Lesson 3 in Algebra I for Students of the Classical Liberal Arts Academy
Having laid the foundations in the previous lessons by learning about mathematical terms, propositions, axioms, and definitions, we now enter the study of the symbols and notation used in algebra. This lesson may seem simple at first glance, but it is one of the most important, for symbols are the written language of mathematics. Without understanding the meaning and use of these symbols, we cannot think or speak clearly about any mathematical truth.
Algebra, like Latin or Greek, is a language—one not spoken aloud but written with symbols and expressions. Just as Latin can say a great deal with a few well-chosen words, so algebra allows us to express complex truths with simple forms. Each symbol we encounter is a doorway to understanding, and when these symbols are mastered, they become the elegant shorthand of the rational soul.
Moreover, in the light of Catholic teaching, we know that God is the author of all truth, and mathematical truths are part of the rational order He has woven into creation. As the Catechism teaches, human reason is capable of attaining truth through its own light, because it is created in the image of God (CCC 35). When we learn mathematics rightly, with wonder and reverence, we draw nearer to that divine Wisdom that orders all things well (cf. Wisdom 8:1).
I. Symbols as a Shorthand Language
Imagine trying to work with numbers if you had to write everything out in full sentences. If a man has five apples and receives three more, he has eight in total. That sentence is correct and clear—but in algebra, we can write it much more simply:
5 + 3 = 8
(Translation: five plus three equals eight.)
This symbolic form is not a replacement for understanding—it is the fruit of it. Once we have understood what “plus” means, and what equality means, we can write and think quickly and clearly using symbols.
Let’s consider another example: “The sum of a number and five is twelve.” This is written as:
x + 5 = 12
(Translation: x plus five equals twelve.)
Each part of this expression has meaning:
- x: A symbol for a number whose value we do not yet know.
- +: The operation of addition.
- 5: A known constant value.
- =: A sign of equality, meaning the two sides are the same in value.
- 12: The total value of the sum.
This equation tells a story: there is a number (x), which, when five is added to it, gives a total of twelve. Learning to read and write this way allows us to reason with great clarity.
St. Augustine taught that all signs point us to something greater than themselves (De Doctrina Christiana, II). Algebraic symbols are like that—they are not the truth, but signs that point our minds to the relationships that exist in the realm of number.
II. Letters in Algebra: Variables and Constants
Variables
When we do not know the value of a number, we can still think about it and reason with it. We simply give it a name: a letter. In algebra, letters like x, y, and n are used to represent numbers whose values may change or are not yet known. These letters are called variables, because their values may vary. For example:
x + 2 = 7
(Translation: x plus two equals seven.)
This tells us that some number, when added to two, equals seven. Our job is to discover what that number is.
Constants
Other letters stand for fixed, unchanging values. These are called constants. Sometimes constants are simply numbers, like 2 or 5. But sometimes mathematicians use special letters to represent important numbers that are used again and again. Here are two examples of constants. You are not expected to understand them now.
- π (“pi:): The number that represents the ratio of a circle’s circumference to its diameter. It is approximately 3.14159.
- e: The base of the natural logarithm, used in advanced mathematics. It is approximately 2.71828.
When we write something like 3x, it means “three times x”—that is, the number x taken three times in multiplication.
In ancient times, Nicomachus of Gerasa, a philosopher admired by the Church Fathers, taught that numbers are not just useful, but beautiful. In his Introduction to Arithmetic, he spoke of how number reveals the order and harmony of creation. By using letters to represent general numbers, we follow in this tradition of thoughtful, philosophical mathematics.
III. The Equals Sign: A Statement of Identity
The symbol = is perhaps the most important of all. It is the equals sign, and it tells us that the two things on either side are the same in value.
For example:
4 + 3 = 7
(Translation: four plus three equals seven.)
This is a statement of truth. It says that what we find on the left (four plus three) is exactly equal in value to what we find on the right (seven).
This may seem obvious at first, but it is the very foundation of all algebraic reasoning. An equation is a statement of equality. It is a proposition, which can be judged true or false, and which can be used to discover unknown truths.
St. Thomas Aquinas said that truth is found in the “conformity between the mind and reality” (veritas est adaequatio intellectus et rei, ST I, q. 16). In mathematics, the equals sign expresses that conformity. It tells us that two different expressions—perhaps one complex, one simple—are really the same.
IV. The Plus and Minus Signs: Addition and Subtraction
The Plus Sign (+)
The plus sign means to add. If you have two apples and someone gives you three more, you now have five. This is written:
2 + 3 = 5
(Translation: two plus three equals five.)
Addition has a property called commutativity, which means that it does not matter in which order we add numbers:
a + b = b + a
(Translation: a plus b equals b plus a.)
The Minus Sign (-)
The minus sign means to subtract. If you have ten coins and you spend four, you now have six:
10 – 4 = 6
(Translation: ten minus four equals six.)
