Unit 1: The Metric System

The Metric System

Introduction to the Metric System

  • The metric system is an internationally standardized system of units of measurement.

  • It is based on a base unit and prefixes.

Customary Units of Measurement

  • The English System:

    • A collection of functionally unrelated units.

    • Difficult to convert from one unit to another.

    • Example of conversions in the English system:

    • 1 ft = 12 inches = 0.33 yard = 1/5280 miles.

  • Customary Units:

    • Length: inch, foot, yard, mile.

    • Weight/Mass: ounce, pound.

    • Volume: teaspoon, cup, quart, gallon.

    • Temperature: degrees Fahrenheit.

    • Time: minutes, hours.

Advantages of Using the Metric System

  • Universal: Used everywhere, by all scientists to communicate, and predominantly in all industrialized nations (except the United States).

  • Simplicity:

    • A few base units make up all measurements:

    • Length: meter (m).

    • Mass: grams (g).

    • Volume: liters (L).

    • Temperature: degrees Celsius (°C).

    • Time: seconds.

  • Unique Units: There is only one unit of measurement for each type of quantity, simplifying conversions.

    • For very small and very large numbers, multiples of the base unit are used with prefixes.

  • Decimal System: Easy to convert between units; shifting the decimal point either to the right (increase) or to the left (decrease).

  • Common Prefixes: The same set of prefixes applies to all units.

    • Greek Prefixes:

    • kilo (k): Represents 1000 × the base unit.

    • hecto (h): Represents 100 × the base unit.

    • deka (da): Represents 10 × the base unit.

    • Latin Prefixes:

    • deci (d): Represents tenths of the base.

    • centi (c): Represents hundredths of the base.

    • milli (m): Represents thousandths of the base.

    • Mnemonic for remembering prefixes: “Kids Have Dropped Over Dead Converting Metrics.”

Metric Prefixes

Table 1.1: Some Common Prefixes Used in the Metric System

Prefix

Multiple

Decimal Equivalent

mega (M)

$10^6$

1,000,000

kilo (k)

$10^3$

1,000

deka (da)

$10^1$

10

deci (d)

$10^{-1}$

0.1

centi (c)

$10^{-2}$

0.01

milli (m)

$10^{-3}$

0.001

micro (μ)

$10^{-6}$

0.000001

nano (n)

$10^{-9}$

0.000000001

Units of Length

  • Definition: Length is defined as the distance between two points.

  • Standard Unit: The standard unit of length is the meter (m).

  • Long Distances: Measured in kilometers (km).

  • Measurement Tools: Length can be measured using a meter stick or ruler.

Prefixes and Units of Length

  • Common Units:

    • centimeter (cm): $1 m = 100 cm$, $1 cm = 1/100$ m.

    • millimeter (mm): $1 m = 1000 mm$, $1 mm = 1/1000$ m, $10 mm = 1 cm$ (measures very small lengths).

    • kilometer (km): $1 km = 1000 m$, $1 m = 1/1000$ km (measures long distances).

Measuring Mass

  • Definition: Mass is the quantity of matter in an object.

  • Standard Unit: The standard unit of mass is the gram (g).

  • Measurement Tools: Mass is measured using a digital scale or triple beam balance.

Measuring Volume and Capacity

  • Definition: Volume is the amount of space occupied by an object.

  • Standard Unit: The standard unit of volume is the liter (L).

    • Equivalences: $1 L = 1000 ext{ ml} = 1000 ext{ cm}^3 = 1 ext{ dm}^3$.

  • Measurement Tool: Measured using a graduated cylinder.

  • Definition of Capacity: Capacity is a measure of the volume inside a container.

Units of Volume

  • Common Units:

    • Liter (L): $1 L = 1000$ milliliters.

    • milliliter (mL): Measures small volumes, $1 mL = 1 ext{ cm}^3$, $1000 mL = 1 L$.

    • kiloliter (kL): Measures large volumes, $1 kL = 1000 L$.

Measuring Temperature

  • Definition: Temperature is the degree of "hotness" of an object.

