Chapter 37: Large Numbers and Scientific Notation
Many people have difficulty grasping and comparing very large numbers. This is a
shame, for this is not too hard a topic to introduce and explain. Students need only learn some
vocabulary in order to name and write many large numbers. For even bigger numbers, there is
scientific notation, which offers an easy introduction to exponents.
When a number has a large number of digits, it is traditional to use commas to separate
the digits into groups of three, starting with the decimal point. For example, the number
2900459821.64 will usually be written as 2,900,459,821.64. This is the proper notation for fancy
occasions like final answers, but is not necessary in the middle of working out a problem. For
example, if the above number were multiplied by 10,000, it is much easier to move the decimal
point to the right four spaces with the commas taken out:
before putting the commas back in for the final answer: 29,004,598,216,400.
If a student attempted to multiply 2,900,459,821.64 × 10,000 with the commas, the result
can be quite confusing:
because some students will want to keep the commas where they are, for a final answer of
2,900,459,821,6400 or something like that.
The importance of the commas is not mathematical but rather linguistic – they help us to
pronounce the number. Each group of three digits (to the left of the decimal point) names a
different category of number, starting with the three closest to the decimal point and moving to
the left. The first group are the ones, measured in hundreds, tens, and ones (although in both
cases, we do not pronounce the "ones"). The next group are the thousands, also measured in
hundreds, tens, and ones. The next group are the millions, also measured in hundreds, tens, and
ones. The next groups are the billions, trillions, and quadrillions, each measured in hundreds,
tens, and ones. The next group is usually called the quintillions, but some suggest that pentillion
might be a better name. The group names go on, but this is quite enough for any practical use.
A student who feels empowered by a knowledge of names can be encouraged to research this
topic and present the results to the class, but the mathematical benefits of such work are
negligible.
We can write out the group names with commas in the following way:
quadrillions , trillions , billions , millions , thousands , (ones)
The ones are written in parentheses as a reminder that these are not pronounced. The names of
cycles can help to teach students the order of these. A "bi"-cycle has two wheels, just as
"billion" is the second sort of "illion". A "tri"-cycle has three wheels, just as "trillion" is the third
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sort of "illion". A "quad" bike has four wheels, just as "quadrillion" is the fourth sort of "illion".
Explaining the "m" in "million" can be a little more difficult for students because a one-wheeled
cycle is called a "uni-cycle" and not a "mono-cycle." Similarly, monocles (a single eye-lens,
famously used in caricatures of rich people), monorails (trains using a single rail), and monoliths
(single-stone monuments) are not in the common vocabulary of most school children.
Fortunately, the word "million" is found in everyday speech, so a teacher need only show the
class what it means.
If we expand our base-ten number columns to include large numbers, it will look
something like this:
However, it might be more useful to write it as:
For example, the number 29,004,598,216,400 will fit in the columns as:
This number is pronounced "twenty-nine trillion, four billion, five-hundred ninety-eight
million, two-hundred sixteen thousand, and four-hundred." If we write this half with numbers,
the pattern becomes even easier to see. The number 29,004,598,216,400 is pronounced "29
trillion, 4 billion, 598 million, 216 thousand, and 400." In other words, we read each set of
numbers between the parentheses as if they were less than a thousand, and then say the order of
that set (trillions, billions, millions, thousands, etc.).
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Students should know how to read out numbers as big as quadrillions. They should also
be able to write out a number that is read aloud to them. This process is even easier – write out
the numbers as they are read and separate them with parentheses. For example, a student who
hears "sixty-four quadrillion..." should write "64,". When the number continues "two-hundred
and seventeen trillion," the student should add to the number to make "64,217,". If the number
continues "fifty billion," the student should know to put in a zero before the "50" (because the
digits are being set down in threes) so that the number currently stands at "64,217,050,". The
only tricky part to this process is to know to put in three zeros when an entire order of number
(or more) is skipped. For example, if our number then finishes "and eight thousand," our student
ought to know that there are no millions and no ones, so that the number is
64,217,050,000,008,000. The students should be encouraged to read their numbers back, to
make sure that nothing has been skipped. A student who accidentally writes
"64,217,050,008,000" should be able to catch that this is approximately 64 trillion and not the 64
quadrillion requested. This example, do note, is about as difficult as possible. It is best to start
of with smaller numbers (in the thousands and millions) before moving up to larger numbers.