Subtraction is not commutative. Changing the order changes the meaning:
5 – 3 ≠ 3 – 5
(Translation: five minus three does not equal three minus five.)
The minus sign is also used to show negative numbers, which are less than zero. For example, -6 means “negative six.” These are used often in algebra to represent values below a given standard.
V. Grouping Symbols and the Order of Operations
In algebra, we often need to show which operations should be performed first. We use grouping symbols to do this:
- Parentheses ( )
- Brackets [ ]
- Braces { }
Consider the difference between these two expressions:
2 × (3 + 4) = 14
(Translation: two times the sum of three and four equals fourteen.)
2 × 3 + 4 = 10
(Translation: two times three plus four equals ten.)
In the first, we add before multiplying. In the second, we multiply before adding. The grouping symbols tell us what to do first.
To help us remember the order in which to do things, we use the acronym PEMDAS:
- Parentheses
- Exponents
- Multiplication and Division (from left to right)
- Addition and Subtraction (from left to right)
Boethius, in his On Music, reminds us that order is the foundation of harmony—whether in music, in the heavens, or in mathematics. When we respect the proper order in our expressions, we preserve the beauty and clarity of mathematical truth.
VI. Translating Language into Algebra
As students of algebra, one of the most valuable skills we can develop is the ability to take a problem stated in words and turn it into an algebraic expression or equation. This is like translating a passage from English into Latin—we must preserve the meaning exactly, while changing the form.
Here are a few examples:
- “The sum of a number and eight” becomes x + 8
(Translation: x plus eight.) - “A number decreased by four” becomes x – 4
(Translation: x minus four.) - “Twice a number” becomes 2x
(Translation: two times x.) - “The product of a number and five” becomes 5x
(Translation: five times x.) - “A number divided by three” becomes x ÷ 3 or x/3
(Translation: x divided by three.) - “The difference between a number and ten is two” becomes x – 10 = 2
(Translation: x minus ten equals two.)
Practicing this kind of translation not only strengthens our skills in algebra, but also sharpens our minds in logic, language, and reasoning.
VII. The Concept of Identity in Algebra
Some expressions in algebra are true no matter what number you use for the variable. These are called identities, because they express truths that are always valid.
Examples:
- a + 0 = a
(Translation: a plus zero equals a.) - a × 1 = a
(Translation: a times one equals a.) - a – a = 0
(Translation: a minus a equals zero.)
These identities show how numbers behave. They are like the laws of nature in the world of mathematics. They never change.
If we solve an equation and end up with something like x = x, that means the equation is true for every value of x. This is what we call an identity equation.
These simple truths remind us of the unchanging nature of truth itself. As Scripture teaches, “with God there is no variation or shadow due to change” (James 1:17). In the unchanging identities of mathematics, we catch a glimpse of the eternal constancy of God’s own reason.
Conclusion
We have now learned how symbols serve as the essential language of algebra. We have seen how variables and constants, the equals sign, the plus and minus signs, and grouping symbols allow us to express mathematical truths with elegance and order.
To study these things is not just to become better at solving problems. It is to train the mind to think with clarity, order, and truthfulness. It is to participate in the reason of the Creator, who made the world according to number and measure and weight (cf. Wisdom 11:20).
When we master these symbols and their meanings, we are ready to move on to the greater work of solving equations, reasoning about unknowns, and discovering the hidden structure of quantity. Let us therefore give our attention to these simple signs, knowing that they open the way to higher things.
Memory Work
1. What is the purpose of algebraic symbols?
Algebraic symbols allow us to express mathematical ideas clearly and efficiently, using a language of reason and order.
2. What is a variable, and why do we use it?
A variable is a letter used to represent a number whose value is unknown or can change. It allows us to think about general quantities and relationships.
3. What is a constant?
A constant is a fixed, unchanging number. It can be a known value like 5 or a special value like π (pi).
4. What does the equals sign (=) mean in algebra?
The equals sign shows that two expressions have the same value; it is a sign of mathematical identity and truth.
5. How do the plus and minus signs function in algebra?
The plus sign (+) indicates addition, and the minus sign (-) indicates subtraction or a negative value.
6. What are grouping symbols, and why are they important?
Grouping symbols like parentheses ( ), brackets [ ], and braces { } show which operations should be performed first, helping us preserve order and clarity.
7. What is the order of operations in algebra?
Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right)—abbreviated as PEMDAS.
8. What is an identity in algebra?
An identity is an equation that is true for all values of the variable, such as a + 0 = a.
9. Translate: “The difference between a number and ten is two.”
x – 10 = 2 (Translation: x minus ten equals two.)
10. Why does learning algebra matter for the Catholic student?
Because algebra trains the mind in truth, clarity, and logical thinking, reflecting the divine order placed by God in creation.
If you have any questions, please contact us.
Mr. William C. Michael, O.P.
Headmaster
Classical Liberal Arts Academy
mail@classicalliberalarts.com
