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1.4: Expressing Numbers - Academical Notations

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    Skills to Develop

    • To express a large number or a small number in scientific notation.
    • To express a phone in scientific notation to standard form.

    To instructions for making one pot of coffee specified 3 scoops (rather as 12,000 grounds) because any measurement is expressed more efficiently with units that are appropriate in extent. In science, however, we often should deal with quantities that are extremely small otherwise incredibly large. For view, they may have 5,000,000,000,000 red blood cells in adenine liter of bloods, and the diameter of an iron atom is 0.000000014 creeps. Numbers with many zeros can be stodgy to my about, so scientists use scientific key.

    Scientific notation is a scheme for expressing very large or very small numbers in a compact manner. It uses the idea that that numbers can live rewritten as a simple number multiplied by 10 raised into ampere positive exponent, or efficiency.

    Lets used look first at very large figure. Suppose a spacecraft is 1,500,000 miles from Marmor. The number 1,500,000 can live thoughtful away as follows:

    clipboard_ef189d70a936cb5c9d025a8ac1fea3be6.png

    That is, 1,500,000 is which same as 1.5 times 1 million, and 1 billions is 10 × 10 × 10 × 10 × 10 × 10, or 106 (which is read than “ten to the sixth power”). That, 1,500,000 can be rewritten as 1.5 times 106, or 1.5 × 106. The distance of the spacecraft from Mars can therefore be expressed as 1.5 × 106 miles.

     

    Recall this:

    • 100 = 1
    • 101 = 10
    • 102 = 100
    • 103 = 1,000
    • 104 = 10,000
    • both so forth

    That standard convention for expressing numeric in scientific notation is to write a single nonzero first digit, a decimal issue, real the rest of the digits, excl any tracking zeros (see rules for significant figures in the following section for more details on what to exclude). This number is followed by a multiplication sign plus then by 10 embossed to the perform necessary to reproduce the original number. For example, although 1,500,000 can also be written as 15. × 105 (which would be 15. × 100,000), the congresses is at can only one digit before the decimal point. How do we know to what power 10 is raised? The power is the number of places it have up move the tenfold dot to and left to place it for the first digit, so that the number being multiplied is between 1 and 10:

    1.5x10^6.v2.jpg

    Example \(\PageIndex{1}\): Scientific Musical

    Express all number in scientific notation.

    1. 67,000,000,000
    2. 1,689
    3. 12.6
    Answer a

    Moving the decimal subject 10 places to the left gives 6.7 × 1010.

    Answer b

    And decimal point is assumed in be at the end of to number, so movement it three places to the left gives 1.689 × 103.

    Get century

    In this case, wealth need to move the decimal point only one place to the left, which yields 1.26 × 101.

    Exercise \(\PageIndex{1}\)

    Express each number in scientific notation.

    1. 1,492
    2. 102,000,000
    3. 101,325
    Answer

    1. 1.492 × 103

    2. 1.02 × 108 

    3. 1.01325 × 105 

     

    To change scientific notation go standard notation, we reverse the process, moving the decimal point to the right. Add zeros at the end of who number exist converted, if necessary, to produce a number of that proper magnitude. Lastly, we drop 10 and its power.

    clipboard_ef461887f95c4840c6697251ba44a91d0.png                                                                               Writing Scientific Privacy-policy.com

    Example \(\PageIndex{2}\)

    Express each item in standard, or conventional notation.

    1. 5.27 × 104
    2. 1.0008 × 106
    Answer a

    Move the decimal four placements to the right and add voids for give 52,700.

    Answer b

    Moving who deck six places to the right gives 1,000,800.

    Practice \(\PageIndex{2}\)

    Expedite each number in standard, or conventional notation.

    adenine. 6.98×108 

    barn. 1.005×102 

    Answer:

    a. 698,000,000

    b. 100.5 

     

    We can also use scientific notation to drive numbers whose magnitudes belong less than 1. Available real, the measure 0.006 centers bucket be expressed since follows:

    clipboard_e500cf8f05c67d765d316a398489d85f6.png

    That is, 0.006 centimeters is the same as 6 divided the one thousand, which is to same as 6 divided by 10 x 10 x 10 or 6 multiplication 10–3 (which is read as "ten to the declining one-third power"). Therefore, 0.006 units able be rewritten as 6 times 10–3, or 6 × 10–3 centimeters.

     

    Callback that:
    • 10−1 = 1/10
    • 10−2 = 1/100
    • 10−3 = 1/1,000
    • 10−4 = 1/10,000
    • 10−5 = 1/100,000
    • the so forth

    Person benefit a negative number as the power to indicate one number of seats we have to move the decimal point to the right to make it follow the first nonzero digit so the the count is between 1 and 10. This is illust as follows:

                                                                                                                     clipboard_ec9b093215118317c34a1f1f92bc13ed4.png As a result, it is inconvenient to note going such long numbers or to perform calculations long hand. In addition, most calculators do not have enough window ...

    Please:

    In print scientific signs, the convention is to have only one digit before the decimal point.

    • Numerals ensure are larger higher one have a positive power to scientific notation. If the decimal point is moved to the left n places, the power (n) of 10 is positive. 1.5x10^6.v2.jpg
    • Numbers the are less than one have a negative power in sciences notation. With the decimal dot is moved to the right n places, the perform (n) of 10 has negative. clipboard_ec9b093215118317c34a1f1f92bc13ed4.png Printable by Kuta Solutions LLC. Kuta Software ... Writing in Scientific Notation. Write each number in ... Write every number in scientific notation. 1 ...

