PHYSICS 101 Laboratory
Scientific Notation and Numerical Estimates
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Introduction: In order to obtain the "final answer" to problems in physics, numerical calculations are
generally required. This lab will introduce you to the idea of scientific notation of numbers - a way of
writing and simplifying arithmetic with very large and very small numbers. You will also get some
practice in estimating numerical answers.
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1.
Scientific notation is a way to conveniently write very large or very small numbers. This also makes
calculations with these types of numbers easier. The general process is to make a number more
compact by moving the decimal point, which eliminates zeros acting as placeholders. The correct
value is maintained by multiplying by ten raised to a power that equals to the number of places the
decimal point was moved.
When the decimal point is moved to the left, the exponent on the power of ten is positive. For
example, consider the number 10,100,000. Move the decimal point to the left seven (7) places to
obtain 1.01 and then multiply by 107. This yields 1.01 x 107. We conventionally move the decimal
point to provide one digit to the left of the decimal point. Sometimes it is convenient to do otherwise.
We might for example prefer to use 10.1 x 106.
When the decimal point is moved to the right, the exponent of the power of ten is negative. For
example, consider the number 0.0000312. Move the decimal point five (5) places to the right to get
3.12 and multiply by 10-5. This yields 3.12 x 10-5. Again, we could move the decimal point to
provide more or less than one digit to the left of the decimal point if convenient. We could use 31.2
x 10-6 or 0.312 x 10-4 if we wished. It is interesting to note that many calculators express large or
small numbers in scientific notation.
2.
Write the following numbers in scientific notation.
Q. 53100 =
Q. 981,000,000,000 =
Q. 125 =
Q. 0.0003 =
Q. 0.00000000000314 =
Q. -456,000 =
Q. -0.0000056 =
3.
To do arithmetic with scientific notation we only have to follow a few rules. We must first note that
scientific notation doesn't help us with addition and subtraction, except when using a calculator. We
usually must rely on conventional decimal notation to effect addition or subtraction by hand
calculations. When multiplying two numbers written in scientific notation, add the powers of ten (the
exponents). For example, to multiply 6,000 by 3,000 in conventional notation we clearly get
18,000,000. In scientific notation we have (6 x 103)(3 x 103) = 18 x 10(3+3) = 18 x 106 = 1.8 x 107,
where the last step was to get one digit before the decimal point.
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PHYSICS 101 Laboratory
Scientific Notation and Numerical Estimates
When dividing, we must subtract the power in the denominator from the power in the numerator. If
we have 6,000/3,000 we certainly know the answer is 2. To do this problem in scientific notation we
have (6 x 103)/(3 x 103) = 6/3 x 10(3-3) = 2 x 100 = 2. This example points out the important property
that 100 = 1. For that matter, any number to the "zero power" is one!
4.
Write the following numbers in scientific notation.
Q. (2 x 104)(3 x 102) =
Q. (106)(5 x 10-3) =
Q. (6 x 1023)(4 x 1028) =
Q. (9 x 103)/(3 x 109) =
Q. (6 x 10-3)/(3 x 103) =
Q. (8 x 1012)/(4 x 1015) =
Q. (-3 x 10-2)(2 x 10-3) =
Q. (6 x 103)/(-3 x 10-6) =
5.
It is interesting to investigate your calculator to find out what is the smallest "large number" that it will
display in scientific notation when you enter it without using scientific notation. What is it? Express
you answer with and without scientific notation.
6.
What is the largest "small number" that your calculator expresses in scientific notation when you
enter it without using scientific notation? Express your answer with and without scientific notation.
7.
If we a have complex computation to perform, it is often helpful to make a quick estimate of the
answer. We make rough calculations all the time, often just guessing at the answer. For example,
you may wish to know if a grocery bill seems to be correct. We often just guess by the size of the
pile rather than actually looking at the numbers. Or, we may wish to check if a restaurant tally is
about right. Or, better still, you may wish to check if a tedious calculator solution to an exam
question is "in the right ball park." A good way to estimate an answer is to imagine or write down the
numbers in scientific notation rounded to just one digit or in unusual cases to maybe two. For
example, you wish to estimate the answer for 501 x 290 / 15,125. Write this as (5 x 102)(3 x 102)/(15
x 103) = (15 x 104)/(15 x 103) = 10(4-3) = 101 =10. Certainly not an exact answer but it is a good
approximation.
Q. What is the exact answer to the problem?
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PHYSICS 101 Laboratory
8.
Scientific Notation and Numerical Estimates
For the following, first get your approximate estimate (show your work here) and then get the exact
answer to determine how good your estimates are. Comment on the "goodness" of each of your
estimates.
Q. 5016 x 314 =
Q. 314 x 5913 / 625
Q. 12,229,314 / (421 x 301)
Q. 0.000125 / (41 x 205)
Q. -12,167,123 / (-42,398 x 0.00309)
Q. -0.0001245 x (-6,125) / 414
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