Also, let the class have some experience writing out numbers like "seventy-five trillion" before
more complicated ones like "seventy-five trillion, two-hundred thousand."
It is very important for people to have an appreciation for the relative sizes of large
numbers. In politics, for example, a debate might involve the numbers 1.4 trillion, 500 million,
and 70 billion. One of the best ways to compare numbers is either to set them out in number
columns or otherwise line up their digits so that the places correspond. The three numbers in our
current example will look like:
Notice that the number 1.4 trillion has a 1 in the trillion's place and the 4 in the hundredbillions place. This is because the 1 in 1.4 trillion represents 1 trillion. The .4 represents a "tenth
of a trillion," which is what you get when you divide a trillion by ten – a number in the next
column to the right. Another way to look at this is to say that 1.4 trillion = 1.4 × 1 trillion = 1.4
× 1,000,000,000,000 = 1,400,000,000,000.
As another example, 2.75 billion is written with a 2 in the billions place, then the 7 in the
next place to the right, and the 5 in the place after that, resulting in: 2,750,000,000. It helps to
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think about it like this: 2.75 is more than 2 and less than 3. Thus 2.75 billion ought to be
between 2 billion and 3 billion, as is 2,750,000,000.
In any case, to compare large numbers, it helps to set them in common terms. For
example, to compare 500 million and 70 billion, we look at them on the number-line chart:
Rather than compare 500,000,000 and 70,000,000,000 directly, we can look at them as
numbers with the same units. If we set our units to millions, then these two numbers are 500
million and 70,000 million:
The relationship between 500,000,000 and 70,000,000,000 is the same as that between
500 and 70,000. This can be seen as viewing the two numbers as a fraction:
500,000,000
500 u 1,000,000
500
. In fact, we can reduce this fraction even more
70,000,000,000 70,000 u 1,000,000 70,000
500
5 u 100
5
. We could also see this on the number chart, by using hundred70,000 700 u 100 700
millions as our units (instead of ones or millions):
by:
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It certainly is weird to change "70 billion" to "700 hundred million," but mathematically
these numbers are the same.
Now we know that the relationship between 500 million and 70 billion is the same as that
between 5 and 700. If we divide 700 ÷ 5, we get 140, which means that 70 billion is 140 times
bigger than 500 million. In other words, a government budget of $70 billion could be used to
fund 140 projects that each cost $500 million.
As another example, let us compare 1.4 trillion to 70 billion. First, we write out the
numbers as 1,400,000,000,000 and 70,000,000,000. Next, let's skip the base-ten number
columns and just write the numbers so that their digits line up:
If we cut off the same number of zeros from both numbers (dividing by the same power
of ten), the remainders of the numbers will have the same relationship as 1.4 trillion and 70
billion. We can make the numbers we need to compare as small as possible by cutting them off
right after the 7, as illustrated below:
This puts the numbers both in terms of "ten-billions": 1.4 trillion is 140 ten-billion and 70
billion is 7 ten-billion. However, this is not important for our calculations. All we need to do is
line up the numbers like this to see that the relationship between 1.4 trillion and 70 billion is the
same as that between 140 and 7. If we divide 140 ÷ 7 we get 20, thus 1.4 trillion is 20 times as
big as 70 billion. A sum of $1.4 trillion, for example, would be able to fund 20 projects that each
cost $70 billion.
Already, we can see that writing large numbers can require a large number of zeros. This
can be troublesome. For one thing, if we accidentally forget a zero or put an extra one in, this
changes the number by a factor of ten. While a lazy student might feel that there is no
substantial difference between 5000000 and 50000000, a factor of ten is a very large amount. A
yearly salary of $50000 is respectible, for example, whereas $500000 is enormous and $5000 is
well below the poverty line.
