2.2 Expressing Numbers

Quantities have two parts: the number and the unit. The number tells “how many.” It is important to be able to express numbers properly so that the quantities can be communicated properly.

Standard notation is the straightforward expression of a number. Numbers such as 17, 101.5, and 0.00446 are expressed in standard notation. For relatively small numbers, standard notation is fine. However, for very large numbers, such as 306,000,000, or for very small numbers, such as 0.000000419, standard notation can be cumbersome because of the number of zeros needed to place nonzero numbers in the proper position.

Scientific notation is an expression of a number using powers of 10. Powers of 10 are used to express numbers that have many zeros:

Table 2.1 Powers of 10
[latex]10^0[/latex] = [latex]1[/latex]
[latex]10^1[/latex] = [latex]10[/latex]
[latex]10^2[/latex] = [latex]100 = 10\times 10[/latex]
[latex]10^3[/latex] = [latex]1,000 = 10\times 10\times 10[/latex]
[latex]10^4[/latex] = [latex]10,000 = 10\times 10\times 10\times 10[/latex]

and so forth. The raised number to the right of the 10 indicating the number of factors of 10 in the original number is the exponent. (Scientific notation is sometimes called exponential notation.) The exponent’s value is equal to the number of zeros in the number expressed in standard notation.

Small numbers can also be expressed in scientific notation but with negative exponents:

Table 2.2 Powers of Negative 10
[latex]10^{-1}[/latex] [latex]= 0.1 = \dfrac{1}{10}[/latex]
[latex]10^{-2}[/latex] [latex]= 0.01 = \dfrac{1}{100}[/latex]
[latex]10^{-3}[/latex] [latex]= 0.001 = \dfrac{1}{1,000}[/latex]
[latex]10^{-4}[/latex] [latex]= 0.0001 = \dfrac{1}{10,000}[/latex]

and so forth. Again, the value of the exponent is equal to the number of zeros in the denominator of the associated fraction. A negative exponent implies a decimal number less than one.

A number is expressed in scientific notation by writing the first nonzero digit, then a decimal point, and then the rest of the digits. The part of a number in scientific notation that is multiplied by a power of 10 is called the coefficient. Then determine the power of 10 needed to make that number into the original number and multiply the written number by the proper power of 10. For example, to write 79,345 in scientific notation,

[latex]79,345 = 7.9345\times 10,000 = 7.9345\times 10^4[/latex]

Thus, the number in scientific notation is [latex]7.9345\times 10^4[/latex]. For small numbers, the same process is used, but the exponent for the power of [latex]10[/latex] is negative:

[latex]0.000411 = 4.11\times \dfrac{1}{10,000} = 4.11 × 10^{-4}[/latex]

Typically, the extra zero digits at the end or the beginning of a number are not included.

Example 2.1

Problems

Express these numbers in scientific notation.

  1. [latex]306,000[/latex]
  2. [latex]0.00884[/latex]
  3. [latex]2,760,000[/latex]
  4. [latex]0.000000559[/latex]
Solutions
  1. The number [latex]306,000[/latex] is [latex]3.06[/latex] times [latex]100,000[/latex], or [latex]3.06[/latex] times [latex]10^5[/latex]. In scientific notation, the number is [latex]3.06\times 10^5[/latex].
  2. The number [latex]0.00884[/latex] is [latex]8.84[/latex] times [latex]\frac{1}{1,000}[/latex], which is [latex]8.84[/latex] times [latex]10^{-3}[/latex]. In scientific notation, the number is [latex]8.84\times 10^{-3}[/latex].
  3. The number [latex]2,760,000[/latex] is [latex]2.76[/latex] times [latex]1,000,000[/latex], which is the same as [latex]2.76[/latex] times [latex]10^6[/latex]. In scientific notation, the number is written as [latex]2.76\times 10^6[/latex]. Note that we omit the zeros at the end of the original number.
  4. The number [latex]0.000000559[/latex] is [latex]5.59[/latex] times [latex]\frac{1}{10,000,000}[/latex], which is [latex]5.59[/latex] times [latex]10^{-7}[/latex]. In scientific notation, the number is written as [latex]5.59\times 10^{-7}[/latex].

Exercise 2.1

Express these numbers in scientific notation.

  1. [latex]23,070[/latex]
  2. [latex]0.0009706[/latex]
Solutions
  1. [latex]2.307\times 10^4[/latex]
  2. [latex]9.706\times 10^{-4}[/latex]

Another way to determine the power of 10 in scientific notation is to count the number of places you need to move the decimal point to get a numerical value between 1 and 10. The number of places equals the power of 10. This number is positive if you move the decimal point to the right and negative if you move the decimal point to the left.

Many quantities in chemistry are expressed in scientific notation. When performing calculations, you may have to enter a number in scientific notation into a calculator. Be sure you know how to correctly enter a number in scientific notation into your calculator. Different models of calculators require different actions for properly entering scientific notation. If in doubt, consult your instructor immediately.

