Scientific Notation

Sometimes it is necessary to use very large and very small numbers. These can best be indicated and handled in calculations by use of scientific notation, which is to say by use of exponents. Use of scientific notation requires writing the number so that it is the result of multiplying some whole number power of 10 by a number between 1 and 10. Examples are:

1234 = 1.234 × 10^{3}

0.01234 = 1.234 × 1/100 = 1.234 × 10^{−2}

0.001234 = 1.234 × 1/1000 = 1.234 × 10^{−3}

To convert a number to its equivalent in scientific notation:

Place the decimal point to the right of the first non-zero digit. This will now be a number between 1 and 9.

Multiply this number by a power of 10, the exponent of which is equal to the number of places the decimal point was moved. The exponent is positive if the decimal point was moved to the left, and negative if it was moved to the right. For example:

1,234,000.0 × 0.000072/6000.0 = 1.234 × 10^{6}× 7.2 × 10 ^{−5}/6.0 × 10^{3}

Now, by simply adding or subtracting the exponents of ten, and remembering that moving an exponent from the denominator of the fraction to the numerator changes its sign,

= 1.234 × 10^{6} × 10 ^{−5} × 10 ^{−3} × 7.2/6 = 1.234 × 10^{−2} × 7.2/6

Now, dividing by 6,

= 1.234 × 10^{−2} × 1.2 = 1.4808 × 10^{−2} = 1.4808/100 = 0.014808

The last operation changed 1.4808 × 10^{−2} into the final value, 0.014808, which is not expressed in scientific notation.

SI Units (Système International d'Unités or International System of Units)

This system includes two types of units important in clinical medicine. The *base units* are shown in the first table, derived units in the second table, and derived units with special names in the third table.

SI BASE UNITS

Quantity | Name | Symbol |

Length | meter | m |

Mass | kilogram | kg |

Time | second | s |

Electric current | ampere | A |

Temperature | kelvin | K |

Luminous intensity | candela | cd |

Amount of a substance | mole | mol |

SOME SI DERIVED UNITS

Quantity | Name of Derived Unit | Symbol |

Area | square meter | m^{2} |

Volume | cubic meter | m^{3} |

Speed, velocity | meter per second | m/s |

Acceleration | meter per second squared | m/s^{2} |

Mass density | kilogram per cubic meter | kg/m^{3} |

Concentration of a substance | mole per cubic meter | mol/m^{3} |

Specific volume | cubic meter per kilogram | m^{3} /kg |

Luminescence | candela per square meter | cd/m^{2} |

Body mass index | kilogram per meter squared | kg/m^{2} |

SI DERIVED UNITS WITH SPECIAL NAMES

Quantity | Name | Symbol | Expressed in Terms of Other Units |

Frequency | hertz | Hz | s^{−1} |

Force | newton | N | kg·m·s^{−2} or kg·m/s^{2} |

Pressure | pascal | Pa | N·m^{−2} or N/m^{2} |

Energy, work, amount of heat | joule | J | kg·m^{2}· s^{−2} or N·m |

Power | watt | W | J·s or J/s |

Quantity of electricity | coulomb | C | A·s |

Electromotive force | volt | V | W/A |

Capacitance | farad | F | C/V |

Electrical resistance | ohm | Ω | V/a |

Conductance | siemens | S | A/V |

Inductance | henry | H | Wφ/A |

Illuminance | lux | lx | ln/m^{2} |

Absorbed (radiation) dose | gray | Gy | J/kg |

Dose equivalent (radiation) | sievert | Sv | J/kg |

Activity (radiation) | becquerel | Bq | s^{−1} |

PREFIXES AND MULTIPLES USED IN SI

Prefix | Symbol | Power | Multiple or Portion of a Multiple |

tera | T | 10^{12} | 1,000,000,000,000. |

giga | G | 10^{9} | 1,000,000,000. |

mega | M | 10^{6} | 1,000,000. |

kilo | k | 10^{3} | 1,000. |

hecto | h | 10^{2} | 100. |

deca | da | 10^{1} | 10. |

unity | 1 | ||

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 |

pico | p | 10 ^{−12} | 0.000000000001 |

femto | f | 10^{−15} | 0.000000000000001 |

atto | a | 10^{−18} | 0.000000000000000001 |

Metric System

MASSES

Mass | Grams | |

1 Kilogram | = | 1000.0 |

1 Hectogram | = | 100.0 |

1 Decagram (Dekagram) | = | 10.0 |

1 gram | = | 1.0 |

1 decigram | = | 0.1 |

1 centigram | = | 0.01 |

1 milligram | = | 0.001 |

1 microgram | = | 10^{−6} |

1 nanogram | = | 10^{−9} |

1 picogram | = | 10^{−12} |

1 femtogram | = | 10^{−15} |

1 attogram | = | 10^{−18} |

Arabic numbers are used with masses and measures, as 10 g, or 3 ml. Portions of masses and measures are usually expressed decimally, e.g., 10^{−1} indicates 0.1; 10^{−6} = 0.000001. SEE: Appendix 7-1: Scientific Notation Appendix

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Weights and Measures

Arabic numerals are used with masses and measures, as 10 g, or 3 mL. Portions of masses and measures are usually expressed decimally. For practical purposes, 1 cm^{3} (cubic centimeter) is equivalent to 1 mL (milliliter) and 1 drop (gtt.) of water is equivalent to a minim (m).

