2.3: Numbers

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Introduction

Scientists use three basic types of numbers when quantifying matter; counted numbers, measured numbers and defined numbers.

Counted Numbers: A counted number is an exact number in the sense that you have an entity, and you count the number of entities. So an exact number has two parts, the entity, and the number of entities. If you have 16 rocks in your hand, you have sixteen rocks, not more, not less. If you break one of the rocks into two, you now have 17 rocks in your hand, not more, not less. These are exact numbers. Counted numbers are exact and there is no uncertainty about their value.

\[\underbrace{16}_{number}\underbrace{rocks}_{entity} \nonumber\]Counted Numbers are Exact

Measured Numbers: Now the mass of the rocks is a measured number and requires a third aspect to describe, which is the unit, defined by the scale used to measure it. So a measured number has 3 parts; magnitude, unit and entity. Sixteen one pound rocks weighs less than one 500 pound rock, although the number 16 is more than the number one, and this is because different units are being used to describe the mass. If you break each of the sixteen one pound rocks in half, you have 32 rocks, but they still weigh the same. Measured numbers have uncertainty that is indicated by their number of significant digits (section 2.5).

\[\underbrace{16}_{number} \; \underbrace{grams}_{unit} \; \; \underbrace{rock}_{entity} \nonumber\]Measured Numbers have Uncertainty

Defined Numbers: A defined number has a value inherent in its definition, and like a counted number is an exact number, but it may, or may not have uncertainty. Defined numbers are often used in measurements, for example, twelve inches is defined to be one foot and there is no uncertainty in that equivalence.
12inches ≡ 1foot
But not all defined numbers are integers, like the number \(\pi\), which is defined to be ratio of the circumference to the diameter of a circle.
\(\pi\)≡\(\frac{circumference}{diameter}\)
\(\pi\) is an exact number in the sense that it is exactly the ratio of the circumference to the diameter of a circle, but it is an irrational number and like a measured number, when we write down the value of \(\pi\) we give it a precision (3.14, 3.142, 3.1416, 3.14159,...), which defines how precisely we express the number.

Understanding defined numbers is of importance, and as we shall learn, as of May 20, 2019, the basic units of the metric system became based on defined numbers (2019 SI unit definitions).

Expressing Numbers

There are four common ways to express numbers, and some of these are preferable when the numbers are very large or very small.

Floating Decimal Notation Number
Scientific Notation
SI Prefixes
Logarithms

Floating Decimal Numbers

Floating numbers are common numbers where the "magnitude" is determined by the position of the decimal point. Digits to the left of the decimal are larger than 1 and digits to the right are less than one, or a fraction. So 12.121200 and 121212.00 are two numbers of identical digits but the second is 10⁴ times larger than the first . A fraction typically has a zero in front of the decimal, so 0.25 is \(\frac{25}{100}\) or one quarter \(\frac{1}{4}\).

Scientific Notation

Unlike a floating numbers scientific notation has two parts, first a number with only one digit to the left of the decimal and a variable number of digits to the right of the decimal, which is multiplied by 10 to a power, such that it equals the value of the desired number. We call the power of 10 its exponent.

\[\text{_ . _ _ _ } \times 10^{n} \]

Where n is the power and the underscore represents the number of digits in the number. Note, due to this format, scientific notation always has an unambiguous number of significant digits.

Q&A: What is Exponentiation?

Exponentiation is the replication of multiplication the way multiplication is the replication of addition

\[ \begin{align*} 10^0 &= 1 \\[4pt] 10^1 &= 10 \quad\quad &&10^{-1} =\frac{1}{10} = 0.1 \\[4pt] 10^2 &= 10(10) = 100 \quad\quad &&10^{-2} =\frac{1}{10^{2}} =\frac{1}{100}=0.01 \\[4pt] 10^3 &=10(10)(10) = 1000 \quad\quad &&10^{-3} =\frac{1}{10^{3}}=\frac{1}{1000}=0.001 \end{align*}\]

Q&A: Can the exponent have units?

No, the exponent can not have units, it is the number of times you multiplied or divided the number by 10, but you write the units behind the number expressed in scientific notation.

\[1.333 \times 10^{4} \; L = 13,330 \; L\]

Advantages of Scientific Notation

Allows Awkwardly Large and Small Numbers to Be Expressed in Term of Compact and Easily Written Numbers
Allows Accurate Representation of the Number of Significant Figures in a Number, That Is a Measurement’s Precision, the “Certainty” of Our Measurements

SI Prefixes

SI prefixes are similar to scientific notation, where it is common to express a number with between 1 and 3 digits to the left of the decimal point, times 10 to the power of 3, but use the appropriate SI prefix to represent 10 to the power of 3.

\[ \begin{align} \_ . & \_ \; \_ \; \_ \times 10^{3n} \\ \_ \; \_. & \_ \; \_ \; \_ \times 10^{3n} \\ \_ \; \_ \;\_ . & \_ \; \_ \; \_\times10^{3n} \end{align}\]

Table \(\PageIndex{1}\): Some common SI prefixes.
(n=1,2,3,-1,-2,-3)
10³ⁿ	Name	Abbreviation		Name	10^-3n
10⁺³	kilo	k	m	milli	10^-3
10⁺⁶	mega	M	m	micro	10^-6
10⁺⁹	giga	G	n	nano	10^-9

Note, it is common to express floating numbers with a comma every three decimal places (power of 10³). So the numbers

\[ \begin{align} 1,330,000,000m & =1 .33 \times 10^{9}m =1.33 \; gigameters \\ 13,300,000,000m & =13.3 \times 10^{9}m =13.3 \;gigameters \\ 133,000,000,000m & =133 \times 10^{9}m = \; 133 \; gigameters \end{align}\]

Note

Not all SI prefixes are in magnitudes of 10³, the complete list is in section 2.7 and you will be responsible for memorizing the ones in red. This includes common ones between a thousand and a thousandth, like centi, deci... You should make flash cards of the prefixes in section 2.7 now, so you do not have to memorize them when we cover them.

Logarithms

Logarithms are very similar to scientific notation, but the pre-exponential is expressed as 10 to the power of a fraction, and the log of a number, is the power of 10 that equals a number. Logarithms are often not covered in general chemistry 1, but they are extensively used in general chemistry 2.

Deeper Look: Logarithms

This is optional material for general chemistry 1, but you will need to understand this for general chemistry 2.

Let’s consider the number 26794. It can be expressed in scientific notation or as a log

So logarithms are very similar to scientific notation, where the number being multiplied by 10 to a power is expressed as 10 to the power of a fraction. Please see the math review in Chapter 14 prelude for

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