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Hướng dẫn học tập > Prealgebra

Scientific Notation

Learning Outcomes

  • Convert from decimal notation to scientific notation
  • Convert from scientific notation to decimal notation
 

Convert from Decimal Notation to 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 \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 \frac{1}{1000}\hfill \\ 4\times {10}^{3}\hfill & & & 4\times \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.

Scientific Notation

A number is expressed in scientific notation when it is of the form a×10na\times {10}^{n} where a≥1a\ge 1 and a<10a<10 and nn 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. Scientific notation is a useful way of writing very large or very small numbers. It is used often in the sciences to make calculations easier. If we look at what happened 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.
  • The power of 1010 is positive when the number is larger than 1:4000=4×1031\text{:}4000=4\times {10}^{3}.
  • The power of 1010 is negative when the number is between 00 and 1:0.004=4×10−31\text{:}0.004=4\times {10}^{-3}.
 

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}
 

try it

[ohm_question]146311[/ohm_question]
 

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.
 

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 \frac{1}{{10}^{3}}\hfill \\ \\ \\ \hfill 5.2\times \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}

 

try it

[ohm_question]146312[/ohm_question]
 

Convert 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. 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.  

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]
 

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.
   

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]
 

Multiply and Divide Using Scientific Notation

We use the Properties of Exponents to multiply and divide numbers in scientific notation.  

example

Multiply. Write answers in decimal form: (4×105)(2×10−7)\left(4\times {10}^{5}\right)\left(2\times {10}^{-7}\right).

Answer: Solution

(4×105)(2×10−7)\left(4\times {10}^{5}\right)\left(2\times {10}^{-7}\right)
Use the Commutative Property to rearrange the factors. 4⋅2⋅105⋅10−74\cdot 2\cdot {10}^{5}\cdot {10}^{-7}
Multiply 44 by 22 and use the Product Property to multiply 105{10}^{5} by 10−7{10}^{-7}. 8×10−28\times {10}^{-2}
Change to decimal form by moving the decimal two places left. 0.080.08

 

try it

[ohm_question]146318[/ohm_question]
   

example

Divide. Write answers in decimal form: 9×1033×10−2\frac{9\times {10}^{3}}{3\times {10}^{-2}}.

Answer: Solution

9×1033×10−2\frac{9\times {10}^{3}}{3\times {10}^{-2}}
Separate the factors. 93×10310−2\frac{9}{3}\times \frac{{10}^{3}}{{10}^{-2}}
Divide 99 by 33 and use the Quotient Property to divide 103{10}^{3} by 10−2{10}^{-2} . 3×1053\times {10}^{5}
Change to decimal form by moving the decimal five places right. 300,000300,000

 

try it

[ohm_question]146319[/ohm_question]
The following video is a mini-lesson on how to convert decimals to scientific notation, and back to a decimal. Additionally, you will see more examples of how to multiply and divide numbers given in scientific notation. https://youtu.be/hY-ecKyZ244  

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