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Chemistry: Scientific (Exponential) Notation


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About this Lesson

  • Type: Video Tutorial
  • Length: 11:40
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 125 MB
  • Posted: 01/28/2009

This lesson is part of the following series:

Chemistry: Full Course (303 lessons, $198.00)
Chemistry Review (25 lessons, $49.50)
Chemistry: Introduction to Matter and Measurement (13 lessons, $14.85)
Chemistry: Mathematics of Chemistry (2 lessons, $2.97)

Scientific Notation is a shorthand method for expressing extremely large or small numbers. Prof. Yee uses Avogadro's number (6.022137 x 10^23) as an example of commonly used scientific notation. Scientific notation is a number that is expressed as r * 10^t, when r is a number greater than or equal to 1 and less than 10. The exponent will be positive if the actual number is greater than 1 and negative if the actual number is smaller than 1. Scientific notation is also a way to ensure that it is not ambiguous as to how many significant figures are in the number. Prof. Yee then instructs you how to add and subtract with numbers in scientific notation, first expressing both numbers to the same power of 10. Then he will teach you how to multiply and divide numbers in scientific notation. Finally, he will show you how to compute numbers in scientific notation with powers and roots.

Taught by Professor Yee, this lesson was selected from a broader, comprehensive course, Chemistry. This course and others are available from Thinkwell, Inc. The full course can be found at The full course covers atoms, molecules and ions, stoichiometry, reactions in aqueous solutions, gases, thermochemistry, Modern Atomic Theory, electron configurations, periodicity, chemical bonding, molecular geometry, bonding theory, oxidation-reduction reactions, condensed phases, solution properties, kinetics, acids and bases, organic reactions, thermodynamics, nuclear chemistry, metals, nonmetals, biochemistry, organic chemistry, and more."

Gordon Yee is an associate professor of chemistry at Virginia Tech in Blacksburg, VA. He received his Ph.D. from Stanford University and completed postdoctoral work at DuPont. A widely published author, Professor Yee studies molecule-based magnetism.

