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Algebra

Cartoon depicting a feamle teacher with an algebraic equation on the board. It has been solved with a solution of x equals three. A small boy is challenging the teacher and saying, Just a darned minute. Yesterday you said that x equalled two.Algebra is one of the main branches of pure mathematics, along with geometry, analysis, topology, combinatorics, and number theory.  It is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures. Algebra is much broader than elementary (school) algebra and studies what happens when different rules of operations are used and when operations are devised for things other than numbers. Addition and multiplication can be generalized and their precise definitions lead to structures such as groups, rings and fields, studied in the area of mathematics called abstract algebra.  See this article from the University of Arkansas and the Wikipedia article for further reading.

Algebra has been developed over a period of many centuries but one of the seminal works describing this way of approaching and analysing problems was a book written in about 830 AD by an Arabic scholar named Al-Khwarizmi.  His books on mathematics were so influential that his name even became a mathematical term, algorithm, which describes a method or process for solving a problem! His second book, al-Kitab al-mukhtasar fi hisab al-jabr wa’l-muqabala, was a treatise describing algebra.  The term algebra comes from the expression al-jabr in the title of his book.  Keith Devlin, in his marvellous book, The Man of Numbers, translates the full title as The Abridged Book on Calculation by Restoration and Confrontation. Restoration (al-jabr) and confrontation (al-muqabala) refer to two processes used in balancing equations:

  • Restoration (or ‘completion’) refers to the removing of all negative terms by transposing them to the other side of the equation to make them positive.  For example, ‘restoring’ the equation 2x = 11 – 4x would result in 6x = 11.
  • Confrontation (or ‘balancing’) refers to the process of eliminating identical quantities from both sides of an equation.  For example, 8x + 1 = 3x + 9 would become 5x + 1 = 9 after removing 3x from both sides of the equation, and then 5x = 8 after removing 1 from both sides of the equation.

These two processes form the foundation for solving any algebraic equation and are often taught to students in the first year or two of High School.

Al-Khwarizmi did not use modern symbols, however (as they had not yet been invented).  He used words to describe the concepts … and unknown quantities (what we would call x and y today), were known by his contemporaries as shay (thing) and jidhr (root).  The term jidhr also means ‘the origin’ or ‘the root of a tree’ and may be the origin of our modern expression ‘the root of an equation.’  Note that the ‘root’ of a number comes from a different source … the Latin word radix.

Linear Algebra

Simultaneous Equations

Quadratic Equations

Polynomials

Matrices

Four Principles that Govern How You Learn Formulae

This material is reproduced on my How to Study page along with a great deal of other information that will help you with your learning.

It is difficult to learn any mathematical formula when it has no meaning or context. Although there are many quite meritorious methods for learning formulae, I have found that the best method for me has been to practise deriving each formula from mathematics that I already know and understand. In this way, I can not only write each formula, but I also understand where it comes from, what it is used for, and how to use it.

Of course, it is also vitally necessary that you practise using each formula as well!

The four principles that I share in this video explain how to focus on learning, how to practise deriving and using a formula, and how to use our understanding of massed practice and distributed practice to ensure that each formula can be remembered and used months later when you face your exams.

If you want to learn a bit about memory techniques, you might watch this TED Talk video by Joshua Foer and this inspiring video by the eccentric memory expert, Ed Cooke! I encourage you to go exploring on YouTube if you wish to embark on a new adventure of developing your memory skills! Please note that I do not necessarily recommend any of these techniques. I suggest that you explore and find what works for you.

Thank you so much! Its like you’ve flipped a switch in my brain somehow. Its my 2nd time around in calc1, I just couldn’t wrap my mind around calc. concepts until I found you. And that’s saying a lot because I had to take trig. 3 times before I felt competent enough to take calc. Please accept my sincere thanks. You Sir, are a God send.
Jeff S (on a CCM YouTube video about the Chain Rule)

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