# 1.8: Measured Equalities and Prefix Modifiers

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- 226510

While temperature units are converted using equations, units for **mass**, **volume**, **length**, and **time **are changed using **conversion factors**, which are based on equalities. Conversion factors will be discussed in the next section of this chapter. Equalities do not contain variables and instead include a number and a unit on both sides of an equal sign. There are two types of equalities: Measured equalities and prefix modifier equalities.

## Measured Equalities

*Measured equalities* are developed by taking the same measurement with two different tools, where the tools each have different units. For example, a block of wood could be measured in centimeters, using the metric side of a ruler, and in inches, using the U.S. side of a ruler. Examples of measured equalities are shown below in Table \(\PageIndex{1}\).

Time |
Mass |
Length |
---|---|---|

1 week (wk) = 7 days (d) | 1 pound (lb) = 16 ounces (oz) | 1 mile (mi) = 5,280 feet (ft) |

1 d = 24 hours (h) | 1 kilogram (kg) = 2.2 lb | 1 yard (yd) = 3 ft |

1 h = 60 minutes (min) | 1 ft = 12 inches (in) | |

1 min = 60 seconds (s) | 1 in = 2.54 centimeters (cm) |

Note that each of these equalities is exact, so its numbers are considered to have infinitely-many significant figures. These equalities will ultimately be used in conversion factors, which will involve multiplication and division. Since the answer must be limited to the lesser count of significant figures when multiplying and dividing, these equalities will never limit the number of significant figures in a calculated answer.

## Prefix Modifiers

A **prefix modifier** is used to change a base unit by a power of 10. The base units for mass, volume, length, and time** **are given below in Table \(\PageIndex{2}\).

Measurable Quantity |
Base Unit |
---|---|

Mass | gram (g) |

Volume | liter (L) |

Length | meter (m) |

Time | second (s) |

Remember that it is important to use appropriate capitalization for abbreviations! Each of these base units can be changed by a power of 10 to be made larger or smaller. A positive power is used to make a unit larger, and a negative power is used to make a unit smaller. This system parallels the rules for powers when writing numbers in scientific notation. Table \(\PageIndex{3}\) lists the most common prefix modifiers and their meanings. Note that all prefix modifiers are exact, by definition.

Prefix |
Abbreviation |
Meaning |
---|---|---|

"Larger" Prefix Modifiers |
||

tera | \(\text{T}\) | 10^{12} |

giga | \(\text{G}\) | 10^{9} |

mega | \(\text{M}\) | 10^{6} |

kilo | \(\text{k}\) | 10^{3} |

"Smaller" Prefix Modifiers |
||

deci | \(\text{d}\) | 10^{-1} |

centi | \(\text{c}\) | 10^{-2} |

milli | \(\text{m}\) | 10^{-3} |

micro | \(\mu\) | 10^{-6} |

nano | \(\text{n}\) | 10^{-9} |

pico | \(\text{p}\) | 10^{-12} |

Each of these prefix modifiers can be inserted before a base unit to make a new unit. For example, combining the prefix modifier "kilo" with the base unit "gram" would create the new unit "kilogram."

Finally, each new unit can be related back to its corresponding base unit by replacing the prefix modifier with its meaning. For example, the kilogram (kg) can be related to the gram (g), as shown below.

kg = 10^{3} g

Since the prefix modifier "kilo" (k) means "10^{3}," these two quantities can be interchanged. The base unit, gram (g), is written with both the prefix modifier *and *its meaning. Since their base units (grams) match, and the prefix modifier and its meaning are equivalent, these two quantities are equal to one another. If no numerical value is explicitly-written in an equality, an unwritten "1" is understood to be present. Note that, while not absolutely necessary, a prefix modifier equality can be simplified by rewriting the numerical value in decimal format.

kg = 1,000 g