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Study Guides > ALGEBRA / TRIG I

Converting Between Scientific Notation and Decimal Notation

Learning Outcomes

  • Define decimal and scientific notations
  • Convert from scientific notation to decimal notation
  • Convert from decimal notation to scientific notation

Definition of Scientific Notation

Remember working with place value for whole numbers and decimals? Our number system is based on powers of 1010. We use tens, hundreds, thousands, and so on. Our decimal numbers are also based on powers of tens—tenths, hundredths, thousandths, and so on. Consider the numbers 40004000 and 0.0040.004. We know that 40004000 means 4×10004\times 1000 and 0.0040.004 means 4×110004\times {\Large\frac{1}{1000}}. If we write the 10001000 as a power of ten in exponential form, we can rewrite these numbers in this way:

40000.0044×10004×110004×1034×11034×10−3\begin{array}{cccc}4000\hfill & & & 0.004\hfill \\ 4\times 1000\hfill & & & 4\times {\Large\frac{1}{1000}}\hfill \\ 4\times {10}^{3}\hfill & & & 4\times {\Large\frac{1}{{10}^{3}}}\hfill \\ & & & \hfill 4\times {10}^{-3}\hfill \end{array}

When a number is written as a product of two numbers, where the first factor is a number greater than or equal to one but less than 1010, and the second factor is a power of 1010 written in exponential form, it is said to be in scientific notation. Before we can convert between scientific and decimal notation, we need to know the difference between the two. Scientific notation is used by scientists, mathematicians, and engineers when they are working with very large or very small numbers. Using exponential notation, large and small numbers can be written in a way that is easier to read. When a number is written in scientific notation, the exponent tells you if the term is a large or a small number. A positive exponent indicates a large number and a negative exponent indicates a small number that is between 00 and 11. It is difficult to understand just how big a billion or a trillion is. Here is a way to help you think about it.
Word How many thousands Number Scientific Notation
million 10001000 x 10001000 = a thousand thousands 1,000,0001,000,000  10610^6
billion (1000(1000 x 1000)1000) x 1000 = a thousand millions 1,000,000,0001,000,000,000   10910^9
trillion (1000(1000 x 10001000 x 10001000) x 10001000 = a thousand billions  1,000,000,000,0001,000,000,000,000   101210^{12}
11 billion can be written as 1,000,000,0001,000,000,000 or represented as 10910^9. How would 22 billion be represented? Since 22 billion is 22 times 11 billion, then 22 billion can be written as 2×1092\times10^9. A light year is the number of miles light travels in one year, about 5,880,000,000,0005,880,000,000,000.  That's a lot of zeros, and it is easy to lose count when trying to figure out the place value of the number. Using scientific notation, the distance is 5.88×10125.88\times10^{12} miles. The exponent of 1212 tells us how many places to count to the left of the decimal. Another example of how scientific notation can make numbers easier to read is the diameter of a hydrogen atom, which is about 0.00000005mm0.00000005 mm, and in scientific notation is 5×10−85\times10^{-8} mm. In this case the −8-8 tells us how many places to count to the right of the decimal. Outlined in the box below are some important conventions of scientific notation format.

Scientific Notation

A positive number is written in scientific notation if it is written as a×10na\times10^{n} where the coefficient a is 1≤a<101\leq{a}<10, and n is an integer.
It is customary in scientific notation to use ×\times as the multiplication sign, even though we avoid using this sign elsewhere in algebra. Look at the numbers below. Which of the numbers is written in scientific notation?
Number Scientific Notation? Explanation
1.85×10−21.85\times10^{-2} yes 1≤1.85<101\leq1.85<10 −2-2 is an integer
1.083×1012 \displaystyle 1.083\times {{10}^{\frac{1}{2}}} no 12 \displaystyle \frac{1}{2} is not an integer
0.82×10140.82\times10^{14} no 0.82 is not ≥1\geq1
10×10310\times10^{3} no 10 is not < 10

Convert from decimal notation to scientific notation

Now let’s compare some numbers expressed in both scientific notation and standard decimal notation in order to understand how to convert from one form to the other. Take a look at the tables below. Pay close attention to the exponent in the scientific notation and the position of the decimal point in the decimal notation.