  • Standard Unit: The standard unit of temperature is Celsius (°C).

  • Measurement Tool: Temperature is measured using a thermometer.

Temperature Conversions

  • Conversions between Fahrenheit, Celsius, and Kelvin.

  • Example Conversions:

    • Convert $75^ ext{o}C$ to $^ ext{o}F$.

    • Convert $-10^ ext{o}F$ to $^ ext{o}C$.

Measurement Unit Conversion

  • You can convert between units of measurement:

    • Within the metric system.

    • Between the English system and metric system.

Dimensional Analysis

  • Definition: Dimensional Analysis (also called Factor-Label Method or the Unit Factor Method) is a problem-solving method that uses the fact that any number or expression can be multiplied by one without changing its value (Multiplication Property of 1 – the Magic One).

  • Usage: Use the units to dictate the form of the Magic One.

Equivalences

Units of Weight

  • $1 ext{ oz} = 28.35 ext{ g}$

  • $1 ext{ lb} = 454 ext{ g}$

  • $2.2 ext{ lb} = 1 ext{ kg}$

Units of Capacity

  • $1.06 ext{ qt} = 1 ext{ L}$

  • $1 ext{ gal} = 3.79 ext{ L}$

Units of Length

  • $1 ext{ in} = 2.54 ext{ cm}$

  • $3.28 ext{ ft} = 1 ext{ m}$

  • $1.09 ext{ yd} = 1 ext{ m}$

  • $1 ext{ mi} = 1.61 ext{ km}$

Example Conversions

Convert 130 lbs to kg (round to the nearest whole number)

  • Process:

    • Write the original measurement as a unit fraction.

    • Multiply the unit fraction by a magic one dictated by the units.

    • The numerator unit is the unit you want; the denominator unit is the unit you want to eliminate.

  • Calculation:

    • $130 ext{ lbs} = 130 rac{ ext{kg}}{2.2 ext{ lbs}} = 59.0 ext{ kg}$

Convert 60 km to mi (round to the nearest whole number)

  • Calculation:

    • $60 ext{ km} = 60 ext{ km} rac{1 ext{ mi}}{1.61 ext{ km}} = 37.2 ext{ mi}$

    • Rounded: $37 ext{ mi}$

Convert 5.4 kg to lb (round to the nearest tenth place)

  • Calculation:

    • $5.4 ext{ kg} = 5.4 ext{ kg} rac{2.2 ext{ lb}}{1 ext{ kg}} = 11.88 ext{ lb}$

    • Rounded: $11.9 ext{ lb}$

Convert 45 cm to in (round to the nearest tenth place)

  • Calculation:

    • $45 ext{ cm} = 45 ext{ cm} rac{1 ext{ in}}{2.54 ext{ cm}} = 17.71 ext{ in} $

    • Rounded: $17.7 ext{ in}$

Conversions Within the Metric System

  • Conversions involve writing the prefixes in order from largest to smallest: k h da d c m.

  • Conversion Directions:

    • To convert from a smaller unit to a larger unit, shift left.

    • To convert from a larger unit to a smaller unit, shift right.

Conversion Examples

  • Convert 1600 cm to m:

    • Move 2 places to the left to convert from cm to m, thus $1600 ext{ cm} = 16.00 ext{ m}$.

  • Convert 2 kL to L:

    • Move 3 places to the right to convert from kL to L, thus $2 ext{ kL} = 2000 ext{ L}$.

  • Convert 241 g to mg:

    • Move 3 places to the right to convert from g to mg, thus $241 ext{ g} = 241,000 ext{ mg}$.

  • Convert 3 mL to L:

    • Move 3 places to the left to convert from mL to L, thus $3 ext{ mL} = 0.003 ext{ L}$.

  • Convert 45 cm to km:

    • Move 5 places to the left to convert from cm to km, thus $45 ext{ cm} = 0.00045 ext{ km}$.

  • Convert 5.4 kg to dg:

    • Move 4 places to the right to convert from kg to dg, thus $5.4 ext{ kg} = 54000 ext{ dg}$.