Scientific Notation
To express very large or very small numbers, scientists express values in terms of "a x 10b", where “a” is a
number between 1 and 10 and “b” is number of places the decimal place had to move in order to express
“a” as a number between 1 and 10. This type of expression is called scientific notation.
1 = 1x100
10 = 1x101
100 = 1x102
1000 = 1x103
etc..
The value in the exponent place describes how many zeroes there are in the number being represented. The
number 100 has 2 zeroes; it's scientific notation is 1X102.
In the case of numbers smaller than one, the exponent becomes negative, and that negative value represents
how many zeroes there are between the number and the decimal place:
0.1 = 1x10-1
0.01 = 1x10-2
0.001 = 1x10-3
etc.
Metric Units (SI units)
The standards metric terms for commonly measured characteristics are listed in below.
Property
Unit
Symbol
Mass
kilograms
kg
Length
meters
m
Volume
liters
L
Time
seconds
s
Electric Current
ampere
A
Temperature
Kelvin, Celsius K, C
Intensity of light
candela
cd
Amount of a Substance mole
mol
Notes: Temperature in the table above is expressed in both Kelvin (K) and Celsius (C). One kelvin is
exactly the same as one C, except that the Kelvin scale starts at absolute zero, or -273.15 C, the lowest
temperature possible, instead of the freezing point of water. Therefore water freezes at 273.15 K or 0 C.
Prefixes for Units
In addition to the basic metric units described above, there are prefixes to indicate larger or smaller
quantities. For example, a meter refers to a standard metric measure of length. A millimeter refers to a
measure that is one-thousandth the size of a meter (one thousand millimeters fit into a meter); a kilometer
refers to a distance one thousand times longer than a meter (one thousand meters fit into a kilometer). By
wisely using these prefixes, you can avoid having to use huge numbers or having to resort to scientific
notation. The table below shows the most commonly encountered prefixes:
Prefix Symbol Value Description
Femto f
10-15
1 femtoliter (fL)= 0.000000000000001
pico
p
10-12
1 picoliter, (pL) = 0.000000000001 l
nano
n
10-9
1 nanogram, (ng) = 0.0000000001 g
micro µ or u 10-6
1 micrometer (µm) = 0.000001 m
milli
m
10-3
1 milliliter (mL) = 0.001 L
centi
c
10-2
1 centimeter (cm) = 0.01 m
deci
d
10-1
1 decigram (dg) = 0.1 g
none
none
1
normal units without prefixes
kilo
k
103
1 kilogram (kg) = 1000 g
mega M
106
1 megagram (Mg) = 1,000,000 g
giga
G
109
1 gigameter (Gm) = 1,000,000,000 m
tera
T
1012
1 teraliter (TL) = 1,000,000,000,000 L
Significant Figures
No experimental measurement can possibly be perfectly precise. Take, for example, a wooden stick that is
approximately two meters long. If a scientist were to measure that stick with a ruler marked only with
meters, then he could only conclude with certainty that the stick measured 1 meter (though of course he
would recognize that his measurement was inexact). If his ruler was marked with decimeters, then he could
see with certainty that the stick measured 1.1 meters. If he could measure centimeters, he might see that the
stick actually measured 1.12 meters. Using a ruler with millimeters he could see the stick is actually 1.121
meters long. Each smaller measurement allows the scientist to determine the length of the stick with a bit
more accuracy. But no scientist can use a ruler to great effect for distances much smaller than a millimeter;
such small distances are simply beyond the ability of the scientist's ability to see. At some point his
measurements will necessarily become slightly inaccurate.
Scientists account for this unavoidable uncertainty in measurement through the use of significant digits.
Significant digits do not remove the uncertainty; instead they alert others as to where the uncertainty lies. In
the case of our measurement of the stick, the value 1.121 meters alerts the next scientist to come along that
the last 1 digit on the right might be slightly inaccurate.
Five rules govern significant figures:
1.
Non-zero digits are always significant; 1.121 has four significant digits.
2.
Any zeros between two significant digits are significant; 1.08701 has six significant digits.
3.
Zeros before the decimal point are placeholders and not significant; in the number .00254, only the
2,5 and 4 are significant, meaning the number has 3 significant figures.
4.
Zeros after the decimal point and after figures are significant; in the number 0.2540, the 2, 4, 5 and
last 0 are significant.
5.
Exponential digits in scientific notation are not significant; 1.12x106 has three significant digits, 1,
1, and 2.
These rules ensure accurate representation and interpretation of data. If, for example, you were to read of
an experimental reaction in which the resulting chemical weighed 0.0254 g, you would know that the
measurement is accurate to 0.0001 g and contains 3 significant figures.
Significant Figures in Operations
When making calculations, significant figures become very important. You must always be careful to
remember how many significant figures your separate values have. The rules governing addition and
subtraction, and those governing multiplication and division are a little different.
Addition and Subtraction of Significant Figures
Addition and subtraction of significant figures follows a simple rule:
The final value must have only as many decimals as the original value with the least number of decimal
places.
Significant Figures in Multiplication and Division
The rule governing multiplication and division of significant figures is slightly different than that for
addition and subtraction, but just as simple:
The final value can only have as many significant figures as the original value with the least significant
figures.