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 [latex]10[/latex]. 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 [latex]4000[/latex] and [latex]0.004[/latex]. We know that [latex]4000[/latex] means [latex]4\times 1000[/latex] and [latex]0.004[/latex] means [latex]4\times {\Large\frac{1}{1000}}[/latex]. If we write the [latex]1000[/latex] as a power of ten in exponential form, we can rewrite these numbers in this way:[latex]\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}[/latex]
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 [latex]10[/latex], and the second factor is a power of [latex]10[/latex] 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 [latex]0[/latex] and [latex]1[/latex]. 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  [latex]1000[/latex] x [latex]1000[/latex] = a thousand thousands  [latex]1,000,000[/latex]  [latex]10^6[/latex] 
billion  [latex](1000[/latex] x [latex]1000)[/latex] x 1000 = a thousand millions  [latex]1,000,000,000[/latex]  [latex]10^9[/latex] 
trillion  [latex](1000[/latex] x [latex]1000[/latex] x [latex]1000[/latex]) x [latex]1000[/latex] = a thousand billions  [latex]1,000,000,000,000[/latex]  [latex]10^{12}[/latex] 
Scientific Notation
A positive number is written in scientific notation if it is written as [latex]a\times10^{n}[/latex] where the coefficient a is [latex]1\leq{a}<10[/latex], and n is an integer.Number  Scientific Notation?  Explanation 
[latex]1.85\times10^{2}[/latex]  yes  [latex]1\leq1.85<10[/latex] [latex]2[/latex] is an integer 
[latex] \displaystyle 1.083\times {{10}^{\frac{1}{2}}}[/latex]  no  [latex] \displaystyle \frac{1}{2}[/latex] is not an integer 
[latex]0.82\times10^{14}[/latex]  no  0.82 is not [latex]\geq1[/latex] 
[latex]10\times10^{3}[/latex]  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  
[latex]500.0[/latex]  [latex]5\times10^{2}[/latex]  [latex]0.05[/latex]  [latex]5\times10^{2}[/latex]  
[latex]80,000.0[/latex]  [latex]8\times10^{4}[/latex]  [latex]0.0008[/latex]  [latex]8\times10^{4}[/latex]  
[latex]43,000,000.0[/latex]  [latex]4.3\times10^{7}[/latex]  [latex]0.00000043[/latex]  [latex]4.3\times10^{7}[/latex]  
[latex]62,500,000,000.0[/latex]  [latex]6.25\times10^{10}[/latex]  [latex]0.000000000625[/latex]  [latex]6.25\times10^{10}[/latex] 
[latex]\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}[/latex]
Notice that the decimal point was moved [latex]5[/latex] places to the left, and the exponent is [latex]5[/latex]. 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
 Move the decimal point so that the first factor is greater than or equal to [latex]1[/latex] but less than [latex]10[/latex].
 Count the number of decimal places, [latex]n[/latex], that the decimal point was moved.
 Write the number as a product with a power of [latex]10[/latex].
 If the original number is:
 greater than [latex]1[/latex], the power of [latex]10[/latex] will be [latex]{10}^{n}[/latex].
 between [latex]0[/latex] and [latex]1[/latex], the power of [latex]10[/latex] will be [latex]{10}^{n}[/latex].
 If the original number is:
 Check.
example
Write [latex]37,000[/latex] in scientific notation. SolutionStep 1: Move the decimal point so that the first factor is greater than or equal to [latex]1[/latex] but less than [latex]10[/latex].  
Step 2: Count the number of decimal places, [latex]n[/latex] , that the decimal point was moved.  [latex]3.70000[/latex] [latex]4[/latex] places 
Step 3: Write the number as a product with a power of [latex]10[/latex].  [latex]3.7\times {10}^{4}[/latex] 
If the original number is:


Step 4: Check.  
[latex]{10}^{4}[/latex] is [latex]10,000[/latex] and [latex]10,000[/latex] times [latex]3.7[/latex] will be [latex]37,000[/latex].  
[latex]37,000=3.7\times {10}^{4}[/latex] 
Example
Write the following numbers in scientific notation. [latex]920,000,000[/latex]
 [latex]10,200,000[/latex]
 [latex]100,000,000,000[/latex]
Answer:
 [latex]\underset{\longleftarrow}{920,000,000}[/latex] 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 [latex]1[/latex] and [latex]10[/latex]. [latex]\underset{\longleftarrow}{920,000,000}=920,000,000.0[/latex], move the decimal point [latex]8[/latex] times to the left and you will have [latex]9.20,000,000[/latex], now we can replace the zeros with an exponent of [latex]8[/latex], [latex]9.2\times10^{8}[/latex]
 [latex]\underset{\longleftarrow}{10,200,000}=10,200,000.0=1.02\times10^{7}[/latex], note here how we included the [latex]0[/latex] and the [latex]2[/latex] 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.
 [latex]\underset{\longleftarrow}{100,000,000,000}=100,000,000,000.0=1.0\times10^{11}[/latex]
try it
[ohm_question]146311[/ohm_question][latex]\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}[/latex]
You may notice that the decimal point was moved five places to the right until you got to the number 4, which is between [latex]1[/latex] and [latex]10 [/latex]. The exponent is [latex]−5[/latex].example
Write in scientific notation: [latex]0.0052[/latex]Answer: Solution
[latex]0.0052[/latex]  
Move the decimal point to get [latex]5.2[/latex], a number between [latex]1[/latex] and [latex]10[/latex].  
