# 1.2 Expressing Numbers

Learning Objectives

- Express numbers using scientific notation.
- Apply the concept of significant figures to limit a measurement to the proper number of digits.
- Recognise the number of significant figures in a given quantity.
- Limit mathematical results to the proper number of significant figures.

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

# Scientific Notation

**Standard notation** is the straightforward expression of a number. Numbers such as [latex]17, 101.5[/latex], and [latex]0.00446[/latex] are expressed in standard notation. For relatively small numbers, standard notation is fine. However, for very large numbers, such as [latex]306,000,000[/latex], or for very small numbers, such as [latex]0.000000419[/latex], 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 [latex]10[/latex]. Powers of [latex]10[/latex] are used to express numbers that have many zeros (see Table 1.2.1).

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 [latex]10[/latex] is the exponent, and it indicates the number of factors of 10 in the original number. Therefore, 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; see Table 1.2.2.

10^{−1} |
$=0.1={\displaystyle \frac{1}{10}}$ |

10^{−2} |
$=0.01={\displaystyle \frac{1}{100}}$ |

10^{−3} |
$=0.001={\displaystyle \frac{1}{1,000}}$ |

10^{−4} |
$=0.0001={\displaystyle \frac{1}{10,000}}$ |

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 [latex]{10}[/latex] is called the coefficient. Then, determine the power of [latex]{10}[/latex] needed to make that number into the original number and multiply the written number by the proper power of [latex]{10}[/latex].

For example, to write [latex]79,345[/latex] in scientific notation. For example, [latex]79,345[/latex] is written in scientific notation in this way:

[latex]{79,345 = 7.9345 × 10,000 = 7.9345 × 10^{4}}[/latex]

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

[latex]0.000411 = 4.11×(1/10000) = 4.11×10^{-4}[/latex]

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

Another way to determine the power of [latex]10[/latex] in scientific notation is to count the number of places you need to move the decimal point to get a numerical value between [latex]1[/latex] and [latex]10[/latex]. 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. An important skill is learning how to correctly enter a number in scientific notation into your calculator. This will not be covered in this book because different models of calculators require different actions for properly entering scientific notation.

Examples 1.2.1

# Problems

Express these numbers in scientific notation.

- [latex]306,000[/latex]
- [latex]0.00884[/latex]
- [latex]2,760,000[/latex]
- [latex]0.000000559[/latex]

## Solutions

- 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 × 10^{5}[/latex].
- The number [latex]0.00884[/latex] is [latex]8.84[/latex] times [latex]1/1000[/latex], which is [latex]8.84[/latex] times [latex]10^{-3}[/latex]. In scientific notation, the number is [latex]8.84 × 10^{-3}[/latex].
- 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 × 10^{6}[/latex]. Note that we omit the zeros at the end of the original number.
- The number [latex]0.000000559[/latex] is [latex]5.59[/latex] times [latex](1/1000000)[/latex], which is [latex]5.59[/latex] times [latex]10^{-7}[/latex]. In scientific notation, the number is written as [latex]5.59 × 10^{-7}[/latex].

$\frac{\mathrm{}}{}$# Test Yourself

Express these numbers in scientific notation.

- [latex]23,070[/latex]
- [latex]0.0009706[/latex]

**Answers**- [latex]2.307 × 10^{4}[/latex]
- [latex]9.706 × 10^{-4}[/latex]

# Significant Figures

If you use a calculator to evaluate the expression [latex](337÷217)[/latex]

you will get the following:

[latex]337 ÷ 217 = 1.55299539171[/latex]

and so on for many more digits. Although this answer is correct, it is somewhat presumptuous. You start with two values that each have three digits, and the answer has twelve digits. That does not make much sense from a strict numerical point of view. Do we need to report all the digits that come after the decimal?

This concept of reporting the proper number of digits in a measurement or a calculation is called significant figures. Significant figures (sometimes called significant digits) represent the limits of what values of a measurement or a calculation we are sure of. The convention for a measurement is that the quantity reported should be all known values and the first estimated value. The conventions for calculations are discussed as follows.

In many cases, you will be given a measurement. How can you tell by looking what digits are significant? For example, the reported population of the United States is [latex]306,000,000[/latex]. Does that mean that it is exactly three hundred six million or is some estimation occurring?

The following conventions dictate which numbers in a reported measurement are significant and which are not significant:

- Any nonzero digit is significant.
- Any zeros between nonzero digits (i.e., embedded zeros) are significant.
- Zeros at the end of a number without a decimal point (i.e., trailing zeros) are not significant; they serve only to put the significant digits in the correct positions. However, zeros at the end of any number with a decimal point are significant.
- Zeros at the beginning of a decimal number (i.e., leading zeros) are not significant; again, they serve only to put the significant digits in the correct positions.