Large Numbers

  

Small Numbers
Decimal Notation Scientific Notation   Decimal Notation Scientific Notation
500.0500.0 5×1025\times10^{2} 0.050.05 5×10−25\times10^{-2}
80,000.080,000.0 8×1048\times10^{4} 0.00080.0008 8×10−48\times10^{-4}
43,000,000.043,000,000.0 4.3×1074.3\times10^{7} 0.000000430.00000043 4.3×10−74.3\times10^{-7}
62,500,000,000.062,500,000,000.0 6.25×10106.25\times10^{10} 0.0000000006250.000000000625 6.25×10−106.25\times10^{-10}
If we look at what happens to the decimal point, we can see a method to easily convert from decimal notation to scientific notation. On the left, we see 4000 equals 4 times 10 cubed. Beneath that is the same thing, but there is an arrow from after the last 0 in 4000 to between the 4 and the first 0. Beneath, it says, In both cases, the decimal was moved 33 places to get the first factor, 44, by itself. To write a large number in scientific notation, move the decimal point to the left to obtain a number between 11 and 1010. Since moving the decimal point changes the value, you have to multiply the decimal by a power of 1010 so that the expression has the same value. Let’s look at an example.

180,000.=18,000.0×1011,800.00×102180.000×10318.0000×1041.80000×105180,000=1.8×105\begin{array}{r}180,000.=18,000.0\times10^{1}\\1,800.00\times10^{2}\\180.000\times10^{3}\\18.0000\times10^{4}\\1.80000\times10^{5}\\180,000=1.8\times10^{5}\end{array}

Notice that the decimal point was moved 55 places to the left, and the exponent is 55. In the examples shown below, we will be following this general strategy for converting from decimals to scientific notation:

Convert from decimal notation to scientific notation

  1. Move the decimal point so that the first factor is greater than or equal to 11 but less than 1010.
  2. Count the number of decimal places, nn, that the decimal point was moved.
  3. Write the number as a product with a power of 1010.
    • If the original number is:
      • greater than 11, the power of 1010 will be 10n{10}^{n}.
      • between 00 and 11, the power of 1010 will be 10−n{10}^{-n}.
  4. Check.
In the examples below, we are converting large decimal values to scientific notation.

example

Write 37,00037,000 in scientific notation. Solution
Step 1: Move the decimal point so that the first factor is greater than or equal to 11 but less than 1010. .
Step 2: Count the number of decimal places, nn , that the decimal point was moved. 3.700003.70000 44 places
Step 3: Write the number as a product with a power of 1010. 3.7×1043.7\times {10}^{4}
If the original number is:
  • greater than 1, the power of 10 will be 10n{10}^{n} .
  • between 0 and 1, the power of 10 will be 10−n{10}^{\mathrm{-n}}
Step 4: Check.
104{10}^{4} is 10,00010,000 and 10,00010,000 times 3.73.7 will be 37,00037,000.
37,000=3.7×10437,000=3.7\times {10}^{4}

Example

Write the following numbers in scientific notation.
  1. 920,000,000920,000,000
  2. 10,200,00010,200,000
  3. 100,000,000,000100,000,000,000

Answer:

  1.  920,000,000⟵\underset{\longleftarrow}{920,000,000}  We will move the decimal point to the left, it helps to place it at the end of the number and then count how many times you move it to get one number before it that is between 11 and 1010. 920,000,000⟵=920,000,000.0\underset{\longleftarrow}{920,000,000}=920,000,000.0, move the decimal point 88 times to the left and you will have 9.20,000,0009.20,000,000, now we can replace the zeros with an exponent of 88, 9.2×1089.2\times10^{8}
  2. 10,200,000⟵=10,200,000.0=1.02×107\underset{\longleftarrow}{10,200,000}=10,200,000.0=1.02\times10^{7}, note here how we included the 00 and the 22 after the decimal point.  In some disciplines, you may learn about when to include both of these.  Follow instructions from your teacher on rounding rules.
  3. 100,000,000,000⟵=100,000,000,000.0=1.0×1011\underset{\longleftarrow}{100,000,000,000}=100,000,000,000.0=1.0\times10^{11}

try it

[ohm_question]146311[/ohm_question]
  Now let's look at converting a very small decimal number to scientific notation.  To write a small number (between 00 and 11) in scientific notation, you move the decimal to the right and the exponent will have to be negative, as in the following example.

