1.2 Handling Numbers in Science

Learning Objective

By the end of this section, you will be able to:

• Learn to express numbers in scientific notation.

Throughout this course, you will frequently encounter very large and very small numbers.  Handling these numbers can be quiet cumbersome, especially when performing calculations.

For example, later on in this course we will be working with Avogadro’s number (the number of elementary entities in 1 mole of a substance).

Avogadro’s number, as show above, is expressed in standard notation. For relatively small numbers, standard notation is fine. However, for very large numbers, such as Avogadro’s number, or for very small numbers, such as 0.000000419, using standard notation can be cumbersome because of the number of zeros needed to place nonzero numbers in the proper position. In order to make it easy to handle large or small numbers scientist use a method called scientific notation. Regardless of the magnitude, all numbers can be expressed in the form of scientific notation, N × 10n, where N is a number between 1 and 9, and n is either a positive or negative whole number.

Avogadro’s number in scientific notation form:  6.022 × 1023

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

 $\large 10^{0}=1$ $\large 10^{1}=10$ $\large 10^{2}=10\times10=100$ $\large 10^{3}=10\times10\times10=1000$ $\large 10^{3}=10\times10\times10\times10=10000$

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:

 $\large 10^{-1}=\frac{1}{10}=0.1$ $\large 10^{-2}=\frac{1}{100}=0.01$ $\large 10^{-3}=\frac{1}{1000}=0.001$ $\large 10^{-4}=\frac{1}{10000}=0.0001$

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,

79,345 = 7.9345 × 10,000 = 7.9345 × 104

Thus, the number in scientific notation is 7.9345 × 104. For small numbers, the same process is used, but the exponent for the power of 10 is negative:

$\large 0.000411 = 4.11\times \frac{1}{10000} = 4.11\times 10^{-4}$

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

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.  Examples of scientific notation are listed in Table 1.

Table 1. Examples of Scientific Notation
453 4.53 × 102
0.0022 2.2 × 10−3
80031575 8.0031575 × 107
700.1 7.001 × 102
0.334 3.34 × 10−1
50000 5 × 104
0.00000000000065 2.2 × 10−13

Example 1: Writing Numbers in scientific notation

As of 2022, the United States population was estimated to be 338,977,986 people.  What is this population expressed in scientific notation?

Express the following numbers in scientific notation.

1. 306,000
2. 0.00884
3. 2,760,000
4. 0.000000559
5. 23,070
6. 0.0009706

Example 2: Converting Scientific notation to standard form

The atomic radius of gold (Au) is 1.46 × 10-8 cm.  What is the radius of gold expressed in standard form?

Express the following numbers in standard form:

1. 1.84 × 10−5
2. 5.4 × 103
3. 7.311 × 10−9
4. 9.10 × 106
5. 1.00 × 101
6. 8 × 10−4

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.

Key Takeaways

• Standard notation expresses a number normally.
• Scientific notation expresses a number as a coefficient times a power of 10.
• The power of 10 is positive for numbers greater than 1 and negative for numbers between 0 and 1.

Exercises

1. Express these numbers in scientific notation.

a)  56.9

b)  563,100

c)  0.0804

d)  0.00000667

2.  Express these numbers in scientific notation.

a)  890,000

b)  602,000,000,000

c)  0.0000004099

d)  0.000000000000011

3.  Express these numbers in scientific notation.

a)  0.00656

b)  65,600

c)  4,567,000

d)  0.000005507

4.  Express these numbers in scientific notation.

a)  65

b)  321.09

c)  0.000077099

d)  0.000000000218

5.  Express these numbers in standard notation.

a)  1.381 × 105

b)  5.22 × 10−7

c)  9.998 × 104

6.  Express these numbers in standard notation.

a)  7.11 × 10−2

b)  9.18 × 102

c)  3.09 × 10−10

7.  Express these numbers in standard notation.

a)  8.09 × 100

b)  3.088 × 10−5

c)  4.239 × 102

8.  Express these numbers in standard notation.

a)  2.87 × 10−8

b)  1.78 × 1011

c)  1.381 × 10−23

9. These numbers are not written in proper scientific notation. Rewrite them so that they are in proper scientific notation.

a)  72.44 × 103

b)  9,943 × 10−5

c)  588,399 × 102

10.  These numbers are not written in proper scientific notation. Rewrite them so that they are in proper scientific notation.

a)  0.000077 × 10−7

b)  0.000111 × 108

c)  602,000 × 1018

11.  These numbers are not written in proper scientific notation. Rewrite them so that they are in proper scientific notation.

a)  345.1 × 102

b)  0.234 × 10−3

c)  1,800 × 10−2

12.  These numbers are not written in proper scientific notation. Rewrite them so that they are in proper scientific notation.

a)  8,099 × 10−8

b)  34.5 × 100

c)  0.000332 × 104

13.  Write these numbers in scientific notation by counting the number of places the decimal point is moved.

a)  123,456.78

b)  98,490

c)  0.000000445

14.  Write these numbers in scientific notation by counting the number of places the decimal point is moved.

a)  0.000552

b)  1,987

c)  0.00000000887

15.  Use your calculator to evaluate these expressions. Express the final answer in proper scientific notation.

a)  456 × (7.4 × 108) = ?

b)  (3.02 × 105) ÷ (9.04 × 1015) = ?

c)  0.0044 × 0.000833 = ?

16.  Use your calculator to evaluate these expressions. Express the final answer in proper scientific notation.

a)  98,000 × 23,000 = ?

b)  98,000 ÷ 23,000 = ?

c)  (4.6 × 10−5) × (2.09 × 103) = ?

17.  Use your calculator to evaluate these expressions. Express the final answer in proper scientific notation.

a)  45 × 132 ÷ 882 = ?

b) [(6.37 × 104) × (8.44 × 10−4)] ÷ (3.2209 × 1015) = ?

18.  Use your calculator to evaluate these expressions. Express the final answer in proper scientific notation.

a)  (9.09 × 108) ÷ [(6.33 × 109) × (4.066 × 10−7)] = ?

b)  9,345 × 34.866 ÷ 0.00665 = ?

Glossary

Scientific Notation: Numerical expression of a large or small number using a coefficient (1 through 9, N) followed by a power of 10 raised to a positive/negative whole number (n). Scientific notation form: N × 10n.