# numbers expressed using exponents are called

## Numbers Expressed in Numerals

there were 15 psychologists at the clinic

students were in the third, sixth, eighth, 10th, and 12th grades

However, there are exceptions to this general guideline for number usage. There are cases in which you should always use numerals to express numbers, even numbers zero through nine, and likewise, there are cases in which you should always use words to express numbers, even numbers 10 and above.

## Cases in which to always use numerals for numbers

Always use numerals to express numbers in the following cases, even numbers zero through nine:

Case

Example

Numbers that immediately precede a unit of measurement 5-mg dose

3 cm

Statistical or mathematical functions multiplied by 2
Fractions or decimals (except common fractions) 1.5

2.27

Percentages 50%

75%–80%

Ratios 4:1 ratio
Percentiles and quartiles the 5th percentile, the 95th percentile

the 3rd quartile

Times and dates (including approximations of time) 30 s

10 min

3 hr

2 days

approximately 4 months

2 years

12:30 a.m.

6 p.m. (or 6:00 p.m.)

Ages 5 years old, 18 years old

5-year-olds, 18-year-olds

Scores and points on a scale scored 6 on a 7-point scale
Exact sums of money \$10

\$50 in U.S. dollars

Numerals as numerals the numeral 2 on the keyboard

## Numbers in a series

Also use numerals to write numbers that denote a specific place in a numbered series when the number comes after the noun (e.g., Step 1). The noun before the number is also capitalized. This guideline applies to parts of books and tables as well (e.g., Chapter 1).

However, when the number comes before the noun, the usual guidelines for number use apply, as in the following examples.

Number after a noun

Number before a noun

Year 5 the 5th year
Step 1 the first step
Level 4 the fourth level
Items 3 and 5 the third and fifth items
Question 2, Question 25 the second question, the 25th question
Table 2, Figure 5 the second table, the fifth figure
Column 8, Row 7 the eighth column, the seventh row
Chapter 6, Chapter 14 the sixth chapter, the 14th chapter

## Numbers Expressed in Words

However, there are exceptions to this general guideline for number usage. There are cases in which you should always use words to express numbers, even numbers 10 and above, and likewise, there are cases in which you should always use numerals to express numbers, even numbers zero through nine.

## Cases in which to always use words for numbers

Always use words to express numbers in the following cases, even numbers 10 and above:

Case

Example

Numbers that begin a sentence, title, or heading (when possible, reword the sentence to avoid beginning with a number) Fifty percent of the students received the intervention, and the other 50% were part of a control condition.

Twenty people enrolled in the class, but 15 dropped out.

Common fractions one fifth of the class

two-thirds majority

Certain universally accepted phrases Twelve Apostles

Five Pillars of Islam

## 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:

 100 = 1 101 = 10 102 = 100 = 10 × 10 103 = 1,000 = 10 × 10 × 10 104 = 10,000 = 10 × 10 × 10 × 10

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:

 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. 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:

0.000411 = 4.11 × 1/10,000 = 4.11 × 10−4

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

### Example 1

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.

Test Yourself

Express these numbers in scientific notation.

1. 23,070
2. 0.0009706

1. 2.307 × 104
2. 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:

56↖4,9↖30↖20↖1=5.69 × 1040.0↗10↗20↗30↗42↗58=2.8 × 10−5

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.

### Exercises

1. Express these numbers in scientific notation.

1. 56.9
2. 563,100
3. 0.0804
4. 0.00000667
2. Express these numbers in scientific notation.

1. −890,000
2. 602,000,000,000
3. 0.0000004099
4. 0.000000000000011
3. Express these numbers in scientific notation.

1. 0.00656
2. 65,600
3. 4,567,000
4. 0.000005507
4. Express these numbers in scientific notation.

1. 65
2. −321.09
3. 0.000077099
4. 0.000000000218
5. Express these numbers in standard notation.

1. 1.381 × 105
2. 5.22 × 10−7
3. 9.998 × 104
6. Express these numbers in standard notation.

1. 7.11 × 10−2
2. 9.18 × 102
3. 3.09 × 10−10
7. Express these numbers in standard notation.

1. 8.09 × 100
2. 3.088 × 10−5
3. −4.239 × 102
8. Express these numbers in standard notation.

1. 2.87 × 10−8
2. 1.78 × 1011
3. 1.381 × 10−23
9. These numbers are not written in proper scientific notation. Rewrite them so that they are in proper scientific notation.