**Note: **Traditionally, the word “weights” is used in these tables, but “masses” is the correct term.

LENGTH

Millimeters (mm) | Centimeters (cm) | Inches (in) | Feet (ft) | Yards (yd) | Meters (m) |

1.0 | 0.1 | 0.03937 | 0.00328 | 0.0011 | 0.001 |

10.0 | 1.0 | 0.3937 | 0.03281 | 0.0109 | 0.01 |

25.4 | 2.54 | 1.0 | 0.0833 | 0.0278 | 0.0254 |

304.8 | 30.48 | 12.0 | 1.0 | 0.333 | 0.3048 |

914.40 | 91.44 | 36.0 | 3.0 | 1.0 | 0.9144 |

1000.0 | 100.0 | 39.37 | 3.2808 | 1.0936 | 1.0 |

- 1 μm = 1 micrometer = 0.001 millimeter. 1 mm = 100 μm.
- 1 km = 1 kilometer = 1000 meters = 0.62137 statute mile.
- 1 statute mile = 5280 feet = 1.609 kilometers.
- 1 nautical mile = 6076.042 feet = 1852.276 meters.

VOLUME (FLUID)

Milliliters (mL) | Cubic Inches (in^{3} ) | U.S. Fluid Quarts (qt) | Liters (L) |

1.0 | 0.061 | 0.00106 | 0.001 |

3.697 | 0.226 | 0.00391 | 0.00369 |

16.3866 | 1.0 | 0.0173 | 0.01639 |

29.573 | 1.8047 | 0.03125 | 0.02957 |

946.332 | 57.75 | 1.0 | 0.9463 |

1000.0 | 61.025 | 1.0567 | 1.0 |

- 1 gallon = 4 quarts = 8 pints = 3.785 liters.
- 1 pint = 473.16 mL.

WEIGHT

Grains (gr) | Grams (g) | Apothecaries' Ounces (f℥) | Kilograms (kg) |

1.0 | 0.0648 | 0.00208 | 0.000065 |

15.432 | 1.0 | 0.03215 | 0.001 |

480.0 | 31.1 | 1.0 | 0.0311 |

7000.0 | 453.5924 | 14.583 | 0.45359 |

15432.358 | 1000.0 | 32.15 | 1.0 |

- 1 microgram (μg) = 0.001 milligram.
- 1 mg = 1 milligram = 0.001 g; 1000 mg = 1 g.

CIRCULAR MEASURE

60 seconds = 1 minute | 60 minutes = 1 degree |

90 degrees = 1 quadrant | 4 quadrants = 360 degrees = circle |

LIQUID MEASURE

16 ounces = 1 pint | 4 quarts = 1 gallon | 1 quart = 946.35 milliliters |

1000 milliliters = 1 liter | 31.5 gallons = 1 barrel (U.S.) | 1 liter = 1.0566 quart |

4 gills = 1 pint | 2 pints = 1 quart |

A U.S. gallon is equal to 0.8327 British gallon; therefore, a British gallon is equal to 1.201 U.S. gallons. 1 liter is equal to 1.0567 quarts.

LINEAR MEASURE

1 inch = 2.54 centimeters | 40 rods = 1 furlong | 8 furlongs = 1 statute mile |

12 inches = 1 foot | 3 feet = 1 yard | 5.5 yards = 1 rod |

1 statute mile = 5280 feet | 3 statute miles = 1 statute league | 1 nautical mile = 6076.042 feet |

HOUSEHOLD MEASURES AND WEIGHTS^{†}

Approximate Equivalents: 60 gtt. = 1 teaspoonful | |

= 5 ml = 60 minims = ⅛ ounce | |

1 teaspoon = ⅛ fl oz | 16 teaspoons (liquid) = 1 cup |

3 teaspoons = 1 tablespoon | 12 tablespoons (dry) = 1 cup |

1 tablespoon = ½ fl oz | 1 cup = 8 fl oz |

1 tumbler or glass = 8 fl oz; ½ pint |

Conversion Rules and Factors

To convert units of one system into the other, multiply the number of units in column I by the equivalent factor opposite that unit in column II.

WEIGHT

1 gram | = | 0.03527 avoirdupois ounce |

1 gram | = | 0.03215 apothecaries' ounce |

1 kilogram | = | 35.274 avoirdupois ounces |

1 kilogram | = | 32.151 apothecaries' ounces |

1 kilogram | = | 2.2046 avoirdupois pounds |

1 grain | = | 64.7989 milligrams |

1 grain | = | 0.0648 gram |

1 avoirdupois ounce | = | 28.3495 grams |

1 apothecaries' ounce | = | 31.1035 grams |

1 avoirdupois pound | = | 453.5924 grams |

VOLUME (AIR OR GAS)

