1.1 Numbers and Data

Introduction

We use information constantly in our daily lives at home and at work to make decisions and future plans. To make these decisions, the information we receive must be translated into data to organize, process and visualize it.  At its heart, data are numbers and the art of organizing, processing and visualizing  data is done with mathematics.

Having a good sense of what numbers mean and how to manage with mathematics will enrich and ease our experiences with them. We will be able to make decisions with information with confidence and discernment. As a first step, this section reviews number sets, order of operations, exponential and scientific notation.

Learning Objectives

• Identify sets of real numbers: natural, whole, integer, rational or irrational
• Apply order of operations
• Identify and use properties of real numbers including
  • associative
  • commutative
  • distributive
  • identity
  • inverse
• Simplify expressions using properties of exponents
• Understand zero exponents
• Use negative exponents
• Use scientific notation
• Simplify algebraic expressions
• Evaluate algebraic expressions

Identify Sets of Real Numbers

The set of natural numbers are also called the counting numbers. There is no zero, no fractions or decimals and no negative numbers. Let’s look at the set notation used to describe the natural numbers.

[latex]\{{1,2,3,4,...}\}[/latex]

Note: The { symbol is called a brace. Pairs of braces surround a set. The ellipsis symbol (…) is used to show that the number set continues to infinity.

Whole numbers can be considered a mathematical advancement over the natural numbers since zero is included along with the set of natural numbers. We denote the set of whole numbers as

[latex]\{{0, 1, 2, 3, 4, ...}\}[/latex]

Again, note the braces communicates a set is contained within and the ellipsis shows that the set continues to infinity.

The set of integers brings the opposite values of the natural numbers into play along with the whole number set as shown below. Ellipses are employed at either end.

[latex]\{{...,-4,-3,-2,-1,0,1,2,3,4...}\}[/latex]

We write the set of rational numbers to show that the set is defined as a ratio of integers.

[latex]\{{\frac{m}{n}|\mathrm{m\; and\; n\; are\; integers\; and\; n} \neq 0}\}[/latex]

A number is considered a rational number if it can be written as a fraction and that includes fractions that create a repeating decimal such as [latex]\frac{1}{3}=0.\overline{33}[/latex] .

And, finally,  the numbers that do not form terminating or repeating decimals fall into the irrational numbers. We can identify this set as below:

[latex]\{{h|\mathrm{h\;is\;not\;a\;rational\;number}}\}[/latex]

Probably the most well known irrational number is π. Competitions exist world-wide in which contestents calculate and memorize π to an astounding number of decimal places. Square roots that are not perfect squares also are categorized as irrational numbers.

Order of Operations (PEMDAS)

Grammar rules let us know how to properly create a sentence. Similarly, Order of Operations is a set of rules that tells us how to evaluate mathematical expressions. Order of Operations are especially helpful if the expression is not clearly defined by parentheses or other grouping symbols.

Example 1.1.1

Simplify the following expression using the Order of Operations Rules

[latex](2 \cdot 2)^2 \div 2 + 9 \cdot 2[/latex]

To use the Order of Operations, you want to simplify the expression following the order of operations in sequence.

Step 1: (P) Look for any parentheses or brackets. Simplify inside the expression.

Step 2: (E) Look for any exponents or radicals and evaluate them.

Step 3: (M/D) Identify multiplication and/or division operations and simplify them from left to right.

Step 4: (A/S) Identify addition and/or subtraction operations and simplify them left to right.

We will perform each step for our expression now.

Step 1: (P) We multiply [latex]2\cdot2[/latex] to get 4.

[latex](4)^2 \div 2 + 9 \cdot 2[/latex]

Step 2: (E) We now raise 4 to the power of 2.

[latex]16 \div 2 + 9 \cdot 2[/latex]

Step 3: (M/D) From left to right we first divide 16 by 2 to get 8 and then we multiply 9 by 2 to get 18.

[latex]8 + 18[/latex]

Step 4: (A/S) Finally we complete the addition of 8+18

[latex]8+18=26[/latex]

If you have been on social media, you most likely have seen the viral arguments over ambiguously written math problems. See the video below for an excellent explanation of the controversial problem below:

A great takeaway from this video and the surrounding controversy is to avoid ambiguity when setting up your math problems. Use your grouping symbols for clarity!

Properties of Real Number

The properties of real numbers help us decide what can and cannot be done as we simplify mathematical expressions. Understanding the properties of real numbers sets the foundation for solving all types of math problems. It is similar to understanding the rules of driving. No sane driver would drive through a red light since a red light is a foundational rule to stop.

The Commutative Property

The word commute means “to move”. Implicitly the commutative property tells us how we can move numbers in a mathematical expression.

The commutative property of addition states that numbers may be added in any order without affecting the sum.

[latex]a + b = b + a[/latex]

Let’s use some real numbers to demonstrate.

1. [latex]2 + 3 = 3 + 2[/latex]
2. [latex](-2) + 5 = 5 + (-2)[/latex]

Notice in the second example that the parentheses were used to avoid any confusion with moving the negative value. The commutative property is not valid with subtraction!