Count the number of decimal places the point was moved.  [latex]3[/latex] places 
Write as a product with a power of [latex]10[/latex].  [latex]5.2\times {10}^{3}[/latex] 
Check your answer: [latex]\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}[/latex]  
[latex]0.0052=5.2\times {10}^{3}[/latex] 
Example
Write the following numbers in scientific notation. [latex]0.0000000000035[/latex]
 [latex]0.0000000102[/latex]
 [latex]0.00000000000000793[/latex]
Answer:
 [latex]\underset{\longrightarrow}{0.0000000000035}=3.5\times10^{12}[/latex], we moved the decimal 12 times to get to a number between 1 and 10
 [latex]\underset{\longrightarrow}{0.0000000102}=1.02\times10^{8}[/latex]
 [latex]\underset{\longrightarrow}{0.00000000000000793}=7.93\times10^{15}[/latex]
try it
[ohm_question]146312[/ohm_question]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.[latex]\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}[/latex]
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. 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 [latex]5.88\times10^{12}[/latex], and a hydrogen atom has a diameter of [latex]5\times10^{8}[/latex] 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.[latex]\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}[/latex]
For each power o f[latex]10[/latex], 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 [latex]1[/latex] less than the exponent because it takes one power of [latex]10[/latex] 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
 Determine the exponent, [latex]n[/latex], on the factor [latex]10[/latex].
 Move the decimal [latex]n[/latex] places, adding zeros if needed.
 If the exponent is positive, move the decimal point [latex]n[/latex] places to the right.
 If the exponent is negative, move the decimal point [latex]n[/latex] places to the left.
 Check.
example
Convert to decimal form: [latex]6.2\times {10}^{3}[/latex]Answer: Solution
Step 1: Determine the exponent, [latex]n[/latex] , on the factor [latex]10[/latex].  [latex]6.2\times {10}^{3}[/latex] 
Step 2: Move the decimal point [latex]n[/latex] places, adding zeros if needed.  

[latex]6,200[/latex] 
Step 3: Check to see if your answer makes sense.  
[latex]{10}^{3}[/latex] is [latex]1000[/latex] and [latex]1000[/latex] times [latex]6.2[/latex] will be [latex]6,200[/latex].  [latex]6.2\times {10}^{3}=6,200[/latex] 
try it
[ohm_question]146313[/ohm_question]example
Convert to decimal form: [latex]8.9\times {10}^{2}[/latex]Answer: Solution
[latex]8.9\times {10}^{2}[/latex]  
Determine the exponent [latex]n[/latex] , on the factor [latex]10[/latex].  The exponent is [latex]−2[/latex]. 
Move the decimal point [latex]2[/latex] places to the left.  
Add zeros as needed for placeholders.  [latex]0.089[/latex] 
[latex]8.9\times {10}^{2}=0.089[/latex]  
The Check is left to you. 
try it
[ohm_question]146314[/ohm_question]Example
Write the following in decimal notation. [latex]4.8\times10^{4}[/latex]
 [latex]3.08\times10^{6}[/latex]
Answer:
 [latex]4.8\times10^{4}[/latex], the exponent is negative, so we need to move the decimal to the left. [latex]\underset{\longleftarrow}{4.8\times10^{4}}=\underset{\longleftarrow}{.00048}[/latex]
 [latex]3.08\times10^{6}[/latex], the exponent is positive, so we need to move the decimal to the right. [latex]\underset{\longrightarrow}{3.08\times10^{6}}=\underset{\longrightarrow}{3080000}[/latex]
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 [latex]1[/latex] in scientific notation. Will your exponent be positive or negative? [practicearea rows="1"][/practicearea] 2. You are writing a number that is between [latex]0[/latex] and [latex]1[/latex] in scientific notation. Will your exponent be positive or negative? [practicearea rows="1"][/practicearea] 3. What power do you need to put on [latex]10[/latex] to get a result of [latex]1[/latex]? [practicearea rows="1"][/practicearea]Answer: 1. You are writing a number that is greater than [latex]1[/latex] in scientific notation. Will your exponent be positive or negative? For numbers greater than [latex]1[/latex], the exponent on [latex]10[/latex] will be positive when you are using scientific notation. Refer to the table presented above:
Word  How many thousands  Number  Scientific Notation 
million  [latex]1000[/latex] x [latex]1000[/latex] = a thousand thousands  [latex]1,000,000[/latex]  [latex]10^6[/latex] 
billion  [latex](1000[/latex] x [latex]1000)[/latex] x [latex]1000[/latex] = a thousand millions  [latex]1,000,000,000[/latex]  [latex]10^9[/latex] 
trillion  [latex](1000 x 1000 x 1000) x 1000[/latex] = a thousand billions  [latex] 1,000,000,000,000[/latex]  [latex]10^{12}[/latex] 
[latex]\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}[/latex]
We now have described the notation necessary to write all possible numbers on the number line in scientific notation.
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.Contribute!
Licenses & Attributions
CC licensed content, Original
 Revision and Adaptation. Provided by: Lumen Learning License: CC BY: Attribution.
CC licensed content, Shared previously
 Unit 11: Exponents and Polynomials, from Developmental Math: An Open Program. Provided by: Monterey Institute of Technology and Education Located at: https://www.nroc.org/. License: CC BY: Attribution.
 Examples: Write a Number in Scientific Notation. Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.
 Examples: Writing a Number in Decimal Notation When Given in Scientific Notation. Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.