So, by these rules, the population figure of the United States has only three significant figures: the 3, the 6, and the zero between them. The remaining six zeros simply put the [latex]306[/latex] in the millions position.

Examples 1.2.2

# Problems

Give the number of significant figures in each measurement.

- [latex]36.7 m[/latex]
- [latex]0.006606 s[/latex]
- [latex]2,002 kg[/latex]
- [latex]306,490,000[/latex] people

## Solutions

- By rule 1, all nonzero digits are significant, so this measurement has three significant figures.
- By rule 4, the first three zeros are not significant, but by rule 2 the zero between the sixes is; therefore, this number has four significant figures.
- By rule 2, the two zeros between the twos are significant, so this measurement has four significant figures.
- The four trailing zeros in the number are not significant, but the other five numbers are, so this number has five significant figures.

# Test Yourself

Give the number of significant figures in each measurement.

- [latex]0.000601 m[/latex]
- [latex]65.080 kg[/latex]

## Answers

- three significant figures
- five significant figures

How are significant figures handled in calculations? It depends on what type of calculation is being performed:

- If the calculation is an addition or a subtraction, the rule is as follows: limit the reported answer to the rightmost column so that all numbers have significant figures in common. For example, if you were to add [latex]1.2[/latex] and [latex]4.41[/latex], we note that the first number stops its significant figures in the tenth column, while the second number stops its significant figures in the hundredths column. We, therefore, limit our answer to the tenth column. We drop the last digit — the 1 — because it is not significant to the final answer.
- The dropping of positions in sums and differences brings up the topic of rounding. Although there are several conventions, in this text, we will adopt the following rule: the final answer should be rounded up if the first dropped digit is 5 or greater and rounded down if the first dropped digit is less than 5.
- If the operations being performed are multiplication or division, the rule is as follows: limit the answer to the number of significant figures that the data value with the least number of significant figures has. So, if we are dividing [latex]23[/latex] by [latex]448[/latex], which have two and three significant figures each, we should limit the final reported answer to two significant figures (the lesser of two and three significant figures). The same rounding rules apply in multiplication and division as they do in addition and subtraction.

Examples 1.2.3

# Problems

Express the final answer to the proper number of significant figures.

- [latex]101.2 + 18.702 = {?}[/latex]
- [latex]202.88 − 1.013 = ?[/latex]
- [latex]76.4 × 180.4 = ?[/latex]
- [latex]934.9 ÷ 0.00455 = ?[/latex]

## Solutions

- If we use a calculator to add these two numbers, we would get [latex]119.902[/latex]. However, most calculators do not understand significant figures, and we need to limit the final answer to the tenth’s place. Thus, we drop the [latex]02[/latex] and report a final answer of [latex]119.9[/latex] (rounding down).
- A calculator would answer [latex]201.867[/latex]. However, we have to limit our final answer to the hundredths place. Because the first number being dropped is [latex]7[/latex], which is greater than [latex]6[/latex], we round up and report a final answer of [latex]201.87[/latex].
- The first number has three significant figures, while the second number has four significant figures. Therefore, we limit our final answer to three significant figures: [latex]76.4 × 180.4 = 13,782.56 = 13,800.[/latex]
- The first number has four significant figures, while the second number has three significant figures. Therefore we limit our final answer to three significant figures: [latex]934.9 ÷ 0.00455 = 205,472.5275 = 205,000.[/latex]

# Test Yourself

Express the final answer to the proper number of significant figures.

- [latex]3.445 + 90.83 − 72.4 = ?[/latex]
- [latex]22.4 × 8.314 = ?[/latex]
- [latex]1.381 ÷ 6.02 = ?[/latex]

## Answer

- [latex]21.9[/latex]
- [latex]186[/latex]
- [latex]0.229[/latex]

As you have probably realised by now, the biggest issue in determining the number of significant figures in a value is zero. Is the zero significant or not? One way to unambiguously determine whether a zero is significant or not is to write a number in scientific notation. Scientific notation will include zeros in the coefficient of the number only if they are significant. Thus, the number [latex]8.666 × 10^{6}[/latex] has four significant figures. However, the number [latex]8.6660 × 10^{6}[/latex] has five significant figures. That last zero is significant; if it were not, it would not be written in the coefficient. So, when in doubt about expressing the number of significant figures in quantity, use scientific notation and include the number of zeros that are truly significant.

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.
- Significant figures in a quantity indicate the number of known values plus one place that is estimated.
- There are rules for which numbers in a quantity are significant and which are not significant.
- In calculations involving addition and subtraction, limit significant figures based on the rightmost place that all values have in common.
- In calculations involving multiplication and division, limit significant figures to the least number of significant figures in all the data values.

Exercises