0.00004⟶=00.0004×10−1000.004×10−20000.04×10−300000.4×10−4000004.×10−50.00004=4×10−5\begin{array}{r}\underset{\longrightarrow}{0.00004}=00.0004\times10^{-1}\\000.004\times10^{-2}\\0000.04\times10^{-3}\\00000.4\times10^{-4}\\000004.\times10^{-5}\\0.00004=4\times10^{-5}\end{array}

You may notice that the decimal point was moved five places to the right until you got  to the number 4, which is between 11 and 10 10 . The exponent is −5−5.

example

Write in scientific notation: 0.00520.0052

Answer: Solution

0.00520.0052
Move the decimal point to get 5.25.2, a number between 11 and 1010. .
Count the number of decimal places the point was moved. 33 places
Write as a product with a power of 1010. 5.2×10−35.2\times {10}^{-3}
Check your answer: 5.2×10−35.2×11035.2×110005.2×0.0010.0052\begin{array}{c}\hfill 5.2\times {10}^{-3}\hfill \\ \hfill 5.2\times {\Large\frac{1}{{10}^{3}}}\hfill \\ \\ \\ \hfill 5.2\times {\Large\frac{1}{1000}}\hfill \\ \hfill 5.2\times 0.001\hfill \\ \hfill 0.0052\hfill \end{array}
0.0052=5.2×10−30.0052=5.2\times {10}^{-3}

Example

Write the following numbers in scientific notation.
  1. 0.00000000000350.0000000000035
  2. 0.00000001020.0000000102
  3. 0.000000000000007930.00000000000000793

Answer:

  1. 0.0000000000035⟶=3.5×10−12\underset{\longrightarrow}{0.0000000000035}=3.5\times10^{-12}, we moved the decimal 12 times to get to a number between 1 and 10
  2. 0.0000000102⟶=1.02×10−8\underset{\longrightarrow}{0.0000000102}=1.02\times10^{-8}
  3. 0.00000000000000793⟶=7.93×10−15\underset{\longrightarrow}{0.00000000000000793}=7.93\times10^{-15}

try it

[ohm_question]146312[/ohm_question]
In the following video you are provided with examples of how to convert both a large and a small number in decimal notation to scientific notation. https://youtu.be/fsNu3AdIgdk

Convert from scientific notation to decimal notation

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

9.12×1049.12×10−49.12×10,0009.12×0.000191,2000.000912\begin{array}{cccc}9.12\times {10}^{4}\hfill & & & 9.12\times {10}^{-4}\hfill \\ 9.12\times 10,000\hfill & & & 9.12\times 0.0001\hfill \\ 91,200\hfill & & & 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. On the left, we see 9.12 times 10 to the 4th equals 91,200. Beneath that is 9.12 followed by 2 spaces, with an arrow from the decimal to after the second space, times 10 to the 4th equals 91,200. On the right, we see 9.12 times 10 to the negative 4 equals 0.000912. Beneath that is three spaces followed by 9.12 with an arrow from the decimal to after the first space, times 10 to the negative 4 equals 0.000912. 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. Recall the number of miles that light travels in a year is 5.88×10125.88\times10^{12}, and a hydrogen atom has a diameter of 5×10−85\times10^{-8} mm. To write each of these numbers in decimal notation, you move the decimal point the same number of places as the exponent. If the exponent is positive, move the decimal point to the right. If the exponent is negative, move the decimal point to the left.

5.88×1012=5.880000000000.⟶=5,880,000,000,0005×10−8=0.00000005.⟵=0.00000005\begin{array}{l}5.88\times10^{12}=\underset{\longrightarrow}{5.880000000000.}=5,880,000,000,000\\5\times10^{-8}=\underset{\longleftarrow}{0.00000005.}=0.00000005\end{array}

For each power o f1010, you move the decimal point one place. Be careful here and don’t get carried away with the zeros—the number of zeros after the decimal point will always be 11 less than the exponent because it takes one power of 1010 to shift that first number to the left of the decimal. As we practice converting from scientific notation to decimal form, we will be following these steps:

Convert scientific notation to decimal form

  1. Determine the exponent, nn, on the factor 1010.
  2. Move the decimal nn places, adding zeros if needed.
    • If the exponent is positive, move the decimal point nn places to the right.
    • If the exponent is negative, move the decimal point ∣n∣|n| places to the left.
  3. Check.
Let's start by converting a large number in scientific notation to decimal form.

