More About Calculations and Measurements

Scientific Notation

• To show how very large or very small numbers can be expressed as the product of a number between 1 and 10 and a power of 10.
  • Any number can be represented as the product of a number between 1 and 10 and a power of 10 (either positive or negative.)
  • The easiest way to determine the appropriate power of 10 for scientific notation is to start with the number being represented and count the number places the decimal point must be moved to obtain a number between 1 and 10.
  • The power of 10 depends on the number of places the decimal point is moved and in which direction. The number of places the decimal point is moved determines the power of ten.
  • The direction of the move determines whether the power of 10 is positive or negative. Whenever the decimal point is moved to the left, the exponent is positive. Whenever the decimal point is moved to the right, the exponent is negative.

Units

• To learn the English, metric, and SI systems of measurement.
  • The units part of a measurement tells us what scale or standard is being used to represent the results of the measurement.
  • The two most widely used systems are the English system used in the United States and the metric system used in most of the rest of the industrialized world. The metric system has long been preferred for most scientific work.
  • In 1960 an international agreement set up a comprehensive system of units called the International System (le Systeme Internationale in French), or SI. The SI units are based on the metric system and units derived from the metric system.

Measurements of Length, Volume, and Mass

• To understand the metric system for measuring length, volume, and mass.
  • The fundamental SI unit of length is the meter, which is a little longer than a yard.
  • The fundamental unit of volume in the SI system is based on the volume of a cube that measures 1 meter in length in each of three directions, or a cubic meter.
  • The fundamental SI unit of mass is the kilogram. Because the metric system, which existed before the SI system, used the gram as the fundamental unit, the prefixes for the various mass units are based on the gram.

Uncertainty in Measurement

• To understand how uncertainty in a measurement arises.
• To learn to indicate a measurement’s uncertainty by using significant figures.
  • A measurement always has some degree of uncertainty. The uncertainty of a measurement depends on the measuring device.
  • When one is making a measurement, the custom is to record all of the certain numbers plus the first uncertain number.

Significant Figures

• To learn to determine the number of significant figures in a calculated result.
  • Leading zeros are never significant figures.
  • Captive zeros are always significant figures.
  • Trailing zeros are sometimes significant figures.
  • Exact numbers never limit the number of significant figures in a calculation.
  • Significant figures are easily indicated by scientific notation.
  • Rules for counting significant figures also apply to numbers written in scientific notation.

• Rules for Counting Significant Figures
  1. Nonzero integers always count as significant figures.
  2. There are three classes of zeros:
     1. Leading zeros are zeros that precede all the nonzero digits. They never count as significant figures.
     2. Captive zeros are zeros that fall between nonzero digits. They always count as significant figures.
     3. Trailing zeros are at the right end of the number. They are significant only if the number is written with a decimal point.
  1. Often calculations involve numbers that were not obtained using measuring devices but were determined by counting. Such numbers are called exact numbers. They can be assumed to have an unlimited number of significant figures. Exact numbers can also arise from definitions.

• Rules for Rounding Off
  1. If the digit to be removed:
     1. Is less than 5, the preceding digit stays the same.
     2. Is equal to or greater than 5, the preceding digit is increased by 1.
  1. In a series of calculations, carry the extra digits through to the final result and then round off. This means that you should carry all the digits that show on you calculator until you arrive at the final number (the answer) and then round off, using the procedure in Rule 1.
  2. When rounding off, use only the first number to the right of the last significant figure.

• Rules for Using Significant Figures in Calculations
  1. For multiplication or division, the number of significant figures in the result is the same as that in the measurement with the smallest number of significant figures. We say this measurement is limiting because it limits the number of significant figures in the result.
  2. For addition or subtraction, the limiting term is the one with the smallest number of decimal places.
  • Note that for multiplication and division, significant numbers are counted. For multiplication and subtraction, decimal places are counted.
  • When we multiply and divide in  a problem, perform all calculations before rounding the answer to the correct number of significant figures.
  • When we multiply or divide and then add or subtract in a problem, round the first answer from the first operation before performing the next operation. We need to know the correct number of decimal places.

Key Terms
1. Measurement- a quantitative observation.
2. Scientific notation- simply expresses a number as the product of a number between 1 and 10 and uses the appropriate power of 10
3. Units- what scale or standard is being used to represent the results of a measurement
4. Volume- the amount of three-dimensional space occupied by a substance.
5. Graduated cylinder- commonly used in chemical laboratories for measuring the volume of liquids; marked off in convenient units of volume (usually milliliters)
6. Liter- 1 dm^3, slightly larger than a quart, abbreviated L
7. Milliliter- 1 cm^3, each liter contains 1,000 cm^3, abbreviated mL
8. Mass- the quantity of matter present in an object
9. Significant figures- The numbers recorded in a measurement (all the certain numbers plus the first uncertain number).
10. Rounding off numbers- reducing a number to fewer digits

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