1 cubic centimeter (cm^{3}) | = | 0.06102 cubic inch |

1 cubic meter (m^{3}) | = | 35.314 cubic feet |

1 cubic meter | = | 1.3079 cubic yard |

1 cubic inch (in^{3}) | = | 16.3872 cubic centimeters |

1 cubic foot (ft^{3}) | = | 0.02832 cubic meter |

CAPACITY (FLUID OR LIQUID)

1 liter | = | 2.1134 pints |

1 liter | = | 1.0567 quart |

1 liter | = | 0.2642 gallon |

1 fluid dram | = | 3.697 milliliters |

1 fluid ounce | = | 29.573 milliliters |

1 pint | = | 473.1765 milliliters |

1 quart | = | 946.353 milliliters |

1 gallon | = | 3.785 liters |

TIME

1 millisecond = one thousandth (0.001) of a second | 1 minute = 1/60 of an hour |

1 second = 1/60 of a minute | 1 hour = 1/24 of a day |

TEMPERATURE ^{†}

Given a temperature on the Fahrenheit scale, to convert it to degrees Celsius, subtract 32 and multiply by 5/9. Given a temperature on the Celsius scale, to convert it to degrees Fahrenheit, multiply by 9/5 and add 32. Degrees Celsius are equivalent to degrees Centigrade.

PRESSURE

TO OBTAIN | MULTIPLY | BY |

lb/sq in | atmospheres | 14.696 |

lb/sq in | in of water | 0.03609 |

lb/sq in | ft of water | 0.4335 |

lb/sq in | in of mercury | 0.4912 |

lb/sq in | kg/sq meter | 0.00142 |

lb/sq in | kg/sq cm | 14.22 |

lb/sq in | cm of mercury | 0.1934 |

lb/sq ft | atmospheres | 2116.8 |

lb/sq ft | in of water | 5.204 |

lb/sq ft | ft of water | 62.48 |

lb/sq ft | in of mercury | 70.727 |

lb/sq ft | cm of mercury | 27.845 |

lb/sq ft | kg/sq meter | 0.20482 |

lb/cu in | g/mL | 0.03613 |

lb/cu ft | lb/cu in | 1728.0 |

lb/cu ft | gm/mL | 62.428 |

lb/U.S. gal | gm/L | 8.345 |

in of water | in of mercury | 13.60 |

in of water | cm of mercury | 5.3543 |

ft of water | atmospheres | 33.95 |

ft of water | lb/sq in | 2.307 |

ft of water | kg/sq meter | 0.00328 |

ft of water | in of mercury | 1.133 |

ft of water | cm of mercury | 0.4461 |

atmospheres | ft of water | 0.02947 |

atmospheres | in of mercury | 0.03342 |

atmospheres | kg/sq cm | 0.9678 |

bars | atmospheres | 1.0133 |

in of mercury | atmospheres | 29.921 |

in of mercury | lb/sq in | 2.036 |

mm of mercury | atmospheres | 760.0 |

g/mL | lb/cu in | 27.68 |

g/sq cm | kg/sq meter | 0.1 |

kg/sq meter | lb/sq in | 703.1 |

kg/sq meter | in of water | 25.40 |

kg/sq meter | in of mercury | 345.32 |

kg/sq meter | cm of mercury | 135.95 |

kg/sq meter | atmospheres | 10332.0 |

kg/sq cm | atmospheres | 1.0332 |

FLOW RATE

TO OBTAIN | MULTIPLY | BY |

cu ft/hr | cc/min | 0.00212 |

cu ft/hr | L/min | 2.12 |

L/min | cu ft/hr | 0.472 |

PARTS PER MILLION

Conversion of parts per million (ppm) to percent: |

1 ppm = 0.0001%, 10 ppm = 0.001%, 100 ppm = 0.01%, 1000 ppm = 0.1%, 10,000 ppm = 1%, etc. |

ENERGY

- 1 foot pound = 1.35582 joule
- 1 joule = 0.2389 Calorie (kilocalorie)
- 1 Calorie (kilocalorie) = 1000 calories = 4184 joules
- A large Calorie, or kilocalorie, is always written with a capital C.

pH

The pH scale is simply a series of numbers stating where a given solution would stand in a series of solutions arranged according to acidity or alkalinity. At one extreme (high pH) lies a highly alkaline solution; at the other extreme (low pH) is an acid solution containing 3.65 g of hydrogen chloride per liter of water. Halfway between lies purified water, which is neutral. All other solutions can be arranged on this scale, and their acidity or alkalinity can be stated by giving the numbers that indicate their relative positions. If the pH of a certain solution is 5.3, it falls between gastric juice and urine on the above scale, is moderately acid, and will turn litmus red.

Tenth-normal HCl | −1.00 | Litmus is red in this acid range |

Gastric juice | ^{‡} 1.4 | |

Urine | ^{‡} 6.0 | |

Water | 7.00 | Neutral |

Blood | 7.35-7.45 | Litmus is blue in this alkaline range. |

Bile | ^{‡} 7.5 | |

Pancreatic juice | 8.5 | |

Tenth-normal NaOH | 13.00 | |

^{‡} These body fluids vary rather widely in pH; typical figures have been used for simplicity. Urine samples obtained from healthy individuals may have pH readings anywhere between 4.7 and 8.0. |

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