[latex]5 - 2 \not = 2 - 5[/latex]

If parenthesis were used, the above expression would be written as follows:

[latex]5 + (-2) \not = 2 + (-5)[/latex]

Using the parenthesis clarifies which number is the negative number and clearly shows that the left and right side of the expression are not equal.

Similarly, the commutative property of multiplication tells us that we may move numbers that are being multiplied into any order that we wish.

[latex]2 \cdot 3 \cdot 4 = 4 \cdot 3 \cdot 2 = 3 \cdot 2 \cdot 4[/latex]

There is no commutative property for division. Like subtraction, the order of the numbers in a division problem is directional.

[latex]10 \div  5 \not = 5\div 10[/latex]

In a social setting, we may say that we are “hanging out” with our group of friends. A more formal statement would be that we are associating with our group of friends. The associative property tells us how numbers “hang out” or associate with other groups of numbers in a mathematical expression.

The associative property of addition tells us that we may group numbers involved in addition however we wish.

[latex]15+23+10=(15+23) + 10 = 15 + (23+10)[/latex]

The associative property of multiplication tells us that we may group numbers involved in multiplication however we wish as well. In the below example, it makes the problem simpler to multiple the 3 and 6 together first and then multiply by 10 (since we can do so by just adding a zero to the end of the first result.)

[latex]3\cdot 6 \cdot 10 = (3 \cdot 6) \cdot 10 = (18) \cdot 10 = 180[/latex]

Consider the following examples and then decide if the associative property extends to subtraction and division.

[latex]\begin{align}(5-2) - 4 &\stackrel {?}{=} 5 - (2-4)\\(3)-4 &\stackrel{?}{=}5-(-2)\\-1 &\neq7\end{align}[/latex]

[latex]\begin{align} (30\div5)\div2 &\stackrel{?}{=}30\div(5\div2)\\6\div2 &\stackrel{?}{=}30\div(2.5)\\3&\neq12\end{align}[/latex]

The distributive property states that the product of a factor times a sum is the sum of the factor times each term in the sum.

[latex]a\cdot\left(b+c\right)=a\cdot b+a\cdot c.[/latex]

This property combines both addition and multiplication (and is the only property to do so). Let us consider an example.

[latex]\begin{align} 3 (6-2)&=\\ &=3\cdot6 +3\cdot(-2)\\ &=18-6\\ &=12\end{align}[/latex]

Note that 3 is outside the grouping symbols, so we distribute the 3 by multiplying it by 6, multiplying it by –2, and adding the products.

To be more precise when describing this property, we say that multiplication distributes over addition. The reverse is not true, as we can see in this example.

[latex]\begin{align} 6+(3\cdot5) &\stackrel{?}{=} (6+3)\cdot(6+5)\\ 6+15   &\stackrel{?}{=}9\cdot11\\ 21 &\neq 99 \end{align}[/latex]

A special case of the distributive property to be wary of is given in an example below:

[latex]12-\left(5+3\right).[/latex]

We have the numbers in parentheses being subtracted from 12. To be sure that the negative is distributed correctly, you can rewrite the expression as shown.

[latex]12+\left(-1\right)\cdot\left(5+3\right)[/latex]

Now, distribute  and simplify the result.

[latex]12+\left(-5-3\right)=12-8=4[/latex]

This seems many extra steps, but it illustrates a powerful result that will be useful once we introduce algebraic terms.

The identity property of addition states that there is a unique number, called the additive identity (0) that, when added to a number, results in the original number.

[latex]a+0=a[/latex]

The identity property of multiplication states that there is a unique number, called the multiplicative identity (1) that, when multiplied by a number, results in the original number.

[latex]a\cdot1=a[/latex]

For example, we have [latex]4 + 0 =0[/latex] and [latex]-6\cdot1=-6[/latex].

There are no exceptions for these properties; they work for every real number, including 0 and 1.

Inverse Properties

The inverse property of addition tells us that, for every real number a, there is a unique number, called the additive inverse (or opposite), denoted by (−a), that, when added to the original number, results in the additive identity, 0.

[latex]a+\left(-a\right)=0[/latex]

For example, if [latex]a=7[/latex] then the additive inverse is -7, since [latex]7+(-7)=0[/latex]

The inverse property of multiplication holds for all real numbers except 0 because the reciprocal of 0 is not defined. The property states that, for every real number a, there is a unique number, called the multiplicative inverse (or reciprocal), [latex]1/diva[/latex] that when multiplied by the original number,[latex]a[/latex], results in the multiplicative identity, 1.

[latex]a\cdot\frac{1}{a}=1[/latex]

For example, if [latex]a=8[/latex] the reciprocal is [latex]\frac{1}{8}[/latex] giving the following:

[latex]8\cdot\frac{1}{8} = 1[/latex]

Exponents

Exponents are just a mathematician’s shorthand for repetitive multiplication. Let’s review the anatomy of a number expressed in exponential notation.

