subject to uncertainties What does this mean? What does this mean?

Uncertainties of Measurements Parallax – an apparent shift in position of an object when viewed from different angles Parallax – an apparent shift in position of an object when viewed from different angles Precision – describes the degree of exactness to which a measurement was made Precision – describes the degree of exactness to which a measurement was made The finest division of a scale on a measuring device limits its precision The finest division of a scale on a measuring device limits its precision Accuracy – describes how well a measurement compares to a standard (accepted) value Accuracy – describes how well a measurement compares to a standard (accepted) value Accuracy depends upon the technique of the person making the measurement and requires properly calibrated equipment Accuracy depends upon the technique of the person making the measurement and requires properly calibrated equipment

Scientific Notation Scientific Notation – used for simplifying the writing of very large and very small numbers Scientific Notation – used for simplifying the writing of very large and very small numbers The numerical part of the measurement is expressed as a number between 1 and 10 multiplied by a whole number power of 10 The numerical part of the measurement is expressed as a number between 1 and 10 multiplied by a whole number power of 10 M x 10 n M x 10 n

Scientific Notation First, move the decimal point until only one non-zero digit remains on the left First, move the decimal point until only one non-zero digit remains on the left Next, count the number of places the decimal point was moved; this number is the exponent of 10 Next, count the number of places the decimal point was moved; this number is the exponent of 10

Scientific Notation Direction Decimal Point Moved Exponent 1. To the left (+) larger 2. To the right (-) smaller Let’s do some examples together Let’s do some examples together

Arithmetic Operations with Scientific Notation Addition/Subtraction Addition/Subtraction 1. Make exponent of 10, n, the same for all numbers 2. Add or subtract values of M and keep 10 n Remember – like units must have the same prefix Remember – like units must have the same prefix Let’s do some examples together Let’s do some examples together

Arithmetic Operations with Scientific Notation Multiplication/Division Multiplication/Division 1. Perform the indicated operation on the values of M. 2. If you multiplied, add the exponents 3. If you divided, subtract the exponents

Significant Digits Measured values are only accurate to within the limits of the precision of a measuring instrument. Measured values are only accurate to within the limits of the precision of a measuring instrument. These valid digits are termed significant digits or significant figures These valid digits are termed significant digits or significant figures

Significant Digits Read the instrument to the smallest division, then estimate to 1/10 of that division. Read the instrument to the smallest division, then estimate to 1/10 of that division. The number of significant digits in a measurement is found by counting the questionable/estimated digit, and the digits to the left of it up to and including the last nonzero digit The number of significant digits in a measurement is found by counting the questionable/estimated digit, and the digits to the left of it up to and including the last nonzero digit

Significant Digits Rules for Significant Digits Rules for Significant Digits 1. If the number does not contain a decimal point, begin counting at the left and stop at the last non-zero digit. 2. If the number contains a decimal point, begin counting at the left-most, non-zero digit and stop at the last digit When writing results, use scientific notation to clearly indicate which zeroes are significant When writing results, use scientific notation to clearly indicate which zeroes are significant

Arithmetic Operations with Significant Digits Addition/Subtraction Addition/Subtraction When numbers are added or subtracted the number of decimal places in the result should equal the smallest number of decimal places of any term in the sum or difference When numbers are added or subtracted the number of decimal places in the result should equal the smallest number of decimal places of any term in the sum or difference Let’s try it Let’s try it

Arithmetic Operations with Significant Digits Multiplication/Division Multiplication/Division When multiplying several quantities, the umber of significant figures in the final answer is the same as the number of significant figures in the “least accurate” of the quantities being multiplied, where “least accurate” means having the lowest number of significant figures. The same rule applies to division. When multiplying several quantities, the umber of significant figures in the final answer is the same as the number of significant figures in the “least accurate” of the quantities being multiplied, where “least accurate” means having the lowest number of significant figures. The same rule applies to division.

A Few Notes on Significant Digits DO NOT round off until after completing the mathematical operation DO NOT round off until after completing the mathematical operation The result of any mathematical operation can NEVER be more precise than the least precise measurement. The piece of data with the greatest uncertainty will determine the uncertainty of the end result. The result of any mathematical operation can NEVER be more precise than the least precise measurement. The piece of data with the greatest uncertainty will determine the uncertainty of the end result.

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