Scientific notation
In many disciplines of science and
engineering, very large and very small numerical quantities
must be managed. Some of these quantities are mind-boggling
in their size, either extremely small or extremely large.
Take for example the mass of a proton, one of the
constituent particles of an atom's nucleus:
Proton mass = 0.00000000000000000000000167
grams
Or, consider the number of electrons passing
by a point in a circuit every second with a steady electric
current of 1 amp:
1 amp = 6,250,000,000,000,000,000 electrons
per second
A lot of zeros, isn't it? Obviously, it can
get quite confusing to have to handle so many zero digits in
numbers such as this, even with the help of calculators and
computers.
Take note of those two numbers and of the
relative sparsity of non-zero digits in them. For the mass
of the proton, all we have is a "167" preceded by 23 zeros
before the decimal point. For the number of electrons per
second in 1 amp, we have "625" followed by 16 zeros. We call
the span of non-zero digits (from first to last), plus any
zero digits not merely used for placeholding, the
"significant digits" of any number.
The significant digits in a real-world
measurement are typically reflective of the accuracy of that
measurement. For example, if we were to say that a car
weighs 3,000 pounds, we probably don't mean that the car in
question weighs exactly 3,000 pounds, but that we've
rounded its weight to a value more convenient to say and
remember. That rounded figure of 3,000 has only one
significant digit: the "3" in front -- the zeros merely
serve as placeholders. However, if we were to say that the
car weighed 3,005 pounds, the fact that the weight is not
rounded to the nearest thousand pounds tells us that the two
zeros in the middle aren't just placeholders, but that all
four digits of the number "3,005" are significant to its
representative accuracy. Thus, the number "3,005" is said to
have four significant figures.
In like manner, numbers with many zero
digits are not necessarily representative of a real-world
quantity all the way to the decimal point. When this is
known to be the case, such a number can be written in a kind
of mathematical "shorthand" to make it easier to deal with.
This "shorthand" is called scientific notation.
With scientific notation, a number is
written by representing its significant digits as a quantity
between 1 and 10 (or -1 and -10, for negative numbers), and
the "placeholder" zeros are accounted for by a power-of-ten
multiplier. For example:
1 amp = 6,250,000,000,000,000,000 electrons
per second
. . . can be expressed as . . .
1 amp = 6.25 x 1018 electrons per
second
10 to the 18th power (1018) means
10 multiplied by itself 18 times, or a "1" followed by 18
zeros. Multiplied by 6.25, it looks like "625" followed by
16 zeros (take 6.25 and skip the decimal point 18 places to
the right). The advantages of scientific notation are
obvious: the number isn't as unwieldy when written on paper,
and the significant digits are plain to identify.
But what about very small numbers, like the
mass of the proton in grams? We can still use scientific
notation, except with a negative power-of-ten instead of a
positive one, to shift the decimal point to the left instead
of to the right:
Proton mass = 0.00000000000000000000000167
grams
. . . can be expressed as . . .
Proton mass = 1.67 x 10-24 grams
10 to the -24th power (10-24)
means the inverse (1/x) of 10 multiplied by itself 24 times,
or a "1" preceded by a decimal point and 23 zeros.
Multiplied by 1.67, it looks like "167" preceded by a
decimal point and 23 zeros. Just as in the case with the
very large number, it is a lot easier for a human being to
deal with this "shorthand" notation. As with the prior case,
the significant digits in this quantity are clearly
expressed.
Because the significant digits are
represented "on their own," away from the power-of-ten
multiplier, it is easy to show a level of precision even
when the number looks round. Taking our 3,000 pound car
example, we could express the rounded number of 3,000 in
scientific notation as such:
car weight = 3 x 103 pounds
If the car actually weighed 3,005 pounds
(accurate to the nearest pound) and we wanted to be able to
express that full accuracy of measurement, the scientific
notation figure could be written like this:
car weight = 3.005 x 103 pounds
However, what if the car actually did weight
3,000 pounds, exactly (to the nearest pound)? If we were to
write its weight in "normal" form (3,000 lbs), it wouldn't
necessarily be clear that this number was indeed accurate to
the nearest pound and not just rounded to the nearest
thousand pounds, or to the nearest hundred pounds, or to the
nearest ten pounds. Scientific notation, on the other hand,
allows us to show that all four digits are significant with
no misunderstanding:
car weight = 3.000 x 103 pounds
Since there would be no point in adding
extra zeros to the right of the decimal point (placeholding
zeros being unnecessary with scientific notation), we know
those zeros must be significant to the precision of
the figure. |
