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.
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