2.2: Expressing Numbers

Learning Objective

• Learn to express numbers properly

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

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

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 shows the powers of 10. Left side has 10 to a specific power and right side has the equation for the specified power.

10^0	= 1
10^1	= 10
10^2	= 100 = 10 × 10
10^3	= 1,000 = 10 × 10 × 10
10^4	= 10,000 = 10 × 10 × 10 × 10

and so forth.

The raised number to the right of the 10 indicates the number of factors of 10 in the original number. (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 shows the powers of 10 with negative exponents. Left side has 10 to a specific power and right side has the equation for the specified power.

10^−1	= 0.1 = 1/10
10^−2	= 0.01 = 1/100
10^−3	= 0.001 = 1/1,000
10^−4	= 0.0001 = 1/10,000

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. We 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,

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

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

\[0.000411 = 4.11 \times \dfrac{1}{10,000} = 4.11 \times 10^{−4}\nonumber \]

Typically, the extra zero digits at the end or the beginning of a number are not included (Figure \(\PageIndex{1}\)).

Figure \(\PageIndex{1}\): "The Blue Marble" is a famous photograph of the Earth taken on December 7, 1972, by the crew of the Apollo 17 spacecraft en route to the Moon at a distance of about 29,000 kilometers. This is \(2.9 \times 10^4\, km\). It shows Africa, Antarctica, and the Arabian Peninsula.The earth is about 93,000,000 miles from the sun. In scientific notation, this is \(9.3 \times 10^7\, miles\). (Public Domain; NASA/Apollo 17 crew; taken by either Harrison Schmitt or Ron Evans.)

Example \(\PageIndex{1}\): Expressing Numbers in Scientific Notation

Express these numbers in scientific notation.

1. 306,000
2. 0.00884
3. 2,760,000
4. 0.000000559

Solution

1. The number 306,000 is 3.06 times 100,000, or 3.06 times 10⁵. In scientific notation, the number is 3.06 × 10⁵.
2. The number 0.00884 is 8.84 times 1/1,000, which is 8.84 times 10⁻³. In scientific notation, the number is 8.84 × 10⁻³.
3. The number 2,760,000 is 2.76 times 1,000,000, which is the same as 2.76 times 10⁶. In scientific notation, the number is written as 2.76 × 10⁶. Note that we omit the zeros at the end of the original number.
4. The number 0.000000559 is 5.59 times 1/10,000,000, which is 5.59 times 10⁻⁷. In scientific notation, the number is written as 5.59 × 10⁻⁷.

Exercise \(\PageIndex{1}\)

Express these numbers in scientific notation.

1. 23,070
2. 0.0009706

Answer a

2.307 × 10⁴

Answer b

9.706 × 10⁻⁴

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:

56900 can be written as 5.69 times 10 to the fourth power. 0.000028 can be written as 2.8 times 10 to the negative fifth power.

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 (Figure \(\PageIndex{2}\)).

Figure \(\PageIndex{2}\): This calculator shows only the coefficient and the power of 10 to represent the number in scientific notation. Thus, the number being displayed is \(3.84951 \times 10^{18}\), or \(3,849,510,000,000,000,000\). Source: "Casio" Asim Bijarani is licensed under Creative Commons Attribution 2.0 Generic.

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.