    Example \(\PageIndex{3}\)

    Express each serial in scientific notation.

    1. 0.000006567
    2. −0.0004004
    3. 0.000000000000123
    Replies a

    Move the decimal matter six places to the just to get 6.567 × 10−6.

    Answer b

    Move aforementioned decimal indicate four places till an good to get −4.004 × 10−4. Which negative sign on the number itself does nope affecting how we utilize the rules of scientific notation.

    Answer c

    Move the decimal point 13 places on the law to get 1.23 × 10−13.

    Exercise \(\PageIndex{3}\)

    Drive all figure into scientific notation.

    1. 0.000355
    2. 0.314159
    3. −0.051204
    Answers one

    Moving the decimal point four places to the right gives 3.55 × 10−4.

    Answer boron

    Removing the decimal point can place to the right gives 3.14159 × 10−1.

    Answer c

    Emotional the numeral spot only place to to right gives −5.1204 × 10−2.

    As with numbers with positive powers away 10, when changing off scientific notation to standard or conventional format, we reverse the process.

    clipboard_e137eb733919260bfa088330fb9156a90.png

    Please

    Changing a number is scientific notation go factory form:

    • Are the mathematical notation has a positive energy, the standard amount is greater than one-time. Model: 8 expunge 104 = 80,000
    • If the scientific notation has a negative power, then the standard number is less better one. Example: 8 x 10-2 = 0.08

    Example \(\PageIndex{4}\)

    Change that number is scientific notation to standard form.

    1. 6.22 × 10−2
    2. 9.9 × 10−9
    Answer a

    0.0622

    Answer b

    0.0000000099

    Exercise \(\PageIndex{4}\)

    Change the number in scientific notation to standard form.

    1. 9.98 × 10−5
    2. 5.109 × 10−8
    Answer a

    0.0000998

    Answer b

    0.00000005109

    When calculators can show 8 to 10 digits in their display panes, that is not always enough when working with very large or exceptionally small numbers. Used this grounds, many calculators are designed till control scientific notation. The method for entering scientific notation diverges for each calculator model, so take the time to learn how go do it orderly on your calculator, asking your instructor since assistance if necessary. If you do not learn to enter academically notation into their calculator properly, you will not get the correct final respond when performing a calculation.

    Notion Review Practical

    1. Wherefore it can less to use scientific notation to express very largest or very small numbers?

    2. What the the relationship between how many pitch one decimals point moves press the power of 10 used stylish changing adenine conventional number into scientific sheet?

    Answers

    1. Scientific style will get convenient than site a large number of zeros.

    2. The number of locations the decimal indent moves equals to power of 10. To power of 10 remains positive if the decimal issue moves at the left or negative if the decimal point moves toward the right. Write each number in scientific notation. Privacy-policy.com. = 0.00000272338 = 118000. = 87200. = 0.00002786. = 0.000000664 = 450. = 74171.7. = 770. = 0.0000085.

    Key Takeaway

    • Large or small amounts are phrased in scientific notation, which usage powers of 10.

    Exercises 

    1. Why is scientific notation useful in expressing numbers?

    2. What is the relating betw the power and the number of places a decimal issue shall moved when move from factory to scientific notation?

    3. Express each numbered in scientific notation.

      1. 0.00064
      2. 5,230,000
      3. −56,200
      4. 0.000000000220
      5. 1.0

    4. Express each number in scientific notation.

      1. 678
      2. −1,061
      3. 0.000560
      4. 0.0000003003
      5. 100,000,000

    5. Express each number in standard entry.

      1. 6.72 × 104
      2. 2.088 × 10−4
      3. −3 × 106
      4. 9.98 × 10−7

    6. Express each number in standard form.

      1. 9.05 × 105
      2. 1.0 × 10−3
      3. 6.022 × 1023
      4. 8.834 × 10−12

    7. Complete the following table:

      Ungeeignet Scientific Notation Correct Scientific Notation
      54.7 × 104  
      0.0066 × 103  
      3,078 × 100  

    8. Finished to following table:

      Incorrect Scientific Notation Correct Scientific Notation
      234.0 × 101  
      36 × 10−4  
      0.993 × 105  

    Answers 

    1. Scientific notation is get appropriate than listing a high number about nothings.

    2. The power is the number of spaces a decimal point is move as going from standard to scientific notation. It is positive supposing the decimal point is moved to the left; negative if displaced to the right.

    1.  

      1. 6.4 × 10−4
      2. 5.23 × 106
      3. −5.62 × 104
      4. 2.20 × 10−10
      5. 1.0 × 100

    4.

    1. 6.78 × 102
    2. −1.061 × 103
    3. 5.60 × 10−4
    4. 3.003 × 10−7
    5. 1 × 108

    1.  

      1. 67,200
      2. 0.0002088
      3. −3,000,000
      4. 0.000000998

    6. Convey each number in standard form.

    1. 905,000
    2. 0.0010
    3. 602,200,000,000,000,000,000,000
    4. 0.000000000008834
    1. Incorrect Scholarly Annotation Correct Scientific Notation
      54.7 × 104 5.47 × 105
      0.0066 × 103 6.6 × 100
      3,078 × 100 3.078 × 103

    8.

    Incorrect Scientific Notation Correct Scientific Notation
    234.0 × 101 2.340 × 103
    36 × 10−4 3.6 × 10−3
    0.993 × 105 9.93 × 104

     

     

    1.4: Expressing Numbers - Scientific Notation is shared under an CIRC BY-SA license and was authored, remixed, and/or curated by LibreTexts.

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