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3.3: Scientific Notation

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    In science, measurements are often extremely small or extremely large. It is inefficient to write the many zeroes in very small numbers like 0.00000000000000523. Usually, the order of magnitude and the first few digits of the number are what people are interested in. How should you represent these extreme numbers?

    Scientific Notation

    Scientific notation is a means of representing very large and very small numbers in a more efficient way. The general form of scientific notation is \(a \cdot 10^{b}\)

    The \(a\) is a number between 1 and 10 and most often includes a decimal. The integer \(b\) is called the order of magnitude and is a measure of the general size of the number. If \(b\) is negative then the number is small and if \(b\) is positive then the number is large.

    \(1,240,000=1.24 \cdot 10^{6}\)
    \(0.0000354=3.54 \cdot 10^{-5}\)

    Note that when switching to and from scientific notation the sign of \(b\) indicates which direction and how many places to move the decimal point.

    Take the number 0.000 000 000 000 000 000 000 000 000 000 910 938 22 kg. This is about the mass of an electron. The number is too long to write out all the time so it is best to write it in scientific notation.

    To write this number in scientific notation, count the number of decimal places that you have to move to determine the exponent. Since the decimal is moving to the right, it should be a negative exponent. An electron's mass written in scientific notation is:

    \(9.1093822 \cdot 10^{-31}\)

    Multiplying and dividing numbers that are in scientific notation is just an exercise in exponent rules:

    \(\begin{aligned}\left(a \cdot 10^{x}\right) \cdot\left(b \cdot 10^{y}\right) &=a \cdot b \cdot 10^{x+y} \\\left(a \cdot 10^{x}\right) \div\left(b \cdot 10^{y}\right) &=\frac{a}{b} \cdot 10^{x-y} \end{aligned}\)

    Addition and subtraction require the numbers to have identical order of magnitudes.

    \(1.2 \cdot 10^{6}-5.5 \cdot 10^{5}=12 \cdot 10^{5}-5.5 \cdot 10^{5}=6.5 \cdot 10^{5}\)


    Example 1

    Earlier, you were asked how to represent numbers with a large number of zeroes either before or after the decimal point. In order to represent an extremely large or small number you should count the number of moves necessary for the decimal point to be directly after the first non-zero digit. This count will be the order of magnitude and will be used as the exponent of 10 as a means of representing how large or small the number is.

    Example 2

    The Earth’s circumference is approximately 40,000,000 meters. What is the radius of the earth in scientific notation?

    The relationship between circumference and radius is \(C=2 \pi r\).

    \(\begin{aligned} 4.0 \cdot 10^{7} &=2 \pi r \\ r &=\frac{4.0}{2 \pi} \cdot 10^{7} \approx 0.6366 \ldots 10^{7}=6.366 \cdot 10^{6} \end{aligned}\)

    Note that the number of significant digits required depends on the number of significant digits (or significant figures) in the original measurements. This example is an approximation, therefore the number of significant digits aren't necessarily accurate.

    Example 3

    Order the following numbers from least to greatest.

    \(\begin{array}{lllll}5.411 \cdot 10^{-3} & 7.837 \cdot 10^{-4} & 9.999 \cdot 10^{3} & 9.5983 \cdot 10^{-7} & 8.0984 \cdot 10^{3}\end{array}\)

    First consider the order of magnitude of each number. Small numbers have negative exponents. If two numbers have the same order of magnitude, then compare the actual digits.

    \(9.5983 \cdot 10^{-7}<7.837 \cdot 10^{-4}<5.411 \cdot 10^{-3}<8.0984 \cdot 10^{3}<9.999 \cdot 10^{3}\)

    Example 4

    Compute the following number and use scientific notation.

    \(2,000,000^{3} \cdot 3,000^{4}\)

    First convert each number to scientific notation individually, then process the exponent and multiplication.

    \(\begin{aligned} 2,000,000^{3} \cdot 3,000^{4} &=\left(2 \cdot 10^{6}\right)^{3} \cdot\left(3 \cdot 10^{3}\right)^{4} \\ &=8 \cdot 10^{18} \cdot 81 \cdot 10^{12} \\ &=648 \cdot 10^{30} \\ &=6.48 \cdot 10^{32} \end{aligned}\)

    Example 5

    Simplify the following expression.

    \(\left(4.713 \cdot 10^{7}\right)+\left(8.985 \cdot 10^{5}\right)-\left(4.987 \cdot 10^{2}\right) \cdot\left(7.3 \cdot 10^{-6}\right) \div\left(6.74 \cdot 10^{-9}\right)\)

    Resolve in order of standard order of operations

    \(\left(4.713 \cdot 10^{7}\right)+\left(8.985 \cdot 10^{5}\right)-\left(4.987 \cdot 10^{2}\right) \cdot\left(7.3 \cdot 10^{-6}\right) \div\left(6.74 \cdot 10^{-9}\right)\)
    \(=\left(4.713 \cdot 10^{7}\right)+\left(8.985 \cdot 10^{5}\right)-\left(5.40135 \cdot 10^{5}\right)\)
    \(=\left(471.3 \cdot 10^{5}\right)+\left(8.985 \cdot 10^{5}\right)-\left(5.40135 \cdot 10^{5}\right)\)
    \(=474.8836499 \cdot 10^{5}\)
    \(=4.748836499 \cdot 10^{7}\)


    Write the following numbers in scientific notation.

    1. 152,780

    2. 0.00003256

    3. 56, 320

    4. 0.0821

    5. 1, 000, 000, 000, 000, 000, 000, 000

    6. 7.32

    7. If the federal budget is $1.5 trillion, how much does it cost each individual, on average, if there are 300,000,000 people?

    8. The Library of Congress has about 60,000,000 items. How could you express this number in scientific notation?

    9. The sun develops \(5 \times 10^{23}\) horsepower per second. How much horsepower is developed in a day? In a year with 365 days?

    10. A light-year is about 5,869,713,600 miles. A spacecraft travels \(8.23 \times 10^{4}\) miles per hour. How long will it take the spacecraft to travel a light-year?

    11. Compute the following number and use scientific notation: \(324,000 \cdot 30,000^{3}\).

    12. Compute the following number and use scientific notation: \(14,300 \cdot 20,200^{2}\).

    Simplify the following expressions.

    13. \(\left(3.29 \cdot 10^{4}\right)-\left(3.295 \cdot 10^{5}\right)+\left(1.25 \cdot 10^{2}\right) \cdot\left(3.97 \cdot 10^{15}\right) \cdot\left(5.8 \cdot 10^{-6}\right)\)

    14. \(\left(1.95 \cdot 10^{2}\right)+\left(6.798 \cdot 10^{6}\right)+\left(2.896 \cdot 10^{3}\right) \cdot\left(5.6 \cdot 10^{-3}\right) \div\left(2.89 \cdot 10^{4}\right)\)

    15. \(\left(2.158 \cdot 10^{7}\right) \cdot\left(1.679 \cdot 10^{6}\right)-\left(9.98 \cdot 10^{4}\right) \cdot\left(3.4 \cdot 10^{-2}\right)\)

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