Exercise 2.2

  1. Express these numbers in scientific notation.
    1. [latex]56.9[/latex]
    2. [latex]563,100[/latex]
    3. [latex]0.0804[/latex]
    4. [latex]0.00000667[/latex]
  2. Express these numbers in scientific notation.
    1. [latex]-890,000[/latex]
    2. [latex]602,000,000,000[/latex]
    3. [latex]0.0000004099[/latex]
    4. [latex]0.000000000000011[/latex]
  3. Express these numbers in scientific notation.
    1. [latex]0.00656[/latex]
    2. [latex]65,600[/latex]
    3. [latex]4,567,000[/latex]
    4. [latex]0.000005507[/latex]
  4. Express these numbers in scientific notation.
    1. [latex]65[/latex]
    2. [latex]-321.09[/latex]
    3. [latex]0.000077099[/latex]
    4. [latex]0.000000000218[/latex]
  5. Express these numbers in standard notation.
    1. [latex]1.381\times 10^5[/latex]
    2. [latex]5.22\times 10^{-7}[/latex]
    3. [latex]9.998\times 10^4[/latex]
  6. Express these numbers in standard notation.
    1. [latex]7.11\times 10^{-2}[/latex]
    2. [latex]9.18\times 10^2[/latex]
    3. [latex]3.09\times 10^{-10}[/latex]
  7. Express these numbers in standard notation.
    1. [latex]8.09\times 10^0[/latex]
    2. [latex]3.088\times 10^{-5}[/latex]
    3. [latex]-4.239\times 10^2[/latex]
  8. Express these numbers in standard notation.
    1. [latex]2.87\times 10^{-8}[/latex]
    2. [latex]1.78\times 10^{11}[/latex]
    3. [latex]1.381\times 10^{-23}[/latex]
  9. These numbers are not written in proper scientific notation. Rewrite them so that they are in proper scientific notation.
    1. [latex]72.44\times 10^3[/latex]
    2. [latex]9,943\times 10^{-5}[/latex]
    3. [latex]588,399\times 10^2[/latex]
  10. These numbers are not written in proper scientific notation. Rewrite them so that they are in proper scientific notation.
    1. [latex]0.000077\times 10^{-7}[/latex]
    2. [latex]0.000111\times 10^8[/latex]
    3. [latex]602,000\times 10^18[/latex]
  11. These numbers are not written in proper scientific notation. Rewrite them so that they are in proper scientific notation.
    1. [latex]345.1\times 10^2[/latex]
    2. [latex]0.234\times 10^{-3}[/latex]
    3. [latex]1,800\times 10^{-2}[/latex]
  12. These numbers are not written in proper scientific notation. Rewrite them so that they are in proper scientific notation.
    1. [latex]8,099\times 10^{-8}[/latex]
    2. [latex]34.5\times 10^0[/latex]
    3. [latex]0.000332\times 10^4[/latex]
  13. Write these numbers in scientific notation by counting the number of places the decimal point is moved.
    1. [latex]123,456.78[/latex]
    2. [latex]98,490[/latex]
    3. [latex]0.000000445[/latex]
  14. Write these numbers in scientific notation by counting the number of places the decimal point is moved.
    1. [latex]0.000552[/latex]
    2. [latex]1,987[/latex]
    3. [latex]0.00000000887[/latex]
  15. Use your calculator to evaluate these expressions. Express the final answer in proper scientific notation.
    1. [latex]456\times (7.4\times 10^8) = ?[/latex]
    2. [latex](3.02\times 10^5)\div (9.04\times 10^15) = ?[/latex]
    3. [latex]0.0044\times 0.000833 = ?[/latex]
  16. Use your calculator to evaluate these expressions. Express the final answer in proper scientific notation.
    1. [latex]98,000\times 23,000 = ?[/latex]
    2. [latex]98,000\div 23,000 = ?[/latex]
    3. [latex](4.6\times 10^{-5})\times (2.09\times 10^3) = ?[/latex]
  17. Use your calculator to evaluate these expressions. Express the final answer in proper scientific notation.
    1. [latex]45\times 132\div 882 =?[/latex]
    2. [latex][(6.37\times 10^4)\times (8.44\times 10^{-4})]\div (3.2209\times 10^{15}) = ?[/latex]
  18. Use your calculator to evaluate these expressions. Express the final answer in proper scientific notation.
    1. [latex](9.09\times 10^8)\div [(6.33\times 10^9)\times (4.066\times 10^{-7})] = ?[/latex]
    2. [latex]9,345\times 34.866\div 0.00665 = ?[/latex]
Solutions to Odd-Numbered Questions
    1. [latex]5.69\times 10^1[/latex]
    2. [latex]5.631\times 10^5[/latex]
    3. [latex]8.04\times 10^{-2}[/latex]
    4. [latex]6.67\times 10^{-6}[/latex]
    1. [latex]6.56\times 10^{-3}[/latex]
    2. [latex]6.56\times 10^4[/latex]
    3. [latex]4.567\times 10^6[/latex]
    4. [latex]5.507\times 10^{-6}[/latex]
    1. [latex]138,100[/latex]
    2. [latex]0.000000522[/latex]
    3. [latex]99,980[/latex]
    1. [latex]8.09[/latex]
    2. [latex]0.00003088[/latex]
    3. [latex]-423.9[/latex]
    1. [latex]7.244\times 10^4[/latex]
    2. [latex]9.943\times 10^{-2}[/latex]
    3. [latex]5.88399\times 10^7[/latex]
    1. [latex]3.451\times 10^4[/latex]
    2. [latex]2.34\times 10^{-4}[/latex]
    3. [latex]1.8\times 10^1[/latex]
    1. [latex]1.2345678\times 10^5[/latex]
    2. [latex]9.849\times 10^4[/latex]
    3. [latex]4.45\times 10^{-7}[/latex]
    1. [latex]3.3744\times 10^11[/latex]
    2. [latex]3.3407\times 10^{-11}[/latex]
    3. [latex]3.665\times 10^{-6}[/latex]
    1. [latex]6.7346\times 10^0[/latex]
    2. [latex]1.6691\times 10^{-14}[/latex]

Attribution

“Expressing Numbers” from Introductory Chemistry by David W. Ball and Jessie A. Key is licensed under a Creative Commons Attribution-NonCommercial-Share-Alike 4.0 International License, except where otherwise noted.

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Fanshawe College Astronomy Copyright © 2023 by Dr. Iftekhar Haque is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.