1.1 Real Numbers: Algebra Essentials
Lesson: January 29, 2025. Quiz: February 13, 2025.
- Classify a real number as a natural, whole, integer, rational, or irrational number.
- Perform calculations using order of operations.
- Use the following properties of real numbers: commutative, associative, distributive, inverse, and identity.
- Evaluate algebraic expressions.
- Simplify algebraic expressions.
Assignment
- 1–3, 5–59 odd
- Recommended: all vocabulary and concept blocks copied into notes
How to Use the Textbook
Reading each section is strongly recommended, along with taking notes. A good place to start is defining any vocabulary and copying any asides that sum up information. The first one of those you will run into is called “Sets of Numbers”.
You should also work through the examples, only looking at the solution steps if you are unsure of how to proceed, or to check your answer. The “Try It” problems are there for you to test if you understood the examples, and the answers for them are found at the end of the book, along with answers to the odd exercises.
Classifying a Real Number
Natural numbers are your counting numbers, so ${1, 2, 3, \dots}$ Whole numbers include 0 so we end up with ${0, 1, 2, 3, \dots }$. It a minor distinction and it doesn’t come up much, but if it does we’ll be sure to be extra clear if zero is included or not.
Integers are the whole numbers and their opposites, so ${\dots, -3, -2, -1, 0, 1, 2, 3 , \dots}$. Integers tend to be talked about as positive or negative, but keep in mind that zero is neither positive or negative.
Rational numbers are any number that can be written as the quotient of any two integers, with the one exception being that you can’t divide by 0.
\[\left\{\tfrac{m}{n}∣m\text{ and } n \text{ are integers and } n\neq 0\right\}\]Any integer can be written as a rational number by putting it over 1.
Rational numbers also have the property, that when written in decimal form, either terminate or have a repeating pattern. So, $4.2$ is the rational number $\frac{21}{5}$ and $3.\overline{23}$ is the number $\frac{320}{99}$.
Irrational Numbers
Irrational numbers are all the numbers that don’t fit the above definitions, so they cannot be expressed as a fraction. And when written as a decimal they don’t terminate and also don’t repeat. We don’t deal with them too often because of their strange behavior, but common ones are $\pi$ and $\sqrt{2}$.
Real Numbers
Together, the rational and irrational numbers make up the set of real numbers. Each real number can be placed in a unique spot on the real number line, and conversely each spot on the number line corresponds to a unique real number.
The rational numbers is a larger set including integers, which in turn includes whole numbers, which include the natural numbers. You can see those relationship below with $N$ being the natural numbers, $W$ whole, $I$ integer, $Q$ the rationals, and $Q’$ the irrational numbers.
Performing Calculations Using the Order of Operations
When evaluating a mathematical expression, there is an order of operations that must be followed. After any grouping symbols, the rule is to do the most powerful operations first, from left to right. That means exponents (repeated multiplication), then multiplication (repeated addition), and finally addition.
Radicals (square roots), division, and subtraction were not included above because they are essentially the same operations as their counterparts. A square root is just a fractional exponent, division is the same as multiplying by the inverse, and subtraction is just adding the negative version of a number. We’ll be frequently switching from one operation to the other where it makes our life easier.
This system is typically presented as PEMDAS, which stands for Parentheses, Exponents, Multiplication, Division, Addition, Subtraction. The issue with this acronym is that radicals are skipped and it misses the fact that multiplication and division happen on the same step, as does addition and subtraction. Be careful using this if it’s what you were taught.
Let’s try one or two out.
\[\begin{align} 24 + 6 \cdot \frac{2}{3} - 4^2 &= 24 + 6 \cdot \frac{2}{3} - 16 && \text{Exponents first} \\ &= 24 + 4 - 16 && \text{Multiplication and division from left to right} \\ &= 12 && \text{Addition and subtraction from left to right} \end{align}\]Let’s look at one with grouping symbols, except none of them are actually parentheses. The numerator of our fraction is grouped together, as is what’s under the radical symbol.
\[\begin{align} \frac{5^2-4}{7}-\sqrt{11-2} &= \frac{25-4}{7}-\sqrt{9} && \text{First pass on the groups} \\ &= \frac{21}{7}-\sqrt{9} && \text{Second pass} \\ &= \frac{21}{7}-3 && \text{Radicals (exponents)} \\ &= 3 - 3 && \text{Division} \\ &= 0 && \text{Subtraction} \end{align}\]Using Properties of Real Numbers
Our real numbers have some properties that are useful to keep in mind when working with expressions and equations.
Commutative Properties
Order doesnt matter, but only when dealing with addition or multiplication.
\[\begin{align} a + b &= b + a \\ a\cdot b &= b \cdot a \end{align}\]Even though subtraction and division are technically just modified versions of addition and multiplication, they are different enough that this property won’t work with them.
Associative Properties
The associative properties tell us that we can freely change the grouping of an addition or multiplication problem, like below.
\[\begin{align} a + (b + c) &= (a + b) + c \\ a(bc)&= (ab)c \end{align}\]Again, this only applies to addition and multiplication. As soon as another operation is in play, this will not work.
The commutative and associative properties might seem very similar, but that’s only because they both happen to work with the operations we’ve looked at. Depending on how far you go in your math education, you will run into operations were only one of the properties works, hence the need to differentiate the two.
Distributive Property
The product of a factor and a sum is the same as the sum of the factor times each term in the sum. Hard to parse in words, so here it is in symbols.
\[\begin{align} a(b+c) = ab + bc \end{align}\]Watch out for situations like below.
\[\begin{align} - (a + b) = (-1)(a+b) = -a - b \end{align}\]Identity Properties
Important concepts for later. Adding zero to an expression doesn’t change its value, and the same goes for multiplying by one.
\[\begin{align} a + 0 &= a \\ a \cdot 1 &= a \end{align}\]Inverse Properties
When you take a value and add its additive inverse, you get zero.
\[\begin{align} a + (-a) = 0 \end{align}\]The multiplicative inverse of a number will produce a $1$ when multiplied by the original.
\[\begin{align} a \cdot \frac{1}{a} = 1 \end{align}\]Evaluating Algebraic Expressions
An algebraic expression is a collection of a constants and variables joined together by the algebraic operations: addition, multiplication, and rational exponents1. Constants are your plain numbers, like $5$ or $-\frac{2}{3}$. Variables are typically represented by letters and their values can either change (vary) or need to be derived.
When substituting in a value for a variable, make sure you apply it to each instance of the variable. It’s also good practice to use parentheses so that the order of operations is preserved. For the example below, let $a=11$ and $b=-8$
\[\begin{align} a + ab + b &= (11) + (11)(-8) + (-8) \\ &= 11 - 88 - 8 \\ &= -85 \end{align}\]If we didn’t use parentheses, we would end up writing $11 + 11-8 +-8$ and risk interpreting the multiplication as subtraction.
Formulas
An equation is a statement equating two expressions. Formulas are essentially the same thing: they show the relationship between variables and constants. You work with these similar to how you work with expressions.
Simplifying Algebraic Expressions
Simplifying usually implies combining like terms in an expression, with the implication that any distribution has to happen beforehand.
\[\begin{align} 2r−5(3−r)+4 &= 2r - 15 +5r + 4 \\ &= 7r - 11 \end{align}\]-
Irrational exponents appear in exponential functions, which are technically not algebraic. Despite that, they still fall under the umbrella of “algebra” and you’ve likely worked with them before. ↩