example

Convert to decimal form: 6.2×1036.2\times {10}^{3}

Answer: Solution

Step 1: Determine the exponent, nn , on the factor 1010. 6.2×1036.2\times {10}^{3}
Step 2: Move the decimal point nn places, adding zeros if needed. .
  • If the exponent is positive, move the decimal point nn places to the right.
  • If the exponent is negative, move the decimal point ∣n∣|n| places to the left.
 6,2006,200
Step 3: Check to see if your answer makes sense.
103{10}^{3} is 10001000 and 10001000 times 6.26.2 will be 6,2006,200. 6.2×103=6,2006.2\times {10}^{3}=6,200

try it

[ohm_question]146313[/ohm_question]
Now, let's try converting small numbers to decimal form.

example

Convert to decimal form: 8.9×10−28.9\times {10}^{-2}

Answer: Solution

8.9×10−28.9\times {10}^{-2}
Determine the exponent nn , on the factor 1010. The exponent is −2−2.
Move the decimal point 22 places to the left. .
Add zeros as needed for placeholders. 0.0890.089
8.9×10−2=0.0898.9\times {10}^{-2}=0.089
The Check is left to you.

try it

[ohm_question]146314[/ohm_question]
 

Example

Write the following in decimal notation.
  1. 4.8×10−44.8\times10^{-4}
  2. 3.08×1063.08\times10^{6}

Answer:

  1. 4.8×10−44.8\times10^{-4}, the exponent is negative, so we need to move the decimal to the left.  4.8×10−4⟵=.00048⟵\underset{\longleftarrow}{4.8\times10^{-4}}=\underset{\longleftarrow}{.00048}
  2. 3.08×1063.08\times10^{6}, the exponent is positive, so we need to move the decimal to the right.  3.08×106⟶=3080000⟶\underset{\longrightarrow}{3.08\times10^{6}}=\underset{\longrightarrow}{3080000}

 

Think About It

To help you get a sense of the relationship between the sign of the exponent and the relative size of a number written in scientific notation, answer the following questions. You can use the textbox to wirte your ideas before you reveal the solution. 1. You are writing a number that is greater than 11 in scientific notation.  Will your exponent be positive or negative? [practice-area rows="1"][/practice-area] 2. You are writing a number that is between 00 and 11 in scientific notation.  Will your exponent be positive or negative? [practice-area rows="1"][/practice-area] 3. What power do you need to put on 1010 to get a result of 11? [practice-area rows="1"][/practice-area]

Answer: 1. You are writing a number that is greater than 11 in scientific notation. Will your exponent be positive or negative? For numbers greater than 11, the exponent on 1010 will be positive when you are using scientific notation. Refer to the table presented above:

Word How many thousands Number Scientific Notation
million 10001000 x 10001000 = a thousand thousands 1,000,0001,000,000  10610^6
billion (1000(1000 x 1000)1000) x 10001000 = a thousand millions 1,000,000,0001,000,000,000   10910^9
trillion (1000x1000x1000)x1000(1000 x 1000 x 1000) x 1000 = a thousand billions  1,000,000,000,000 1,000,000,000,000   101210^{12}
2. You are writing a number that is between 00 and 11in scientific notation. Will your exponent be positive or negative? We can reason that since numbers greater than 11 will have a positive exponent, numbers between 00 and  11 will have a negative exponent. Why are we specifying numbers between  00 and  11? The numbers between 00 and 11 represent amounts that are fractional. Recall that we defined numbers with a negative exponent as a−n=1an{a}^{-n}=\frac{1}{{a}^{n}}, so if we have 10−210^{-2} we have 110×10=1100\frac{1}{10\times10}=\frac{1}{100} which is a number between 00 and 11. 3. What power do you need to put on 1010 to get a result of 11? Recall that any number or variable with an exponent of 00 is equal to 11, as in this example:

t8t8=t8t8=1t8t8=t8−8=t0 therefore t0=1\begin{array}{c}\frac{t^{8}}{t^{8}}=\frac{\cancel{t^{8}}}{\cancel{t^{8}}}=1\\\frac{{t}^{8}}{{t}^{8}}={t}^{8 - 8}={t}^{0}\\\text{ therefore }\\{t}^{0}=1\end{array}

We now have described the notation necessary to write all possible numbers on the number line in scientific notation.

  In the next video you will see how to convert a number written in scientific notation into decimal notation. https://youtu.be/8BX0oKUMIjw

Summary

Large and small numbers can be written in scientific notation to make them easier to understand. In the next section, you will see that performing mathematical operations such as multiplication and division on large and small numbers is made easier by scientific notation and the rules of exponents.  

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