Product Rule

When two numbers or variables of the same base but different exponents are being multiplied, we can use the product rule for exponents to quickly accomplish the task. Let’s first see an example if we multiplied by expanding the exponents.

Example 1.2.1

Find the product by expanding each exponent term.

[latex]x^4 \cdot x^2 = x \cdot x \cdot x \cdot x \cdot x \cdot x = x^7[/latex]

We can see that finding products this way can be quite time consuming especially when the exponents are large values. The product rule below simplifies the process greatly.

Example 1.2.2

Using the product rule, write the following as a single base.

1. [latex]x^2\cdot x^3 = x^{2+3} = x^5[/latex]
2. [latex]2^5\cdot  2^4 = 2^{5+4} = 2^9[/latex]
3. [latex]y^2\cdot y^2\cdot y^4 = y^{2+2+4} = y^8[/latex]

Explanation: When multiplying variables or numbers with the same base, add the exponents.

Quotient Rule

When two numbers or variables with the same base but different exponents are being divided, we can use the quotient rule for exponents to simplify it. Let’s look at the example below to see how simplifying would occur without the quotient rule.

Example 1.2.3

Simply the quotient by canceling common factors between the numerator and denominator

[latex]\large \frac{b^6}{b^4} = \frac{b\cdot b\cdot b\cdot b\cdot b\cdot b\cdot}{b\cdot b\cdot b\cdot b\cdot}[/latex]

[latex]\Large = {\frac{\cancel{b}\cdot \cancel{b}\cdot \cancel{ b}\cdot \cancel{b}\cdot b\cdot b}{\cancel{b}\cdot \cancel{b}\cdot \cancel{b}\cdot \cancel{b}\cdot}}[/latex]

[latex]=b \cdot b[/latex]

[latex]= b^2[/latex]

We can see that this process would be very time consuming especially if the exponents were even larger numbers. The quotient rule below greatly expedites the process.

Example 1.2.4

Using the quotient rule, write the following as a single base.

1. [latex]\frac{3^5}{3^2} = 3^{5-2} = 3^3[/latex]
2. [latex]\frac{x^7}{x} = x^{7-1} = x^6[/latex]
3. [latex]\frac{b^{10}}{b^6} = b^{10-6} = b^4[/latex]

Explanation: When dividing variables with the same base, subtract the exponents.

Power Rule 

What happens when an exponent expression is raised to another exponent? Enter the power rule. But first let’s look at how to simplify the expression by expanding the exponents.

Example 1.2.5

Simplify the exponential expression by expanding the exponents.

[latex]{(x^3)}^2 = (x^3) \cdot (x^3)[/latex]

[latex]= (x \cdot x \cdot x) \cdot (x \cdot x \cdot x)[/latex]

[latex]= x \cdot x \cdot x \cdot x \cdot x \cdot x[/latex]

[latex]= x^6[/latex]

Zero Exponent Rule

One situation that often leads to confusion when simplifying with the quotient rule is shown below.

[latex]\frac{x^5}{x^5}=x^{5-5}=x^0=1[/latex]

Whenever the numerator and denominator are the same, we know that the value of the fraction is 1. When seeing the exponent of zero, students often assume that the value of the expression is also zero. This is not true.

Note: [latex]0^0[/latex] is a special case that will be presented in a more advanced math course.

Negative Exponents

Let’s use the quotient rule to simplify the expression below.

[latex]\frac{t^2}{t^4}=t^{2-4}=t^{-2}[/latex]

The quotient rule leads us to the interesting result of a negative exponent. To investigate what this means, we simplify the expression in a different way.

[latex]\large \frac{t^2}{t^4}=\frac{t\cdot t}{t\cdot t\cdot t \cdot t}=\frac{\cancel t\cdot \cancel t}{\cancel t\cdot \cancel t\cdot \ t \cdot \ t}=\frac{1}{t^2}[/latex]

These two expressions are equivalent. A negative exponent represents the reciprocal of an exponent.

Scientific Notation

Imagine being an astrophysicist writing a paper on the distance to the Andromeda galaxy from Earth which is 24,000,000,000,000,000,000,000 m

Yes, that’s 21 zeros every time you need to include the distance in your paper. A short cut is needed!

Enter scientific notation for a large number….

Example 1.2.6

Write 24,000,000,000,000,000,000,000 m in scientific notation

Explanation:

Step 1: If not written assume there is a decimal point at the end of the value. Move the decimal point to left until we have a value between 1 and 10.

Step 2: Count how many decimal places you moved. This will be the power of 10 to use.

Step 3:  Truncate the zeros after this.

[latex]2.4 \times 10^2[/latex]

Examples 1.2.7

The probability of matching 6 of 49 numbers in a lottery is 0.0000000715. Write this value using scientific notation.

Explanation:

Step 1: Move the decimal point to right until we have a value between 1 and 10.

Step 2: Count the decimal places. Use the negative of this value as  the power of 10.

Step 3: Truncate the zeros before this.

[latex]7.15 \times 10^{-8}[/latex]