In scientific calculations, even larger numbers come into play. The distance light travels
in a year, for example, is approximately 9,460,000,000,000,000 meters. To avoid having to write
out all of these zeros, we can use scientific notation. To write a number into scientific notation,
we first break it into two parts. The first part will have a non-zero digit in the one's place and the
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rest of its digits after a decimal point. The second part will be a power of ten. For example, the
number 9,460,000,000,000,000 can be broken into 9.46 × 1,000,000,000,000,000. In other
words, this number is 9.46 quadrillion. Next, we write the power of ten as a ten raised to an
exponent. The exponent gives the number of times that 10 must be multiplied by itself to form
the number. For whole-number powers of 10, this will be the number of zeros after the 1. Thus,
in our example, 9.46 × 1,000,000,000,000, 000 = 9.46 u 1015 (there are 15 zeros after the 1).
Another way to do this is to put a decimal place after the first digit and then count the
number of spaces we must shift the decimal place to arrive at this spot. For example:
This means we need to divide 9,460,000,000,000,000 by 10 fifteen times to turn it into
9.46. Thus, to turn 9.46 back into our number, we need to multiply by 10 fifteen times, which is
represented by 9.46 u 1015 .
As another example, 35,980,000 can be converted into scientific notation as:
Thus 35,980,000 = 3.598 u 10 7 .
Students should also know how to convert from scientific notation back into whole
numbers. The trick is to move the decimal place as many times to the right as given in the
exponent of the number. For example, 2.7 u 10 4 really means 2.7 × 10 × 10 × 10 × 10 because
104 means 10 × 10 × 10 × 10 (ten multiplied by itself 4 times). To
multiply by 10 four times is to move the decimal point to the right
four times, thus 2.7 u 10 4 = 27000 = 27,000.
Similarly, the famous Avogadro's number 6.022 × 1023 is the number of carbon atoms in
12 grams of carbon. To convert this into a whole number, we need to move the decimal point 23
times:
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Thus, Avogadro's number is 602,200,000,000,000,000,000,000. This should show the
advantage of scientific notation: 6.022 × 1023 tells us that the number is basically a 6 followed by
23 zeros, with no need to actually write them all out.
Scientific notation can also represent very small numbers. This is done by using a
negative number as the exponent of 10. Just as subtracting 3 is the opposite of adding 3, for
example, multiplying by 10-3 is the opposite of multiplying by 103. Thus, the number 3.75 × 10-3
represents 3.75 divided by 10 three times. This moves the decimal point to the left three times,
resulting in 0.00375:
Similarly, the number 1.4 u 10 6 is formed by moving the decimal place to the left 6
times, resulting in 0.0000014:
To convert a very small number into scientific notation, we do the same as before. We
put the new decimal place after the first non-zero digit (the non-zero digits are called the
significant digits) and then figure out how many places we need to move the decimal point to
reach the new position. For example, the number 0.000198 is converted as:
This means that 0.000198 = 1.98 u 10 4 .
Similarly, the number "one thousandth" is 0.001, as can be seen with base-ten columns:
To convert this into scientific notation, we move the decimal point to right after the 1 by
shifting it to the right three times:
Thus one-thousandth = 0.001 = 1 u 10 3 .
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It can help to see the base-ten number columns written out with powers of ten:
Questions:
(1) Make a chart of base-ten number columns that goes up to the hundred-quadrillions and use it
to represent each of the following numbers:
(a) 35 million
(b) two-hundred and sixteen billion, four thousand and twelve
(c) 16 quadrillion, 75 trillion, 256 billion, 749 million, 218 thousand, and 15
(d) 8.4 million
(e) 27.91 billion
(f) 0.3 trillion
(g) 38,000 million
(h) 208 ten-thousand
(2) Write out the formal name of each of the following numbers:
(a) 29081
(b) 3100000
(c) 77460241
(d) 3504821222
(e) 991000000435
(f) 391498329785966843
(3) For each pair of numbers, state which one is larger, then say exactly how many times larger:
(a) 4 thousand
25 hundred
(b) 16 million
32 billion
(c) 0.2 trillion
10 billion
(d) 750 million
3 trillion
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(4) Suppose a small college can be run with an annual budget of $50 million. How many
colleges of this size could be funded with a yearly sum of $20 billion?
(5) Write out the following numbers without scientific notation:
(a) 6.49 u 10 6
(b) 2.07 u 10 4
(c) 3.9 u 10 3
(d) 8.197 u 10 2
(6) Write the following numbers in scientific notation:
(a) 480,000
(b) 97 billion
(c) 1 quadrillion
(d) 0.00914
(e) one millionth
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