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An Introduction to Matter and Measurement
Mathematics of Chemistry
Scientic (Exponential) Notation Page [1 of 2]
Scientists have their own special language, and I’m going to share one of the secrets with you know, how we write
down really, really, really big numbers. It’s a shorthand notation that includes all of the information, but it saves a lot
of paper. And that technique is called writing a number in scientific notation.
So here’s a number that some of you may recognize already and, if you don’t recognize it, it’s no big deal, but it is a
very big number. It’s 602213700, followed by 15 more zeros. And, obviously, if every time you had to write down this
number and, for reasons that’ll become clear, it’ll be important to write down this number several times in your study of
chemistry, you’d use a lot of paper. So rather than write it this way, what we’re going to do is we’re going to forget
about all the zeros. Not forget about them, but we’re going to use a shorthand notation for taking care of all of the
zeros. And the reason why we can do this is because the zeros aren’t really significant numbers anyway, they’re
placeholders. And so what we do is we express numbers, either very large numbers or very small numbers, and both
positive and negative numbers, by writing them in two pieces. And the first piece is the piece that includes all of the
non-zero digits and maybe some zeros that happen to be interspersed in there. This part of the number is going to be
greater than or equal to one and less than ten. So, in other words, there will be a number, followed by a decimal
point, followed by a bunch more numbers. It’s possible that you wouldn’t have the decimal point, that it would just be
a single digit, but the point is that digit is going to be one through nine. And then we have a multiplication sign, and
then we’d follow it by ten to some power, where this power tells us how many zeros we have to include in order to get
this number to be exactly the same as this number. And the way to remember whether this number should be positive
or negative is, first of all, big numbers have a positive exponent and small numbers, numbers that are less than one,
have a negative exponent. And to go from this number to this number, recognize that we go from this point, so there’s
an implied decimal point at the end of the zeros here, one, two, three, four, five, six, seven, eight, nine, ten, eleven,
twelve, thirteen, fourteen, fifteen, sixteen, seventeen, eighteen, nineteen, twenty, twenty-one, twenty-two, twentythree.
In order to get the decimal point between the six and the zero, we have to move the decimal point twenty-three
times to the left. And this twenty-three tells us that’s how many places we had to move it in order to get it into what we
call exponential notation. Now, one of the really good benefits of exponential notation is that every digit in this part of
the number is a significant figure. So you have to consider every single digit here as significant.
Now, let’s look at some numbers and convert them into scientific notation and make sure that we understand how we
go about the process of getting from a number that’s just written out with all the zeros to a number that’s in scientific or
exponential notation.
25.0, remember, we want a number that’s between one and 9.9999999, greater than or equal to one or less than ten.
So to go from 25.0 to something in exponential notation, we have to move the decimal place one place to the left,
which means that we put a one here, a positive one, and that number is significant, so it’s 2.50 ´ 101. Here we have
a number that’s less than one, and so we’re going to have to move the decimal place the other way to get it into
scientific notation, one, two, three, four. So it’s 3.71, and then 10-4, because we had to move the decimal place to the
right to get it into scientific notation. And 1,376, again, we’re moving it to the left, one, two, three. So there’s a 103.
And here’s .33167, so we’d have to move that one, one place to the right. And so we have a minus one here.
Now, again, one of the great advantages of scientific notation is that the number of significant figures is unambiguous.
So 100 is somewhat ambiguous. Does it have one significant figure? Does it have two significant figures? Does it
have three significant figures? Some conventions say that, if it has a period here, a decimal place, then it has three
significant figures. And if it doesn’t have a decimal place, then it only has one significant figure. And that’s a
reasonable way to do it, but it’s somewhat ambiguous. It’s entirely unambiguous if we express 100 in scientific
notation, because we can either express it as 1 ´ 102, which has one significant figure, or 1.0 ´ 102, which has two
significant figures, or 1.00 ´ 102, which has three significant figures. If we want to express it to four, or five, or six
significant figures, the point is, we’d just be putting significant zeros after the decimal place and the placeholder part,
the 102, goes along for the ride.
Now, when we’re doing arithmetic with numbers in scientific notation, if you’re using your calculator, you really can
ignore the rest of this lecture, because your calculator is going to be doing what I’m telling you. You can do longhand,
if you don’t happen to have a calculator, or if your teacher doesn’t allow you to use calculators. We know, for
instance, that 3.94 + 0.67 = 4.61. If this is written in scientific notation, it would be 3.41, so you don’t typically write
100 for numbers that are between one and 9.99999. You just would write 3.94 and forget about the fact that 100 is just
one. And then if you’re adding this number written in scientific notation, scientific notation would look like that. And
what you have to do is you have to convert them both to have the same power of ten. So the exponentials must be
An Introduction to Matter and Measurement
Mathematics of Chemistry
Scientic (Exponential) Notation Page [2 of 2]
same when you want to add them. And so, if we, for instance, converted this back into, excuse me, put it in with an
exponent of zero, it means that we would convert this number into this. So this number is equal to this number, 0.67 ´
100, just as this is 3.94 ´ 100. So we can convert this number to here, and then we can get to the answer. And
remember that, when you’re considering the number of significant figures in an addition problem, it is if we have two
figures past the decimal point in this number and two in this number, then the sum is going to have two significant
figures past the decimal point. Whichever number has the fewer number of places past the decimal point, that’s what
determines the number of significant figures in a sum or a difference.
The other way to do this is to convert them both into numbers that have an exponent of minus one. And, of course,
this is not, this number is not in scientific notation. But now that they both have an exponent of minus one, we can
add these two parts and we get 46.1 ´ 10-1. So when the exponents are the same, then it comes along for the ride.
And then converting this back into scientific notation, we get to the same answer, 4.61.
When we’re multiplying numbers, recognize that 10a ´ 10b = 10a+b, where this is the exponent. So 3.94 ´ 102 ´ 6.7 ´
10-3, we don’t have to do this conversion into a common exponential, but we do have to group together the part that
isn’t a power of ten. So, for instance, we group 3.94 and 6.7. And then we group the parts that are a power of ten
and use the fact that 10a ´ 10b = 10a+b. So we would add the exponents to a minus three, and then we would multiply
these parts out, 3.94 ´ 6.7, which, if you work it out on a calculator, looks like 26.398. Then you have to put this part
into the correct number of significant figures, which, in this case, is going to be two, because there are three
significant figures in 3.94, but only two in 6.7. So to the correct number of significant figures, this product is 26 ´ 10-1.
And then, correcting it to scientific notation, we move the decimal place one place to the right and that cancels out the
minus one here. So the answer is 2.6. Plug it into your calculator and you’ll see that the answer is, in fact, 2.6 with a
bunch of other numbers, but rounding, considering the number of significant figures, the answer is 2.6.
And finally, we have to consider powers and roots and, once again, with powers and roots, if you’re using your
electronic calculator, it’s not really an issue. But 10a raised to the B power is 10ab. And so if we have a number that ’s
R ´ 10t raised to the B power, so this whole number raised to the B power, it is Rb ´ 10tb. And so, for instance, 6.7 ´
10-14 = 6.74, and then 10tb, which, in this case, is –1 ´ 4. Now, we multiply this product, and this part is the part that
has the significant figures, and I’ve expressed it with four significant figures here. So (6.7)4, which is 6.7 ´ 6.7 ´ 6.7 ´
6.7 = 2,015 and a bunch of other digits, but this is already too many significant figures. When we round to the correct
number of significant figures, which is two, because we’re multiplying, we get 2.0, and then 10-4, excuse me, we get
back, this is 2,015, so we get back three orders of magnitude or three powers of ten. So those three powers of ten
cancel with most of the four powers of ten here, so we get a minus one. Let me express that; 2.015 ´ 103 ´ 10-4. So
2,015 is 2.015 ´ 103, and then there’s a 10-4, so the three and the four from the previous page become a minus one.
And then, when we round this to the correct number of significant figures, we get 2.0 ´ 10-1.
And to take square roots or n-th roots, this is the b-th root of the number r ´ 10t. It’s R to the one over b power.
Remember that a b-th root is a number raised to the power one over b times ten to the t over b. This is just an
extension of what we had on the previous page. Another way to write this is r ´ 10t, all raised to the one over b power.
And so that’s r to the one over b times 10 to the t over b. And so if we wanted to take the square root of 9.67 ´ 10-4,
it’s 9.67 taken to the half-power times 10-4 raised to the half-power, and that’s 3.11 to the correct number of significant
figures and 10 to the minus four over two, which is 10-2.
So again, all of the stuff about addition and multiplication, the only thing you really need to worry about is rounding to
the correct number of significant figures. Your calculator is going to take care of the rest of it. But, on the other hand,
it would be great if what the calculator is doing was not a great mystery to you; that you were actually aware of the
arithmetic and aware of the algebra and what exactly is going on. Other than that, scientific notation is a way to
express really big numbers, numbers much, much larger than one or much, much smaller than one, in a convenient
way that makes it really crystal clear how big the number is without having to count all of the zeros.

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