0.6: The Mathematical Mantra

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Traditionally, when you learn mathematics, you start with Arithmetic. You learn the axioms that define the "game of mathematics" (e.g., the Associative Property of Addition and the Commutative Property of Multiplication) and you use those, along with the operations of addition, subtraction, multiplication, and division, to perform simple computations.

Once you attain mastery of Arithmetic, you move on to Basic Algebra (this is usually split into Elementary and Intermediate Algebra). You learn how to simplify expressions involving unknown quantities. It is also here that you begin to solve equations involving unknown quantities. You spend time visualizing the infinite number of solutions to equations using graphs, and you apply your knowledge to find solutions to complex real-world problems.

Once you have completed Basic Algebra, you reach a fork in the road of your mathematics education - you could take Trigonometry (this course) or College Algebra. In either case, you must complete both subjects before moving into Differential Calculus (also known as Calculus I), but they can be done in either order. In fact, some institutions combine them into a single course called Precalculus.¹

Trigonometry is the study of angles within triangles and is necessary to master before you can begin Calculus. In Trigonometry, you spend a good deal of time refining your critical-thinking skills by practicing proofs. You also develop skills to help you tackle applications requiring spatial interpretations.

College Algebra, on the other hand, consists of the advanced topics in Algebra that are not covered in previous Algebra courses, but required for a student to understand and succeed in Calculus. You spend time working with transcendental functions like logarithms and exponentials. You further refine your critical-thinking skills by proving statements using both direct logic and mathematical induction. It is also within College Algebra that you investigate conic sections, vectors, and parametric curves.

On and on you travel the path of mathematics, adding to your ever-expanding knowledge-base at each stop. At some point, you have such an abundance of tools within your "mathematical toolkit" that it seems daunting to know what to do when facing problems within mathematics, physics, engineering, other complex fields. It is in these instances that the Mathematical Mantra will be incredibly helpful.

$MapOfMath.png$

Mathematical Mantra

Perform mathematics in the order you learned mathematics. That is, perform

Arithmetic before Basic Algebra,

Basic Algebra before College Algebra,

College Algebra before Trigonometry, and

Trigonometry before Calculus.

Always be willing to ask yourself, "Is there something I learned previously that can simplify my work here?"

I will try to reinforce these concepts in the worked out examples using the following notations\[ \begin{array}{|c|c|}
\hline \mathrm{A} & \textbf{A}\text{rithemtic} \\
\hline \mathrm{BA} & \textbf{B}\text{asic }\textbf{A}\text{lgebra} \\
\hline \mathrm{CA} & \textbf{C}\text{ollege }\textbf{A}\text{lgebra} \\
\hline \mathrm{T} & \textbf{T}\text{rigonometry} \\
\hline
\end{array} \nonumber \]

If you spend a moment reminding yourself of the Mathematical Mantra before jumping into most math problems, you will (hopefully) be surprised at how it simplifies your work into nice little logical chunks.

Footnotes

¹ As mathematics is a subject that requires time to "digest," I do not recommend taking the "single-step" Precalculus course. It contains too much material in too little time and you end up getting a superficial treatment of the necessary mathematics with very little time to make sense of any of it.

For those interesting in learning more about the path of mathematics, I recommend watching this wonderful video created by Dr. Dominic Walliman (Domain of Science).

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