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Converting Scientific Notation to Decimal Form

Converting Scientific Notation to Decimal Form

How can we convert from scientific notation to decimal form? Let’s look at two numbers written in scientific notation and see.

\(\begin{array}{cccc}\hfill 9.12\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{4}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}9.12\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-4}\hfill \\ \hfill 9.12\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}10,000\hfill & & & \hfill \phantom{\rule{4em}{0ex}}9.12\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}0.0001\hfill \\ \hfill 91,200\hfill & & & \hfill \phantom{\rule{4em}{0ex}}0.000912\hfill \end{array}\)

If we look at the location of the decimal point, we can see an easy method to convert a number from scientific notation to decimal form.

\(\begin{array}{cccc}\hfill 9.12\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{4}=91,200\hfill & & & \hfill \phantom{\rule{4em}{0ex}}9.12\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-4}=0.000912\hfill \end{array}\)

This figure has two columns. In the left column is 9.12 times 10 to the fourth power equals 91,200. Below this, the same scientific notation is repeated, with an arrow showing the decimal point in 9.12 being moved four places to the right. Because there are no digits after 2, the final two places are represented by blank spaces. Below this is the text “Move the decimal point four places to the right.” In the right column is 9.12 times 10 to the negative fourth power equals 0.000912. Below this, the same scientific notation is repeated, with an arrow showing the decimal point in 9.12 being moved four places to the left. Because there are no digits before 9, the remaining three places are represented by spaces. Below this is the text “Move the decimal point 4 places to the left.”

In both cases the decimal point moved 4 places. When the exponent was positive, the decimal moved to the right. When the exponent was negative, the decimal point moved to the left.

Example: How to Convert Scientific Notation to Decimal Form

Convert to decimal form: \(6.2\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{3}.\)

Solution

This figure is a table that has three columns and three rows. The first column is a header column, and it contains the names and numbers of each step. The second column contains further written instructions. The third column contains math. On the top row of the table, the first cell on the left reads “Step 1. Determine the exponent, n, on the factor 10.” The second cell reads “The exponent is 3.” The third cell contains 6.2 times 10 cubed.In the second row, the first cell reads “Step 2. Move the decimal n places, adding zeros if needed. If the exponent is positive, move the decimal point n places to the right. If the exponent is negative, move the decimal point absolute value of n places to the left.” The second cell reads “The exponent is positive so move the decimal point 3 places to the right. We need to add two zeros as placeholders.” The third cell contains 6.200, with an arrow showing the decimal point jumping places to the right, from between the 6 and 2 to after the second 00 in 6.200. Below this is the number 6,200.In the third row, the first cell reads “Step 3. Check to see if your answer makes sense.” The second cell is blank. The third reads “10 cubed is 1000 and 1000 times 6.2 will be 6,200.” Beneath this is 6.2 times 10 cubed equals 6,200.

The steps are summarized below.

Convert scientific notation to decimal form.

To convert scientific notation to decimal form:

  1. Determine the exponent, \(n\), on the factor 10.
  2. Move the decimal \(n\) places, adding zeros if needed.
    • If the exponent is positive, move the decimal point \(n\) places to the right.
    • If the exponent is negative, move the decimal point \(|n|\) places to the left.
  3. Check.

Example

Convert to decimal form: \(8.9\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-2}.\)

Solution

 8.9 times 10 to the power of negative 2.
Determine the exponent, n, on the factor 10.The exponent is negative 2.
Since the exponent is negative, move the decimal point 2 places to the left.8.9, with an arrow the decimal place showing the decimal point being moved two places to the left.
Add zeros as needed for placeholders.8.9 times 10 to the power of negative 2 equals 0.089.

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