1. 72.44 × 103
2. 9,943 × 10−5
3. 588,399 × 102
10. These numbers are not written in proper scientific notation. Rewrite them so that they are in proper scientific notation.

1. 0.000077 × 10−7
2. 0.000111 × 108
3. 602,000 × 1018
11. These numbers are not written in proper scientific notation. Rewrite them so that they are in proper scientific notation.

1. 345.1 × 102
2. 0.234 × 10−3
3. 1,800 × 10−2
12. These numbers are not written in proper scientific notation. Rewrite them so that they are in proper scientific notation.

1. 8,099 × 10−8
2. 34.5 × 100
3. 0.000332 × 104
13. Write these numbers in scientific notation by counting the number of places the decimal point is moved.

1. 123,456.78
2. 98,490
3. 0.000000445
14. Write these numbers in scientific notation by counting the number of places the decimal point is moved.

1. 0.000552
2. 1,987
3. 0.00000000887
15. Use your calculator to evaluate these expressions. Express the final answer in proper scientific notation.

1. 456 × (7.4 × 108) = ?
2. (3.02 × 105) ÷ (9.04 × 1015) = ?
3. 0.0044 × 0.000833 = ?
16. Use your calculator to evaluate these expressions. Express the final answer in proper scientific notation.

1. 98,000 × 23,000 = ?
2. 98,000 ÷ 23,000 = ?
3. (4.6 × 10−5) × (2.09 × 103) = ?
17. Use your calculator to evaluate these expressions. Express the final answer in proper scientific notation.

1. 45 × 132 ÷ 882 = ?
2. [(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.

1. (9.09 × 108) ÷ [(6.33 × 109) × (4.066 × 10−7)] = ?
2. 9,345 × 34.866 ÷ 0.00665 = ?

## What are the numbers with exponents called?

Powers.

If you have an expression for a number that is of the form [math]b^e[/math], where [math]b[/math] and [math]e[/math] are simple numbers, [math]b[/math] is called the base, [math]e[/math] is called the exponent (most common in US) or index or occasionally order, and the whole expression (the result) is called the power. [This is exactly parallel to subtraction with minuend minus subtrahend yields difference,] Thus, in 3⁴, 3 is the base, 4 is the exponent or index, and the whole expression 3⁴ is the power, which can be read as “the fourth power of three” or “three to the fourth [power]” and has value 64—the brackets [] indicating that the word is optional and often simply implied.

Powers may focus on some particular fixed value for either the exponent or the base.

For example, the exponent my be fixed at 3, and the base is run through the sequence of positive integers as 1³, 2³, 3³, 4³, 5³, …are third powers, more commonly known by the special name of cubes, so that 1, 8, 27, 64, 125, … are called perfect cubes because they can be expressed as the cube of a positive integer. These numbers are called such because one can construct in geometry figures appearing as lattices—either 2-dimensional or 3-dimensional with the same number [math]n[/math] of points in each dimension—so that they look like squares or cubes, respectively, containing [math]n^2[/math] or [math]n^3[/math], respectively, lattice points.

One can extend this concept beyond human experience and intuition by increasing the exponent beyond 3, such as perfect fourth powers and perfect fifth powers, though there is no generally accepted name for such higher powers. The number 0 as a base is a special case that is not useful from a geometry standpoint as a square or cube containing no lattice points would not appear—it would just be blank. Therefore, from a geometric perspective, the case of 0 is usually excluded from consideration as a figurate number. However, from an algebraic perspective, 0² and 0³ are quite meaningful and useful expressions, causing 0 to be regarded as a perfect square or perfect cube in certain contexts.

On the other hand, we might decide to fix the base rather than the exponent and let the exponent be any nonnegative integer (yes, 0 is included in this scenario):
b⁰, b¹, b², b³, b⁴, … are called the powers of b.
We have for the case of b = 2: 1, 2, 4, 8, 16, … as the powers of 2—particularly useful in digital electronics (including most computers) and binary tree structure analysis. [NOTE: If the word “power” referred merely to the exponent, then all of 0, 1, 2, 3, 4, … (in other words, all nonnegative integers) would be powers of 2, which would make the term equivalent to “nonnegative integers” and totally useless. However, by corresponding to the whole of the expressions 2⁰, 2¹, 2², 2³, 2⁴, … the word “power” is much more useful, so that 3 and 5 are not powers of 2.]